Forensics and Physics 1527584070, 9781527584075

Table of contents :
Table of Contents
List of Illustrations
List of Tables
Introduction
Chapter One
Chapter Two
Chapter Three
Chapter Four
Appendix
References
About the Authors

Citation preview

Forensics and Physics

Forensics and Physics By

Renata Holubova, Jiří Straus and Jana Slezáková

Forensics and Physics By Renata Holubova, Jiří Straus and Jana Slezáková This book first published 2022 Cambridge Scholars Publishing Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Copyright © 2022 by Renata Holubova, Jiří Straus and Jana Slezáková All rights for this book reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN (10): 1-5275-8407-0 ISBN (13): 978-1-5275-8407-5

TABLE OF CONTENTS

List of Illustrations ................................................................................... vii List of Tables .............................................................................................. x Introduction ............................................................................................... xi Chapter One ................................................................................................ 1 Forensic Trasology Introduction ........................................................................................... 1 1.1 Searching for and securing of trasological tracks ........................... 6 1.2 Biomechanical content of trasological traces ................................ 15 1.3 Physics behind trasology ............................................................... 27 1.4 Walking and Physics ..................................................................... 40 Chapter Two ............................................................................................. 48 Biomechanics of Falls Introduction ......................................................................................... 48 2.1 Biomechanical classifications of falls ........................................... 49 2.2 Injuries caused by falls.................................................................. 53 2.3 Analysis and experimental results................................................. 66 2.4 Standing on a pad .......................................................................... 78 2.5 Human reaction time ..................................................................... 85 Chapter Three ......................................................................................... 107 Dactyloscopy Introduction ....................................................................................... 107 3.1 Searching, visibility and securing dactyloscopic traces .............. 110 3.2 Examining of dactyloscopic traces.............................................. 113 3.3 Physics behind dactyloscopy ...................................................... 121 3.3.1 Optics ................................................................................. 121 3.3.2 Molecular physics-adhesion and cohesion ......................... 126

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Table of Contents

Chapter Four ........................................................................................... 138 Forensic Ballistics 4.1 Forensic ballistics as a scientific discipline ................................. 138 4.2 Physics behind the ballistics ....................................................... 146 Appendix: Mathematics .......................................................................... 159 Differential and vector calculus ........................................................ 159 References .............................................................................................. 183 About the Authors .................................................................................. 188

LIST OF ILLUSTRATIONS

Figure 1-1. Types of soles Figure 1-2. Heels and sole Figure 1-3. Traces of vehicles Figure 1-4. 3D trasological trace Figure 1-5. Stride length and two-step locomotion Figure 1-6. Plantogram with significant parameters Figure 1-7. Dependence of body height on significant parameters of the plantogram Figure 1-8. Several types of plantograms Figure 1-9. Bar foot trace Figure 1-10. Kepler´s law of areas Figure 1-11. Gravitation Figure 1-12. Throw vertically upwards Figure 1-13. Horizontal throw Figure 1-14. Oblique up throw Figure 1-15. Movement of the centre of gravity Figure 1-16. Diagram of the human body Figure 2-1. Classification of falls Figure 2-2. The impact of the body and the representation of the primary (triangle) and secondary injury (wheel) Figure 2-3. Evaluating the course of the fall from the height Figure 2-4. Fall schedule Figure 2-5. Biomechanics of falls Figure 2-6. Falling patterns of standing jump and running jump above the jumping level: Running and standing jump are intimated at initial velocities of 9.15 and 2.70 m·s-1 at initial angles of 21 at 38 deg above the jumping level Figure 2-7. Falling patterns intimated at various angles of jump at initial velocities of 2.70 m·s-1 (A) and 9.15 m·s-1 (B): Maximal horizontal movement can be achieved at about 40 deg; the angle at 50 deg or over starts to minimize the horizontal movement Figure 2-8. Falling patterns intimated at various angles of jump at initial velocities of 2.70 m·s-1 (A) and 9.15 m·s-1 (B), falling from height of 100 m

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List of Illustrations

Figure 2-9. Falling patterns intimated at various angles of jump at initial velocities of 2.0 m·s-1 (A) and 9.5 m·s-1 (B), falling from height of 100 m Figure 2-10. Range of maximal horizontal movement of standing jump and running jump at angles between 0 and 40 deg Figure 2-11. Body mass center trajectory comparison as relation of different kind of falls Figure 2-12. Unprotected fall Figure 2-13. Scheme of dropping the body from stand to pad Figure 2-14. Typical course of head velocity and center of gravity over time Figure 2-15. Typical course of the angular velocity of the head and the angular velocity of the resting limb over time, in case of restored stability Figure 2-16. The course of the spontaneous fall of figurant Figure 2-17. Categorization of reaction times Figure 2-18. The process of motor response formation for each type of reaction time, according to Donders Figure 2-19. Structure of the total duration of the action Figure 2-20. Dependence of reaction time on the intensity of the auditory stimulus Figure 2-21. Reaction time dependencies on alcohol level - maximum alcohol level 0.6 ‰ Figure 2-22. Reaction time dependencies on alcohol level - maximum alcohol level of 1.2 ‰ Figure 3-1. Comparison of the secured track and the captured fingerprint Figure 3-2. Ridge characteristics Figure 3-3. Visibility of the dactyloscopic trace with finely groung ferric oxide Figure 3-4. Visibility of the dactyloscopic trace by reaction with ninhydride and by iodine vapour Figure 3-5. Papilary lines Figure 3-6. A compound microscope Figure 3-7. Running rays Figure 3-8. Comparative microscope for forensics Figure 3-9. Ken – a – vision comparative microscope Figure 3-10. Comparison microscope - image Figure 3-11. Measurement of adhesion forces Figure 3-12. Surface tension - forces Figure 3-13. Liquid wetting the solid body Figure 3-14. A liquid that does not wet the solid body

Forensics and Physics

Figure 3-15. A drop of liquid Figure 3-16. Surface tension of a liquid drop on another liquid Figure 3-17. Capillary elevation and depression Figure 3-18. Adhesion of dactyloscopic powder to a dactyloscopic brush Figure 3-19. Adhesion of dactyloscopic powder to the solid surface Figure 3-20. Adhesion of dactyloscopic powder to a surface with a dactyloscopic trace Figure 4-1. Coriolis force Figure 4-2. Air flow around the ball Figure A-1. Magnitude of the instantaneous velocity of the point [x; y], which moves along the curve k at time t Figure A-2. Geometric meaning of the derivative Figure A-3. Tangent and normal of the graph of the function Figure A-4. Geometric interpretation of Rolle's theorem Figure A-5. Geometric interpretation of the Lagrange’s theorem Figure A-6. Local and global extremes of function Figure A-7. Oriented line Figure A-8. Bound geometric vectors Figure A-9. Non-collinear vectors Figure A-10. Collinear vectors (same oriented) Figure A-11. Collinear vectors (not same oriented) Figure A-12. Multiple of the bound geometric vector a) concordant direction b) non concordant direction Figure A-13. Free geometric vectors Figure A-14. The sum of free vectors FigureA-15. Orthonormal bases of vector spaces Figure A-16. The scalar product of the vectors u, v

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LIST OF TABLES

Table 1-1. Linear regression relations Table 1-2. Frequency of walking steps Table 1-3. Estimating the mean stride length for walking Table 1-4. Mechanical power at different movements Table 1-5. Mechanics of walking Table 2-1. Biomechanical studies of standing (swimmer´s) jump and running (long) jump Table 2-2. Average group indicators of the development of psychic properties of top wrestlers with different ways of fighting Table 2-3. Conventional simple reaction time before selected combat actions Table 2-4. Conventional simple reaction time before selected combat actions Table 2-5. Duration of combat actions

INTRODUCTION

Dear readers! The presented publication entitled Forensics and Physics deals with criminological aspects of investigations from the point of view of science, especially physics. The book you hold in your hands deals with solving complex scientific research tasks when examining forensic traces and during forensic identification. The publication is based on extensive suggestions from literary, professional studies, and articles in professional journals. An interesting and key element is the connection between forensic - criminology and physics. Physics can be seen as a scientific field whose content is the study of the most general properties, states, and changes of material objects. Physics comes with its knowledge in three basic ways. The first method is the observation, e.g., observing the fall of a shot person. The following important method is an experiment by which we observe the phenomenon in artificially prepared conditions, such as the fall of a training dummy by shooting. The third method is to create hypotheses based on observations and experiments or based on the knowledge of the phenomenon. Thus, we create a scientifically substantiated idea of the course and causes of the occurrence under investigation. The book is divided into four main chapters. Forensic trasology deals with the method of finding and securing trasological traces. The general principles of securing trasological traces are given here. The biomechanical content of trasological traces is mentioned and the connection between walking and physics is discussed. It is worth mentioning the explanation of mechanical work during ordinary walking and the list of average values of mechanical force during various movements that a person performs. The second chapter deals with the biomechanics of falls. The introductory part explains that the fall of the human body is a compound movement, which consists of movement in the horizontal direction and free fall. The text explains in detail the classification of falls using a training dummy, whose weight parameters, dimensions, and location of the center of gravity are the same as for a living person.

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Dactyloscopy is the content of the third chapter. This is the oldest method of identification in criminology. Dactyloscopy is a scientific field of forensic technology that allows you to identify a specific person under optimal conditions. It shows how dactyloscopic traces are found, made visible and secured, as well as how they are examined based on the evaluation of dactyloscopic features. Attention is paid to applications in the field of optics and molecular physics The last chapter deals with the issue of forensic ballistics, which, due to the nature of the use of weapons, is mostly dedicated to the study of small arms. It is explained in detail that it is not just the science of firearms of all types and kinds. Forensic ballistics is, among other things, a scientific discipline that also examines by-products of firing, objects with traces of impact, etc. Emphasis is placed on the question of the mechanism of criminological traces during the shot itself and after leaving the projectile from the weapon. Of interest is the description, classification, and identification of firearms. In the part concerning physical applications, the reader will encounter, for example, the concepts of energy, kinetic energy, variable force, work, work in the gravitational field, mechanical energy saving, projectile energy, Coriolis force, Magnus effect, and projectile motion. Their meaning and use in connection with the issue of forensic ballistics are always explained. The appendix Differential and Vector Calculus is focused on physical applications in mathematics. This section mentions the historical development of differential and vector calculus. The text contains an overview of basic definitions and theorems about the derivation of a function of one variable. The terms are supplemented by explanatory figures. The subchapter Vector calculus deals with the introduction of the term-oriented line and its size, the definition of bounded and free geometric vectors. The introduction of vector coordinates in plane and space is also shown here, as it is directly applicable in applications in physics and engineering. The reader is acquainted with selected solved problems on physical topics in connection with criminology. We believe that the presented publication will be of benefit to all who are interested in the currently most used examination techniques and their significance from the point of view of physics. Dear readers, we wish you a lot of joy and lots of new knowledge while studying this publication. —Authors

CHAPTER ONE FORENSIC TRASOLOGY

Introduction Trasology is a field of forensic technology that deals with the search for securing and examination of footprints, footwear, means of transport, and traces of other similar objects. Trasology examines the traces of these objects if features of the external structure of the objects (morphological features) are highlighted in the trace. It is a science of traces, which examines a trace as a representation of the outer side of an object to identify those objects or to determine group affiliation and to clarify all the circumstances associated with the emergence of a trasological trace. The objects of research are trasological traces, which can be divided into the following groups: a) b) c) d)

footprints of bare and worn feet, traces of human locomotion traces of means of transport other traces of similar species.

Traces of bare and worn feet are created by contact of the bare foot or the bottom of the shoe with the pad. They are therefore the result of the reflection of the outer structure of the flat feet or sole, heel etc. These traces may have general (typical) so as special (specific) characters. Footprints of shoes are created by contact of the bottom of the shoe with the pad. The bottom of the shoe can be: a) Sole – can be monolithic, bloc or rolled (cut out), injection molded, pinned and quilted. Covers the lower part of the shoe upper (top) from toe to heel. b) Heel – it is either a part of the sole (monolithic) or forms a separate part of the bottom of the shoe. It is made of various materials. From the criminological and technical point of view, the most important is the so-called bollard (the upper part of the heel). Separately pressed

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Figure 1-1. Types of soles (monolithic, block sole without heel, doweled with attached sole and heel, part of the rubber plate from which the rolled sole is cut, rolled sole) (Straus, Porada et al, 2004, p. 26-27).

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bollards give us the opportunity to distinguish new from old-repaired – shoes, although if both cases bollards with different patterns are often used. c) Tread – covers the lower part of the shoe from the toe to the niche, while the sole from the toe to the heel. It is cut by machine or by hand from various materials and can be smooth or variously shaped. The tread is attached to the upper by sewing, doweling, nailing and gluing. The treads are mainly used for repairing worn shoes. Footwear footprints essentially provide the possibility of identification, which, however, depends on the quality of the footprint. In the case of smooth soles, the entire imprint is usually required, in the case of shaped soles, sometimes only the marginal part is sufficient to determine whether it is: a) Men´s, women´s, or children´s footwear b) Footwear of a certain type, shape, size, etc. (group features) c) Certain footwear (individual features).

Figure 1-2. Heels and sole (Straus, Porada et al, 2004, p. 26-27).

Group affiliation can be determined by comparison in the sole collection and in the footwear production catalogs. Individual identification can be carried out only if the footprint contains some peculiarities caused by production, use, and wear or repair (scratches, punch, trampling), which cannot appear in the same arrangement in the second shoe.

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The surface trasological footprint of the bare foot is called as the bare foot plantogram (sometimes called podogram, especially in the medical orthopedic literature). The plantogram is created by the contact of the foot with the mat due to the natural loading of both legs by the body weight during the dynamics of walking. Plantograms are relatively rare in crime scenes, but they can occur, so it is necessary to know their geometric characteristics. Traces of bare feet are examined in trasology only if they do not show usable features of papillary lines (otherwise they are examined in dactyloscopy). Vehicle tracks – this group of tracks includes tire tracks, tracks of rubber, wooden, and metal wheels, tracks of tracked vehicles, and track of skid vehicles.

Figure 1-3. Traces of vehicles (https://pixabay.com/cs/photos/stopy-pneumatik-sn%C3%ADh-silnice-497461/)

Traces of vehicles are created by direct contact of wheels, belts, or sliding surfaces of skis and sleds on a pad (road or open terrain). Under certain conditions, significant information about the technical characteristics of the vehicle, structural elements of the external structure (shape, dimensions) of the object that created the tracks can be obtained (the reflection of the structure of the treads of tires, rims, belts, skis, and sleds).

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There are general and specific features in traces of this type. General features are important in terms of determining the group affiliation of the object (relatively stable and unchanging shapes and dimensions of some parts of the tire pattern, track and wheelbase of the vehicle, etc.). These features are common to a particular type of tire or vehicle. They are therefore material sources for determining such parameters as the shapes and dimensions of patterns and size of tires, type of vehicle, type of the car, etc. Specific features relating to a particular object are material sources for determining the identity of the object that created the track. Under certain conditions, the contact surfaces of the tire treads and the rims of the wheels, the belt, but also the skid surfaces of the skis and sleds may reflect features which are specific only to a particular object. In terms of their origin, the traces of means of transport are divided into groups: 1. 2. 3. 4.

Tires for bicycles, motorcycles, cars, tractors, etc. Rubber or iron wheel rims of agricultural and other machines Tracked vehicles Skid vehicles (skis and sleds).

Traces of tires on bicycles, motorcycles, cars, etc. are created on the ground (road, open terrain) by turning the wheels when the vehicle is moving or when it is standing. Depending on the mechanical properties of the surface, either flat or volumetric traces are created. The tire tread pattern reflects its shape and dimensions in the case of volume marks at the bottom of the tread. In some cases, crushed tracks are created that do not display the specific features of the reflected object in the required quality (e.g., when the tire tread slips on the road, when braking, etc.). Traces of wheel rims of agricultural and other machines occur infrequently. Tracked vehicle tracks are mainly volume tracks (impressions) caused by articulated belts, which usually have a significant shape and depth of impression in the soil due to their considerable weight. Tracks of skid vehicles are created by skidding of the sliding part of skis and sledges on a snowy road or in open terrain. In most cases, there are volume traces. Other traces of similar type include traces of lips, ears, knees, elbows, fists and palms, gloves, socks, luggage, traces of animal feet, crutches, support sticks, etc. if they reflect the morphological features of the object. They arise

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under suitable conditions by contact with other objects, e.g., when the offender is resting on a dusty wall, pad or window frame, on the ground, etc. Mostly these are volumetric traces. In some cases, there are also flat marks, such as traces of knees, elbows, gloves, etc.

1.1 Searching for and securing of trasological tracks Searching for footprints of feet, shoes, means of transport, and other tracks, was carried out by searching for crime scenes in buildings, or, more often, directly in the terrain. Attention must be paid to both isolated tracks, resp. fragments of fit, as well as two sets of tracks, such as a locomotion trail. Depending on the mechanism of traces, surface and volume traces can be found at crime scenes. It is necessary to search for the track systematically from the moment of entering the search area, either in the building or in terrain. The traces found are suitable, marked, and protected against adverse weather conditions or damage, such as trampling. The traces are fixed and secured for expert examination. Trasological traces are provided in the original, by photographing, casting, or removal. When searching for plastic traces of bare feet and human locomotion, it is not possible to limit oneself to the place of the event. However, it is necessary to search for these traces in a wider area. Areas to which special attention should be paid are: x The crime scene in the narrower sense - it is the specific area where the crime was committed. For example, when committing violent crimes, there is often a struggle or other activity that results in a large number of trasological clues at a particular crime scene. Traces can be left in the victim´s blood, clothing, and body, such as a wall as on objects kicked on the floor during the match. When breaking in, there may be objects on the floor, such as safe insulation, paper, or other rubbish, on which trasological traces may be kept. x Place of entry – this is the place where the offender entered the crime scene. Violent entry occurs when the offender enters the object of the unnatural way and usually, he is more likely to step on objects, rubbish, etc. Usually provides a greater opportunity to find clues than if the crime scene where the normal entry occurred, such as the door. Trace locations should also include the outer areas of the site immediately around the point of entry, such as flower beds, verandas, balconies, etc.

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x Path of transit – the path that the offender has passed through the scene is depending on the nature of the crime scene, the identification of the point of entry, the place of the offense, and the point of departure. Wherever the passage is obvious, traces of human locomotion and bare and worn legs should be carefully sought (dusty and dirty surfaces - cellar, back porch). x Route of departure – may be difficult to identify. Often traces can be found around trees, bushes, where the perpetrator hid. Areas covered with snow, soft soil, or sand can provide an extensive amount of traces of human locomotion and bare and worn feet. Immediately after finding the track, its technical and tactical value is preliminarily evaluated. Only tracks or sets of tracks that show significant and appropriate general and specific identifiers shall be provided. The principles of providing trasological tracks are general and special. The general principles define those aspects that must always be observed, regardless of the type of trasological track. The principles that are important only for a certain type of trasological track are special. Failure to comply with even one of the principles below will run the risk of reducing the quality of the secured track, which will lead to the impossibility of the group and thus rather individual identification. Equally important, the track would lose its tactical and criminal law significance. General principles of providing trasological tracks: a) Completeness - we provide all known tracks at the crime scene, as only a forensic expert has the power to decide whether the track is usable or not. b) Integrity - we always provide the entire track, never just a part of it. c) Speed - as tracks are affected by internal and external influences that affect their quality and usability, they must be secured as quickly as possible, considering the precision of securing. d) Track protection - closely related to the principle of speed. The track must be suitably protected from damage until it is secured and handed over for examination. e) Accuracy of documentation of the place of detention - is especially important for the elaboration of an expert opinion, but also for other activities of bodies active in criminal proceedings (forensic experiments, verification of the testimony on the spot, repeated and additional search of the crime scene, etc.).

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f) Priority of nondestructive securing methods - if it is possible to apply nondestructive securing (in nature photography), it always takes precedence over destructive (casting, removal of a dactyloscopic foil, electrostatic scanning etc.). g) Priority of search and seizure - first, tracks should be found and secured at the entrance to the crime scene and on the floor (ground). The special principles of securing trasological tracks are: a) If possible, we always ensure the track in nature. b) We always, without distinction, provide tracks by scaling using appropriate photographic highlighting methods. Because the film may be damaged, we take the picture twice. c) Pour the plastic tracks or, on the surface track, remove them with a dactyloscopic foil. Examination of bare feet and footwear footprints - a trasological track created by a person may contain not only information important for the possibility of determining group affiliation. It may also enable individual identification of a man or footwear according to footprints of the bare foot, resp. worn legs, but may further include substantiated information about the offender's movement behavior and some of its somatic properties. The projection of them on the track is not excluded due to the various mechanical connections between the track-forming object and the track-receiving object. When examining and evaluating bare feet, it is necessary to consider some negative factors that affect the outcome of the examination. These are in particular: a) relativity of the dimensions of the bare foot; b) differences in area and volume track values; c) the mechanism of track formation (free walking, running, jumping, etc.). Examining the track of bare feet usually allows only the determination of group affiliation, especially in cases where the track reflects and examines only the shape and size of the foot and its parts. From this point of view, grouping and different finger shaping are of considerable importance. All elements of the foot and its parts which have been reflected in the track must be measured with each other accurately. The acquired foot shapes and dimensions are group identification marks. In summary, they can make it

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possible to create a closer characteristic of the person who created the researched track. Examining footwear tracks involves determining group affiliation in the first stage, in some cases it is not always an easy task. In some cases, it is difficult to distinguish between men's and women's footwear, as they often match size and patterns. Sometimes the difference between the bottom of men's shoes and women's shoes is significant, e.g. in width, heels, finer patterning, etc.

Figure 1-4. 3D trasological trace (Straus, Porada et al, 2004, p. 38)

In the process of determining the group affiliation of footwear, knowledge of basic production technology, and footwear records (catalogs), photographs of sole patterns are used. Often no track is required to identify a group of shoes. The size of the footwear can also be determined according to the track of the patterned block and especially the monolithic heel. According to the tracks of the designs of these types of shoe soles, it is possible to determine relatively accurately, the size, but usually also the

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type, shape of the shoe sole and upper. According to the track created by the smooth sole, the determination of group affiliation is only possible approximately. For heel tracks, the so-called sowing (i.e., distance and location of circular holes for nails) can be used, which is proportional to the size of the heel. The size of the secured track cannot be compared with the shoe size number without appropriate corrections. It depends on the mechanism of the track and the properties of the footwear as well as the properties of the material in which the track is formed. It also depends on the type of footwear that created the track. In most cases, the track is slightly larger (depending on the size of the shoe framing). The differences can be 1, 2, or more centimeters, which corresponds to one to four size numbers. The technical value and quality of the group identification marks exhibited by the size and shape of the sole and heel and their designs, as well as the method of attachment, are decisive for determining the group membership of the footwear. Individual identification of footwear is based on the existence of specific features reflected in the track of the micro relief of the surface structure of the sole, a sole or heel. The specific character of these features lies, in essence, in the individually unique, completely random, and incomparable unevenness of the bottom surface of each individual shoe. The forensic technical examination of footwear is an examination of the macrostructure of the surface of the bottom of the footwear. This is due to the fact that the material of the base (various types of soil, mud, linoleum, asphalt, etc.) does not have such properties that it can accept or reflect the microstructure of the relief of the shoe sole. Unevenness in the external construction of the shoe sole surface is caused by the manufacture, use, and repair of the shoe. Production-specific features can only occur in footwear with a rubber rolled bottom. Specific features given for use are created by trampling, sitting, penetration of various objects into the bottom of the shoe, etc. Specific features given by the repair are created when attaching the sole, bollard, repairing the sole, sole, heel, etc. The track of a shoe, i.e., each individual track, contains certain information about the shoe that created that track. It therefore contains features that can, in a favorable case, lead to individual identification of the footwear. However, already in the first phase of the examination (until a piece or pair of shoes to be compared has been secured), it is possible to determine not

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only the group belonging to the shoes, but also certain information about the person who wore these shoes (under normal circumstances) - height, albeit statistically on average, i.e. regardless of individual disproportionate variations. When evaluating the tracks of a locomotion trail (walking, running, jumping, or a combination), it should consider that the walking of some people is not regular, but has an individual character, which is displayed in the trail. Therefore, whenever a continuous set of walking tracks is provided at the crime scene, due care must be taken. To evaluate the walking path, it is necessary to carefully measure and draw in particular: a) length of the track (shoe or foot imprint); b) width of the track (shoe or foot imprint); c) step length (this is the longitudinal distance of the heel of the right and left foot in the walking direction); d) the length of the two-step (it is the longitudinal distance of the heel of the right and right foot in the direction of walking or the left and left foot); e) track angle (angle between the axes of the foot and the longitudinal direction of walking). According to research in recent years, the biomechanical content of trasological tracks of locomotion can be decoded in bipedal locomotion tracks. The biomechanical content of trasological tracks created during bipedal locomotion can be understood as the reflection of some biological properties of a person and his movement behavior in the track created during mechanical interaction with the substrate. Examination of tire tracks and various tracks created by means of transport - examination of tire tracks is performed to determine the group affiliation of wheeled vehicles, most often cars. It is based on the existence of characteristic identifiers which are common to a certain type (group) of tires and a certain type (group) of cars. The individual identification of wheeled vehicle tires depends on the existence of specific features reflected in the surface or volume footprint of the tire tread surface of a particular vehicle, such as a car. The bearer of group identification marks is the tread pattern of the tire treads regarding the structural location of individual wheels in connection

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with the wheelbase and track of the car. The following identification features are decisive for the group determination of the membership of wheeled vehicles with tires: tire dimensions, tread shape and dimension, track, and wheelbase of the vehicle. The size of the tire indicates the width of the tire tread and its inner diameter. Width and circumference, resp. tire diameters are important group identifiers. They make it possible to determine the production dimensional markings of the tire, which are typical for only a few types of cars. The width of the tire shown in the track can be obtained by measuring the distance from the edge of the pattern of one side to the edge of the pattern of the other side. To determine the circumference of the tire, it is necessary to look in the track on a continuous route for two consecutive marked specific characteristics, such as a certain type of tire damage. The determined tire size is then determined from the measured tire circumference dimension with the help of specialized catalogs. The size of the basic tread shape is always proportional to the size of the tire. There is a so-called ribbing (gap) between the individual figures of the pattern. The dimension of the basic shape of the tread changes. The tread pattern is created according to precise matrices, so all products in the series are identical. The dimensions of the ribbing depend on the tire pressure, the weight of the load, and the condition of the road. Therefore, the dimensions of the ribbing vary according to the specific conditions. The wheelbase of cars is different. It is determined in feet by measuring the distance from the center of the right wheel track to the center of the left wheel track. In the case of lorries, if they are equipped with twin wheels, the track gauge is determined by measuring from the center of the space between the pair of wheels. Most cars have different front and rear wheelbases. The wheel tracks are best detected in a slight bend, when the tracks of the front and rear wheels do not overlap. The wheelbase of the vehicle does not have a constant value. It depends on the vehicle load, direction and speed of travel, vehicle wear, and whether it is a vehicle with a fixed or split axle. Therefore, when determining the size of the track after measuring specific values, it is necessary to take into account a certain tolerance. The size of the track in connection with the shape of the tread and the dimensions of the tire makes it possible to narrow down the circle of inspected objects to the smallest number. The most important group identification feature is the wheelbase of the vehicle, i.e., the distance between the front and rear axles. This is because

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almost every type of vehicle has a different wheelbase size. The possibility of detecting the wheelbase is only given when turning backwards or during heavy braking, when the locking tracks of the front and rear wheels of the car are created. In the case of reversing rotation, the wheelbase is determined by measuring the distance from the leading edges of the marked front and rear wheel tracks. The measured values can be different between the right and left wheels. The measurement is always subject to a certain error. In the latter case, for the measurement, which is analogous to the previous example, the completed locking tracks of the front and rear wheels are authoritative. When evaluating the wheelbase, deviations can be taken into account, which can be caused by material wear, various chassis repairs or suspension properties, and the degree of tire inflation. All listed features of group affiliation make it possible to determine the type and type of cars in summary. The individual identification of wheeled tires is based on the quantity and quality of irregularities in the surface structure of the tire tread, which could not have arisen in terms of spatio-temporal arrangement of two or more objects of a similar type used in different conditions. These characters shown in the track are therefore naturally considered random and nonrepeatable characters. The origin of their origin and their uniqueness individually, but especially in their summary, are unique to the individual surface of a single, specific tire tread. Specific identification features arise by: a) detrition when the tire is used; b) various repairs. The features caused by tire detrition are different. They can be cuts, markets, cracks, or grooves. Furthermore, various smaller objects, such as nails, stones, fragments of glass, metal, etc., can be pressed into the tire. The features caused by the repair of the tire tread are local and given by the individual repair during the use of the tire. They occurred, for example, as a result of a puncture or other damage to the tire.

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Chapter One

Examination of the tracks of other motor vehicles with tires does not allow as detailed a group delimitation of objects as is generally possible in determining the group affiliation of an automobile. For these vehicles, there are not so many group identification features (structural-technical elements) that, according to their reflection in the tracks, would allow to narrow the range of inspected objects to a certain group (motorcycles, tractors, etc.). Depending on the size of the tires, the shape, and the size of the tread, it is not possible to determine exactly the type or type of vehicle. For some means of transport, e.g., single-track, the possibility of group identification features, such as gauge (not applicable) and wheelbase, cannot be used. In the case of agricultural vehicles, one type of tire is commonly used on the wheels of different types of machinery, which makes it difficult or even impossible to determine the group membership of a particular means of transport. Under certain conditions, the examination of tracks created by means of transport is of tactical importance. It is mainly a comprehensive and consistent evaluation of the track, respectively on-site tracks, as they can help to determine the direction of travel, the approximate speed of the vehicle, and, where appropriate, the type of vehicle. The determination of the group affiliation of tracked vehicles depends on the group identification features, such as: the length and width of the belt and the dimensions of the belt links. According to these features, the given type of tracked vehicle can be determined. The length and width of the belt are determined similarly to tires. The length of the belt is determined by measuring the distance between the repetitive features displayed in the belt track. The width of the belt is given by the distance from the outer edge to the inner edge of the same belt. The dimensions of the belt link are determined by longitudinal and transverse measurements. Individual identification of the tracked vehicle is possible on the basis of specific features reflected in the track. These characters are of different shapes. They are caused by wear, deformation of the belt, or by replacing individual belt links. The determination of the group affiliation of wheeled vehicles with rubber or metal rims is possible on the basis of group identification features such as: the width of the rim tracks, the dimension of the outer circumference of the rim and the track gauge of the vehicle. These characters can be used to

Forensic Trasology

15

further characterize the vehicle with metal rims. Due to the nature of these vehicles, it is not possible to determine their form, shape, or purpose according to the displayed tracks. Individual identification of vehicles with rubber or metal rims can usually be done very rarely, as the reflected traces in most cases show very few usable specific features (e.g., the shape of the rim, weld, significant wear or deformation of the rim, etc.). The determination of the group affiliation of the sledge can usually be performed only according to the displayed profiles and the width of the sledges, their gauge and the sliding surface shown in the track. As with determining the group membership of skis according to the reflected tracks in the snow, there is only a limited possibility of using these tracks to determine the group of these objects, respectively, for their closer characterization. Individual identification of sledges and skis is practically limited to the existence of only such peculiarities exhibited by sliding surfaces of sledges and skis, such as objects in sledge fittings, places of random deformations of sliding surfaces of sledges and skis, etc. These peculiarities can be detected under certain conditions; quality, shape, and size, so that they can be considered as specific features specific to only one specific object (sledges, skis), which were reflected in the trail. In some cases, the tracks of ski poles may also reflect specific features (e.g., features of various repairs or modifications) that can be used for the forensic identification process. Examination of tracks of objects of a similar type can lead to the determination of individual identification, often there are tracks of the outer ear lobe on the door, or tracks of gloves, teeth, and lips. These tracks are examined by scoring, overlaying, or a combined view.

1.2 Biomechanical content of trasological traces Trasological traces of bipedal locomotion are a typical representative of traces that reflect the functional and dynamic properties of the acting object (person). From these traces it is possible to decode the biomechanical content. The biomechanical content of trasological traces is classified into geometric, kinematic and dynamic features. Geometric features of the biomechanical content of trasological tracks are manifested mainly in the spatial arrangement of the track (track set) in the length, the width and area of the track, in the depth (volume) of the plastic

Chapter One

16

track, and in the spatial relationships between tracks in the track set. The basic characteristics of the geometric features of the biomechanical content of trasological tracks include: length and width of footwear, length and width of bare foot, length of right and left foot steps, length of right and left step, left and right foot angle. Among recent works that have broadened the so far sparse basis for the analysis of the biomechanical content of tracks, enhancing the possibilities of criminalistic identification by these means, the most notable is that by Titlbach et al. The authors of this study have treated the question of the existence of relationship, and their numerical expression, between the dimensions of the soles of the feet and body height, between the dimensions of soles and shoes, and between the sizes of shoes and body height. The statistical analysis of this problem involved the following parameters: body height, mass of the body, length of the sole of the foot, width of the sole, shoe length, shoe width, shoe type, age. The individual geometric somatic parameters were measured either by common anthropometric methods or by means of a special device for the measurement of the dimensions of the soles of feet. These experimental data provided the basis for an evaluation of the statistical characteristics of the random variables involved. i.e., their mean values, standard deviations, and the average error in the mean. Furthermore, the length/width ratio of the sole, the difference between the length of the sole and that of the shoe, and the difference between the width of the sole and that of the shoe were computed. Statistical treatment of the final set of data yielded information that seemed to indicate the following correlations: 1. Body height depends on both the length and the width of the sole. 2. With increasing body height, the length of the sole also increases within a certain scatter band with the average rate of this increase being 2.5 cm/cm (increment of height against that of the length of the sole). 3. A simultaneous correlation exists between body height and the width of the sole, the ratio between the increments in body height and the width of the sole being 4.5 cm/cm. The correlations defined above allowed an empirical relationship to be constituted for the prediction of the probable body height of an average individual depending on data on the dimensions of the soles of his feet in the form of vT = 3.1 dn + 4.0 sn + 53 (cm),

Forensic Trasology

17

where vT represents the body height (cm), dn is the sole lengths (cm), and sn is the width of the sole (cm). The probabilistic relationship between body height and shoe size was determined in an analogous manner. This correlation can be expressed as vT = 2.6 do+ 4.3 so + 55 (cm), where do is the shoe size (cm) and so stands for the width of the shoe (cm). These relationships allow the probable body height of an individual to be evaluated on the basis of numerical data on the dimensions of his feet or on the shoe size. The scatter band of these two correlations’ lies within the + 1 cm limits to the mean curve, which represents acceptable accuracy for practical purposes. In subjectively normal walking, the average stride length of 70 cm was experimentally determined and the length of the two-step in the same type of walking is 142 cm. Analytical dependencies vary around these statistical averages as follows: a) step length (dK) – body height (vT) x up to 70 cm step length, the relationship applies vT = 0.297 dK + 153 x over the 70 cm stride length, the relationship applies vT = 0.315 dK + 163 b) length of two steps (dDK) – body height (vT) x up to 142 cm, the two steps length, the relationship applies vT = 0.157 d + 151 x over 142 cm, the two steps length, the relationship applies vT = 0.175 dDK + 155 If at the crime scene a set of at least four consecutive tracks is found, there are several ways to determine the height of the person who created the tracks. It is possible to use the dimensions of the footprint, or it is appropriate to use the relationships given above. If we want to obtain the body height as accurately as possible, it is suitable to use more independent methods. The accuracy of the calculation and prediction of body height can be set to ± 2 cm. The highest accuracy is achieved using the maximum number of input parameters.

18

Chapter One

Several functional dependencies exist for these needs. According to experimental verification, the following two ways of determining body height from walking parameters appear to be optimal:

Figure 1-5: Stride length and two-step locomotion

1. determination of body height from stride length (dK) and two-step (dDK) vT = 0.153 dK + 0.083 dDK + 155.5 (cm) 2. determination of body height from stride length, two-step length, footprint length (dDO) and footprint width (dSO) vT = 0.076 dK + 0.041 dDK + 1.35 dDO + 2.4 dSO + 101.25 (cm) The mentioned functional dependences apply to subjectively natural walking on a flat surface without external influence. From the known equations, we can present a suitable number of different equations for all variants of input variables for the needs of criminalistic practice with a suitable mathematical combination. The body height of the offender can be calculated according to the measured parameters of the locomotion path and the accuracy of the calculation depends only on the number of measured input parameters. For the need of wider use of the indicated dependencies, a large number of experiments were performed for walking in different dispersion environments, in different substrates and in different topographic conditions. For all types of experiments, the step of stride length and two-step length of body height were significant. All measurements showed a higher correlation of the length of two-step to body height than the length of the step-to-to-body height. Linear regressions depending on two variables when walking in different types of substrate are shown in the following table.

Forensic Trasology

19

Table 1-1. Linear regression relations Type of substrate plowed soil snow sand slag asphalt

Linear regression relations vT = 0.278 dK + 0.175 dDK + 134 vT = 0.248 dK + 0.194 dDK + 126 vT = 0.322 dK + 0.196 dDK + 118 vT = 0.384 dK + 0.218 dDK + 109 vT = 0.308 dK + 0.217 dDK + 119

Analogous significant relationships have been shown in the study of the dependence of the length of the two-step and the length of the step on body height, e.g., for running vT = 0.379 dK + 0.161 dDK + 92 for walking just before running was found vT = 0.178 dK + 0.086 dDK + 151 and for steady state running vT = 0.380 dK + 0.190 dDK + 72 Four variables are important for predicting a person´ s body height from the parameters of the walking path, namely, the length of the step, the two-step, the length of the shoe trace, and the width of the shoe trace. In addition to the geometric features of the biomechanical content of trasological tracks, it is possible (with some probability) to decode the kinematic features of trasological tracks, especially the speed of locomotion. Determination of the locomotion speed is currently only possible for movement on a flat, horizontal and rigid surface. From the basic research, several possible expressions of locomotion rate are available. All following formulas require knowledge of the value of the step length, or the length of the jump in the run, which can be deducted from the walking path, as well as knowledge of body height and lower limb length (measured from the mat to the spina iliac anterior superior). Precise determination of the locomotion speed based on the measured parameters of the locomotion path:

20

Chapter One

a) walking speed v (km·h-1) = 11.6 l – 11.61 hDK + 8.54 v (m·s-1) = 3.23 l – 3.14 hDK + 2.31 The above equations apply to walking speeds from 0.88 to 2.2 m·s-1. b) running speed v (km·h-1) = 11.35 l – 8.17 hDK + 6.79 v (m·s-1) = 3.06 l – 2.21 hDK + 1.83 apply to running speed from 2.22 m·s-1 to 3.58 m·s-1. After the start on the first 30 meters, the stride length increases linearly. A simpler basis, also suitable for indicative determination of the locomotion rate of the subject, can be expressed as: v (m·s-1) = 3.89 l – 1.41 v (km·h-1) = 14.01 l – 0.51 Both formulas apply to speeds from 0.83 m·s-1 to 2.7 m·s-1. In all cases, the length of the lower limb and the length of the stride are set in meters. Functional dependencies usable in criminalistic practice must include input variables and such values that are directly and relatively accurately measurable from the locomotion path. Such values are the dimensions of the shoe trace and the length of the stride and two-step. Then it is possible to express the value of the probable locomotion a speed (speed or running) by one of the following equations: walking speed v = 9.314 dK – 2.226 v = 11.962 dK – 1.440 dDK – 1.784 v = 11.962 dK – 26.831 dDO – 34.613 dSO + 7.554 running speed v = 5.761 dK – 5.055

Forensic Trasology

21

v = 11.351 dK – 3.23 dDK + 3.905 v = 11.351 dK – 18.88 dDO – 24.35 dSO + 6.09 where v is the locomotion rate (m·s-1), dK is the step length (m), dDK is the two-step length (m), dDO – footprint length (m), dSO – footprint width. The prediction of the velocity according to the traces of locomotion can expressed more precisely by an exponential or algebraic function. Formulas used for estimating the frequency of walking steps are in the table. Table 1-2. Frequency of walking steps Author, year Sholz, 1953 Dean, 1965 Grive-Gear, 1966 Cavagna-Margaria, 1966

Formula f (steps/min) = (C1 ˜ v + C2)1/2 f (steps/min) = 63 ˜ v0,65 f (steps/min) = 64.8 . v0,57 ௩ f (steps/sec) = , ଴.ଷ଺ଶା଴.ଶହ଻×௩

apply for v  ¢ 0.83 m·s-1; 2.7 m·s-1 ² The formulation for estimating the mean stride length for walking is expressed in the following table: Table 1-3. Estimating the mean stride length for walking Author, year

formula

l = 0.257 v + 0.362, l – step length (m), v - speed (m·s-1), average step length was 0.83 m, apply to speed 2.7 m/s l = 0.0836 v + 0.97 hT – 0.714 Van der Walt l – step length (m), v - speed (km·h-1), hT – lower limb Wyndham (1973) length (m), average step length was 0.88 m, apply to speed 2.2 m/s ଴.ହଶ଼ି଴.ଷଵ௩ ݈ = ሺ଴.ହିଵ.ଵ ௩ሻ×௧೚ Zaciorskij l – step length (m), v - speed (m·s-1), Kajmin (1978) to – support time (s) Cavagna, Margaria (1966)

22

Chapter One

At a running speed of 2.22 to 3.58 m·s-1 applies to the average step length the formula dK = 0.0881 v + 0.720 hT – 0.598, where dK is the step length (m), v is the average running speed (km·h-1) and hT is the lower limb length (m). One of the important information about the biomechanical content of trasological traces is, in addition to body height, the information about the dynamic features of the biomechanical content of trasological traces. For the needs of criminology, it tis the information about the body weight. The weight of the body together with the body height give a precondition for creating an idea of the probable somatotype of the person who created the footprints. For the prediction of the person´s body weight, the bare foot plantogram or bare footprints in a dispersive environment are determined. The footprint of the area trace can be clearly delimited by external tangents. These tangents define four touch points on the plantogram. With a large range of measurements, it was found that only cross-sections in front of the foot correlate with body weight. The correlation of this dimension with body weight was found to be 0.72. For the needs of forensic practice, it is possible to calculate the probable body weight from selected significant plantogram parameters. For a set of men, we can express several relations for the calculation of weight (h), e.g. h = 4.2 p + 5.4 x5 - 90.5, h = 21.6 x2 + 2.6 x4 - 61.2. For the set of women, similar equations for calculating the probable weight apply, e.g.: h = 3.8 p + 8.3 x5 - 94.0, h = 5.7 x2 + 2.9 x4 + 19.2.

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These relationships predict body weight with an accuracy of plus or minus 4 kg. By further modification, we obtain usable equations for the input variables, the result of the calculation of the probable body weight is more accurate, according to the research data in tis based on a tolerance of plus or minus 2.5 kg. For the set of men apply: h = 2.1 p + 10.8 x2 + 1.3 x4 + 2.7 x5 - 75.9 For the set of women apply: h = 1.9 p + 2.8 x2 + 1.4 x4 + 4.2 x5 - 37.4. The given equations are for input values measured in centimeters; the weight of a person is obtained in kilograms. Research focused on the dynamic features of biomechanical content has shown that body weight can be predicted from selected parameters of the bare foot plantogram. The current level of knowledge does not allow estimating the weight of the body from the footprints of shoes, in tis possible only from the footprints of bare feet. The calculation of body weight can be performed from four precisely defined parameters measurable on the plantogram. These are three width dimensions (x2, x4, x5) and one diagonal parameter (p). These four dimensions show a significant relationship with body weight, which is expressed in the table with correlation coefficients. Body weight can be calculated according to a relatively simple linear regression. These formulas are created for the input of all four variables, but it always depends on which dimensions can be measured on the track. Body weight is most accurate when all input variables are available. Previous and current research shows that the footprint of the foot plantogram is not only capable of determining the individual identification of a person, but is an important information needed to predict a person´s weight. The information obtained from the plantogram expands the overall biomechanical content of trasological traces, apart from geometric and kinematic features. These are so far the only important sources of dynamic features of the biomechanical content of criminological tracks. It was shown that it is possible to obtain essential information about the perpetrator even from a single trasological trace.

24

Chapter One

Figure 1- 6. Plantogram with significant parameters

Forensic Trasology

Figure 1-7. Dependence of body height on significant parameters of the plantogram.

25

26

Chapter One

Figure 1-8. Several types of plantograms (http://www.socea.cz/projekty/paloucek/material/TVM/13.pdf. [online]. In: [cit. 2019-05-17]. DOI: https://www.eobuv.cz/boty-brooks-launch-5-120266-1b-003black-teal-green-white.html).

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Figure 1-9. Bar foot trace ([online]. In: [cit. 2019-05-17]. DOI: http://www.krimi-ltsezam.cz/cs/meritkoplast-oboustr-trasolog-cernobile-30x15-cm).

1.3 Physics behind trasology Impulse of force and momentum Image the following situation: there is a box on the floor that someone is pulling with a rope. The force exerted by the rope on the box is constant over a period of time. We define: ‫ܫ‬Ԧ = ‫ܨ‬Ԧ ሺ‫ݐ‬ଶ െ ‫ݐ‬ଵ ሻ = ‫ܨ‬Ԧ ο‫ݐ‬

Chapter One

28

This is the so-called impulse of force. The impulse of a force in a given time is a vector quantity. The unit is kg˜m˜s-1. The numerical value can be written as ‫ܨ = ܫ‬οt Note: The value of the impulse does not depend on whether and how the body moves, or whether other forces act on them. Significance of the force impulse: Consider II. Newton´s law in the form ‫ܨ‬Ԧ = ݉ܽԦ, where ܽԦ = ݉

ሬԦ ο௩ ο௧

. It folows ‫ܨ‬Ԧ =

ο௣Ԧ ο௧

ሬԦ ο௩ ௧

. Then ‫ܨ‬Ԧ =

, ‫ܨ‬Ԧ ο‫ = ݐ‬ο‫݌‬Ԧ.

The change in momentum is equal to the impulse force. If we know that some forces acted on a particle (mass point) from time t2 – t1 with a constant resultant force ‫ܨ‬Ԧ , thus the momentum changed form ‫݌‬Ԧଵ to ‫݌‬Ԧଶ , then we can write ሬሬሬԦሺ‫ݐ‬ଶ െ ‫ݐ‬ଵ ሻimpulse theorem for a mass point ‫݌‬ ሬሬሬሬԦଶ െ ‫݌‬ ሬሬሬԦଵ = ‫ܨ‬ Linear momentum of a system of particles The momentum of a system of particles (in a certain frame of reference) is defined as the sum of the momentums of all its parts ‫݌‬Ԧ = ሬሬሬԦ ‫݌‬ଵ + ሬሬሬሬԦ ‫݌‬ଶ + ‫ ڮ‬+ ሬሬሬሬԦ ‫݌‬௡ Forces acting on a system of particles -

Internal forces – these are the forces by which the individual parts of the system act on each other External forces – forces by which the environment acts on the system

If we add all the forces acting on one selected mass point, we find that the sum of internal forces will be zero (it follows from Newton´s 3rd law) and the sum of external forces holds

Forensic Trasology

ሬሬሬሬሬԦ ‫ܨ‬௩௡ =

29

ο‫݌‬Ԧ ο‫ݐ‬

If ο‫ ݐ‬՜ 0 ‫ܨ‬Ԧ =

d‫݌‬Ԧ d‫ݐ‬

Conservation of linear momentum If the sum of external forces acting on the system is zero (‫ܨ‬Ԧ vn = ሬ0Ԧ ), then ‫݌‬Ԧଶ ሬԦ , i.e. ‫݌‬Ԧଶ = ‫݌‬Ԧଵ . Since ‫݌‬Ԧଵ and ‫݌‬Ԧଶ are the momenta of the system at - ‫݌‬Ԧଵ = 0 two different moments, the momentum of the system is constant, i.e., ‫݌‬ ሬሬሬԦ = ሬሬሬሬሬሬሬሬሬሬሬԦ const. ሬሬሬሬሬሬሬሬሬሬԦ ሬሬሬԦଵ + ሬሬሬሬԦ ‫݌‬ଶ + ‫ ڮ‬+ ሬሬሬሬԦ ‫݌‬୬ = const ‫݌‬ ሬሬሬሬሬሬሬሬሬሬԦ ‫ݒ‬ଵ + ݉ଶ ሬሬሬሬԦ ‫ݒ‬ଶ + ‫ ڮ‬+ ݉௡ ሬሬሬሬԦ ‫ݒ‬௡ = const ݉ଵ ሬሬሬሬԦ The total momentum of an isolated system of mass points with mutual force interaction does not change. The momentum of individual mass points can vary. Example – the impact of a human body on a surface E.g., with an impact lasting approximately 0.006 s and the change in momentum is 2 kg˜m˜s-1, while the magnitude of the applied force is equal to 3.3˜102 N. When two rigid objects collide, the collision time is very short and the force is large, on the contrary, when the object is soft, the interaction time is prolonged, and the forces are reduced. Therefore, for example, falling into sand or loam is less dangerous than falling onto concrete pavement. If a body (human body) falls from a height h, its velocity of impact is equal to v = ඥ2݄݃ , the corresponding change in momentum is mv = mඥ2݄݃ = ଶ௛

FGට . After substituting into the relation for the calculation of the force ௚

impulse, the given force can be expressed by the relation F=

௠ ο௧

ඥ2݄݃.

It is generally assumed that the time of impact of the body on the surface is 10-2 s. The force that causes the bone to break is approximately 104 N·cm2.

Chapter One

30

If a person falls on his heels, the contact area is about 2 cm2. The corresponding force is 2˜104 N. From the previous relation for calculating the force, the corresponding height of the fall can now be expressed h=

ଵ ଶ௚

ிο௧ ଶ

ቀ ௠ ቁ , m = 70 kg, 't =10-2 s, we have h = 41.6 cm.

Center of mass of a system of particles A. Effect of external forces acting on the system: the resulting external force ሬሬሬԦ ‫ = ܨ‬σ ሬሬሬሬԦ ‫ܨ‬௞ . B. The behavior of the system under the action of this force can be described by a single point, the so-called center of mass (= center of gravity) Definition The center of mass is a fictitious point assigned to a system that has the following properties: x It concentrates the weight of the entire system, ݉ ் = σ௡௞ୀଵ ݉௞ , x It moves as if it were affected by the resultant of external forces, x Its momentum ሬሬሬሬԦ ‫ ்݌‬is therefore equal to the total momentum of the system, ሬሬሬሬԦ் = σ௡௞ୀଵ ݉௞ ሬሬሬሬԦ ‫ݒ‬௞ . ݉‫ݒ‬ x The position of the center of mass relative to a system of mass points does not depend on the choice of the reference system. Calculation of the coordinates of the center of mass Designation ‫ݎ‬ሬሬሬԦ் = position vector of the center of mass ‫ݎ‬ሬሬሬԦ் =

‫ݎ‬ሬሬሬԦ݉ ‫ݎ‬ଶ ଶ + ‫ ڮ‬. + ሬሬሬሬԦ݉ ‫ݎ‬௡ ௡ ଵ ଵ + ሬሬሬሬԦ݉ ݉ଵ + ݉ଶ + ‫ ڮ‬+ ݉௡ ‫ݎ‬ሬሬሬԦ் =

σ௡௞ୀଵ ‫ݎ‬ሬሬሬԦ௞ ݉௞ σ௡௞ୀଵ ݉௞

This vector equation can be written using three scalar equations for each coordinate

Forensic Trasology

‫= ்ݔ‬

σ௡௞ୀଵ ݉௞ ‫ݔ‬௞ σ௡௞ୀଵ ݉௞

‫= ்ݕ‬

σ௡௞ୀଵ ݉௞ ‫ݕ‬௞ σ௡௞ୀଵ ݉௞

‫= ்ݖ‬

σ௡௞ୀଵ ݉௞ ‫ݖ‬௞ σ௡௞ୀଵ ݉௞

31

The center of gravity of the human body The center of gravity of an upright human body with outstretched arms lies at approximately 56 % of the height of the body measure from the heel of the figure. The position changes with any movement of the body. The stable position of the human body requires the center of gravity to lie above the feet, otherwise the body falls. Assume a body height of 1.5 m from the feet to the shoulders. A force ሬሬሬԦ ‫ܨ‬௩ will act on the body in the area of the shoulders. The body falls – it rotates around one point (we assume it does not slip). The magnitude of the moment of rotation can be written as Mr = Fv x 1.5 N˜m. This moment of force will be counteracted by the moment caused by the gravitational force of the human body of magnitude MT = FG x 0.1 N˜m. If we consider the body weight 70 kg, then gravity force FG = 70 x 9.8 N = 686 N. The magnitude of the momentum is MT = 68.6 N˜m. The condition of a fall in this case is given by the violation of the equality of both moments, i.e.. Mr = MT. From here you can determine the magnitude of the force Fv, that will cause the body to fall Mr = Fv x 1.5 MT = 68.6 N˜m Fv = 68.6 /1.5 N = 45.7 N The human body can withstand greater force if it leans against the applied force or stretches its legs.

32

Chapter One

Gravitation The gravitational field mediates the force of the Earth on the bodies around it by means of gravitational force. The source of the gravitational field is all material objects. Newton´s law of gravitation describing the interaction of bodies was derived on the basis of Kepler´s law. The empirical basis for understanding the motions of the planets are Kepler´s three laws. The law of orbits: All planets move in elliptical orbits having the Sun at one focus. The law of areas: A line joining any planet to the Sun sweeps out equal areas in equal time. The orbiting body moves more rapidly when located in perihelion (close to the central body), than if it is in the aphelium. https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion#/me dia/File:Kepler-second-law.gif

Figure 1-10. Kepler´s law of areas (http://fyzweb.cz/materials/srazky_a_rotace/k38.php)

The law of periods: The square of the period of any planet about the Sun is proportional to the cube of the planet´s the mean distance from the Sun. ܶଵଶ ܽଵଷ = ܶଶଶ ܽଶଷ The law only applies if the mass of both planets is negligible relative to the Sun.

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The validity of Kepler´s law is not limited to planets but applies more or less to the trajectories of all bodies that move in the radial gravitational field of a central body with a mass many times greater than the mass of the orbiting body (e.g., satellites and planet moons).

Newton´s law of universal gravitation Every particle in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The direction of this force is along the line joining the particles. The forces are an action and reaction, these forces are equal in magnitude but oppositely directed.

Figure 1-11. Gravitation (https://www.google.com/search?q=Newton%C5%AFv+grav.+z%C3%A1kon&rlz =1C1GCEU_csCZ936CZ936&sxsrf=ALeKk03YibEx6WPenEp6_rG_tTfYYnSE Kg:1613566592337&source=lnms&tbm=isch&sa=X&ved=2ahUKEwiPyrr8_DuAhVw-ioKHb08A5cQ_AUoAXoECBAQAw&biw=1296&bih=900 #imgrc=FPF33KuBOnTz2M)

ሬሬሬԦ ሬሬሬԦ ‫ܨ‬௚  -‫ܨ‬ ௚ ‫ܨ‬௚ = ‫ܩ‬

݉ଵ ݉ଶ ‫ݎ‬ଶ

Here G is the so-called gravitational constant. It is a universal constant that has the same value for all pairs of particles.

Chapter One

34

G = 6.67 ˜ 10–11 N ˜ m2 ˜ kg–2 We can also use this shape for nonhomogeneous objects of other shapes than spheres, if their dimensions can be neglected due to their distance, i.e.. we consider them as mass points. The space near the Earth´s surface where the effects of the gravity take effect is called the gravity field. The force of gravity is not the same in all parts of the Earth´s surface. This is due to the unequal magnitude of the inertial force ‫ܨ‬௦ = ݉Zଶ ‫݉ = ݎ‬Zଶ ܴ௓ cos ߮. In the region of the equator is the inertial force the largest and the gravity force is the smallest. The opposite situation is on the poles (the inertial force is zero). By changing the gravitational force, the gravitational acceleration also changes. An agreement was determined between the normal gravitational acceleration gn = 9.80655 m˜s-1. Near the Earth we are talking about a homogeneous field of gravity.

Movement of the Earth´s gravity field Consider a particle located in the homogenous gravitational field of the Earth, which is acted only by the gravitational force ሬሬሬሬԦ ‫ = ீܨ‬m݃Ԧ, where ݃Ԧ is the gravitational acceleration. At time t = 0 we give the particle an initial ሬሬሬԦ. velocity ‫ݒ‬ ௢ The particle performs a compound motion – a uniform rectilinear motion with the velocity ‫ݒ‬ ሬሬሬԦ௢ in the direction of the x-axes and a free fall – a uniformly accelerated motion with acceleration ݃Ԧ in the direction of the y – axes (vertical direction). In a general case, the initial ሬሬሬԦ௢ makes an elevation angle D. with the x – axes. According velocity vector ‫ݒ‬ to the size of the angle D we recognize different types of throws: x x x x x

Throw vertically upwards D = 90° Throw vertical down and free fall D = - 90° Horizontal throw D = 0° Throw sloping upwards 90° ! D ! 0° Throw sloping down 0° ! D ! -90°

The motion is studied in the reference frame connected to the Earth´s surface.

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35

Throw vertically upwards The motion is a compound of moving vertically upwards at an initial ሬሬሬԦ௢ has velocity of a magnitude vo and a free fall. The initial velocity vector ‫ݒ‬ an opposite direction to the direction of the gravitational acceleration vector. The movement upwards of the body is an uniformly slowed motion. The velocity decreases until the top of the trajectory, where it is zero. The body returns to the Earth in a free fall. The velocity at time t (upwards):

v

v0  gt

Height at time t:

s

v0 t 

1 2 gt 2

The greatest height that the body reaches is called the height of the throw h. The velocity is zero at this point and the ascent time is

th

v0 g

The height of the throw is

h

v02 2g

Figure 1-12. Throw vertically upwards

Horizontal throw It is a compound movement, consisting of a movement in the horizontal direction (in the direction of the x axis and the free fall. It is performed by a body to which we give the initial velocity ሬሬሬԦ ‫ݒ‬௢ in the horizontal direction.

36

Chapter One

The trajectory of this movement is a part of a parabola with the vertices at the point of throw. If we draw this parabola in a coordinate system with the vertices at points x = 0; y = h, so the point B at which the body finds itself at time t, has the coordinates: ‫ݒ = ݔ‬଴ ‫ݐ‬ 1 ‫ ݄ = ݕ‬െ ݃‫ ݐ‬ଶ 2 The maximum distance from the place of throw d (point D, in which x = d, y = 0). ݀ = ‫ݒ‬଴ ඨ

2݄ ݃

The trajectory of the motion depends on the size of the initial speed v0 and the height h.

Figure 1-13. Horizontal throw

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37

Oblique up throw motion Motion with two components – the object is thrown with an angle to the horizontal and a free fall (the vertical component of the velocity is at the maximum height zero).

Figure 1-14. Oblique up throw

The initial velocity vector ‫ݒ‬ ሬሬሬԦ௢ grips to the horizontal an elevation angle D. The trajectory of motion is a parabola (only in vacuum), its peak is the highest point of the trajectory. In air, the object moves along a ballistic curve (due to the air resistance). We can write: ‫ݒ = ݔ‬଴ cos ߙ ‫ݒ = ݕ‬଴ ‫ݐ‬sin ߙ െ Distance of the throw: x = d and y = 0 Ÿ ‫ݒ‬௢ ‫ ݐ‬sin ߙ െ Time of impact:

ଵ ଶ

݃‫ ݐ‬ଶ = 0

1 ଶ ݃‫ݐ‬ 2

Chapter One

38

‫ݐ‬ௗ =

ଶ௩೚ ௚

sin ߙ.

After substituting ݀ = ‫ݔ‬ௗ = ‫ݒ‬௢ ‫ݐ‬ௗ cos ߙ =

ଶ௩೚మ ௚

sin ߙ cosߙ =

௩೚మ ௚

sin 2ߙ.

The time for reaching the maximum of the trajectory vy = 0 vy = vo sin D - gtv = 0, ‫ݐ‬௩ =

௩೚ ୱ୧୬ ఈ ௚

.

Maximum height: 1 ‫ݒ = ݕ‬௢ ‫ݐ‬௩ െ ݃‫ݐ‬௩ଶ 2 ݄=

‫ݒ‬௢ଶ sinଶ ߙ 2݃

The maximum height is reached for the elevation angle 45°.

Movement with air resistance Therefore far, we have in our calculations neglected the resistive force acting in the air on the movement of the object. It is easy to prove that the magnitude of the resistive force is proportional to the velocity of the object and the magnitude of its surface. The relation for the calculation of the air resistance force was derived by I. Newton. The formula can be written as Fo = C S v2, where v is the velocity of the object due to the air, S is the surface of the object perpendicular to the direction of motion, and C is the coefficient of friction. The size of coefficient depends on the shape of the object. For the human body is its value C = 0.88 kg·m3. If we now study the fall of the human body, the equation of motion can be written in the same form ma = FG - Fo .

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At the beginning of the fall, the velocity of the body is zero and the only force that acts on the body is the force of gravity. As the velocity of the falling body increases, the magnitude of the resistive force increases and the magnitude of the force that imparts acceleration to the body decreases. If the body falls from a sufficient height, the velocity reaches such a value that the magnitude of the resistive force is equal to the magnitude of the gravitational force. From this moment on, the body is no longer accelerated and falls at a constant velocity. Since the force acting on the body is not constant, the limiting velocity cannot be found by a simple algebraic adjustment. However, you can proceed as follows. According to the previous one applied FG - Fo = 0 mg = C S v2 Speed limit: ிಸ

vm = ට

஼ௌ

.

If we substitute the value for the human body weighing70 kg and the size of the effective area of 0.2 m2 in this relationship, we calculate the value of the speed limit ிಸ

vm = ට

஼ௌ

= ට

଻଴ήଽ.଼

଴.଼଼୶଴.ଶ

m˜s-1 = 62.4 m˜s-1 = 224 km˜h-1

It follows that the limiting velocity of an object having a comparable density and shape is proportional to ξ݈. This statement follows from the following consideration: FG (m) v l3, S v l2 ௟య

vm vට మ =ξ݈ ௟

It is therefore possible to estimate the height an animal can survive a fall. For example, if a person is trained, he can jump from a height 10 m. In this case, the person hits the ground at a velocity of ‫ = ݒ‬ඥ2݄݃ = 14 m˜s-1.

40

Chapter One

1.4 Walking and physics During evolution, only very few modes of terrestrial movement of animals have evolved. Undoubtedly, the most effective of these are walking and running, moving using limb supports. Let us study the most common movement, human walking, in terms of mechanics. Walking differs from running in that at least one foot is in contact with the surface at times. Assume that at the very beginning of the step, the right foot is at the back and the left foot is ahead. The step begins with the right foot bouncing off the ground and then the foot slightly bent flips forward. The body performs an arch on the tense left leg. This moves the right foot forward and the left foots backward. This movement repeats alternately with the right and left foot. In the following text, we will understand the duration of two consecutive steps as a walking period.

Mechanical work while walking The horizontal motion of a body in a homogeneous gravitational field is a typical example of primary and secondary school physics. In this case, no work is done. However, one subjectively evaluates the amount of work done by feeling tired, and therefore, for example, carrying a bag around the classroom cannot convincingly demonstrate such movement. At the same time, with the help of a very simple reasoning, we can show the need to expend mechanical energy when walking, even without considering the ubiquitous resistance forces. Decisive for estimating the mechanical work performed while walking is the movement of the center of gravity of the body, the trajectory of which is schematically shown in Fig. 1-15. The center of gravity moves approximately along a curve composed of parts of arcs and the energy required to maintain this movement is done by the work needed to increase the position of the center of gravity by the cant h.

Forensic Trasology

A

41

B

Figure 1-15. Movement of the center of gravity

In each step, therefore, we must supply the energy that we will not regain when the center of gravity of the body subsequently decreases. From the simple geometry (see Fig. 13) it follows directly ݀ ଶ ݄ = ݈ െ ඨ݈ ଶ െ ൬ ൰ 2 where the meaning of the symbols is clear from the drawing. Substituting typical values of l = 0.95 m, d = 0.80 m, we get h = 0.1 m, which is about a tenth of the step length.

42

Chapter One

Figure 1-16. Diagram of the human body

The (so called) biomechanical principle states then when walking on a horizontal surface, we must perform about 1/15 the necessary mechanical work to lift the body in the vertical direction by the same distance. Our simple estimation provides a slightly higher value, the movement of the center of gravity is more complicated than indicated in Fig. 1-15 and the number is actually smaller. When walking, the whole body moves in a coordinated way, which effectively smoothest the trajectory of the center of gravity, and, in addition, distributes mechanical power into more muscle groups. The main contributors to this are: 1. Bending foot to the ankle. We bounce off the tip of the foot, which prolongs the effective length of the foot at the final phase of the step, and thus prolongs the step. 2. Shooting the pelvis. We always turn the pelvis upwards on the side of the support leg, the rotation is maximal in position B from Fig.115. This will reduce the maximum height of the center of gravity of the body, and thus reduce the value of the cant. During a racing walk, a significant rotation of the pelvis minimizes the undulation of the trajectory of the center of gravity and the associated expenditure of energy.

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3. Limb movement. Stepping accompanied by the movement of the hands shifts the center of gravity of the body upwards with respect to the torso. Thus, in position A, we increase the minimum height of the center of gravity and again decrease the height h. Using the biomechanical principle, it is easy to estimate the average walking power as the 1/15 the power required to climb vertically at walking speed. Thus ܲ=

ଵ ଵହ

ή‫ܨ‬ή‫= ݒ‬

ଵ ଵହ

ή݉ή݃ή‫= ݒ‬

ଵ ଵହ

ή 80 ή 10 ή 1.1 W ൎ 60 W,

where we count on walking speed ‫ = ݒ‬4 km · hିଵ = 1.1 m˜s-1. The following table shows the average values of mechanical power at different movements, and, for comparison, also the thermal power of a person at rest. Table 1-4. Mechanical power at different movements human activity walking running a marathon run 1 500 m run 100 m thermal output at rest

output 60 W 300 W 500 W 1 200 W 80 W

It is interesting to compare walking with a sprint for 100 m. When walking ten times slower, we give twenty times less power. Thus, walking is approximately twice as efficient as running fast. The vertical movement of the center of gravity and therefore also of the eyes, is usually not even realized when walking. It will become clear, for example, if we walk next to another pedestrian, so there will be a phase shift S/2 between our steps. Try to estimate the cant value h of your step.

Activity Glue foil to a mirror with drawn horizontal lines spaced, for example, by 2 cm. Position the mirror vertically, approximately at eye level. Walk towards the mirror and observe through this foil the reflection of your eyes. From

Chapter One

44

the mutual apparent movement of the eye reflection and the orientation lines, you can easily be convinced of the vertical movement of your eyes and, with little attention, you can also estimate the size of h.

Walking velocity From the point of view of mechanics, the swing of the foot forward sis and the swing of a physical pendulum rotatable mounted in the hip joint. Walking will be the most comfortable and least tiring if the frequency of the step is close to the frequency of the leg´s own oscillations. To estimate this frequency, assume that the swing of the foot is a harmonic motion, for the period of which ܶ = 2ߨඨ

‫ܬ‬ ݉ή݃ήܽ

where J is the moment of inertia of the foot with respect to the axis of rotation, a – distance of the center of gravity of the foot from the axis of rotation. The moment of inertia of the foot and the position of the center of gravity with respect to the axis of rotation can only be estimated approximately. To calculate the moment of inertia, assume that the leg is a homogeneous rod. Its narrowing in the lower part is at least partially compensated by the foot, which forms an additional weight at a great distance from the axis of rotation. The center of gravity of the foot is approximately 40% of its length. Under these assumptions for the period of natural (undamped) oscillations we get: ଶ 1 ଶ + ݉ ቀ݈ቁ ଶ ݈݉ 0.95ଶ ඩ 2 = 2ߨඨ ݈ ܶ = 2ߨ 12 = 2ߨඨ s ݉ή݃ήܽ 3ή݃ήܽ 3 ή 10 ή 0.4 ή 0.95

ൎ 1.8 s We have used Steiner´s theorem to calculate the moment of inertia J. With an average stride length of 0.8 m, the walking velocity would be 3.2 km·h-1, close to the common velocity of slow and comfortable walking. If we want to speed up our walking, we must speed up the swing of the leg by the strength of our muscles, which requires an effort that is disproportionate

Forensic Trasology

45

to the increase in velocity. Doubling your walking velocity requires more than double your muscle power.

Why do we wave our hands while walking? Waving the hands, as mentioned earlier, increases the position of the center of gravity at the moment of maximum stepping, and thus reduces the actual value of the cant h. Both upper limbs make up about 10 % of the body weight and by deviating them 30 %, we increase the height of the center of gravity by about 5 % of the previously calculated value h. In fact, the movement of the hand also requires the work of muscles, but when waving our hands, we involve other muscle groups, min activity, and thus we help to relieve the muscles of the legs. A small smoothing of the trajectory of the center of gravity is not the only reason for the movement of the hands. Waving the hands and possibly also turning the upper part of the torso compensates the momentum of the legs relative to the vertical axis of the body. The oscillating movement of the legs during walking is related to the momentum, which, if the body was isolated system, would have to be compensated by the rotational movement of the whole torso or by the movement of the hands in antiphase with the movement of the legs. Of course, our body is not isolated system when walking, so we can walk completely stiffly without any movement of the torso or hands. The torso is then kept at rest by the frictional force between the feet and the surface and the associated moment of force. However, the moment of the frictional force torsionally strains our lower limbs and makes such gait very uncomfortable.

Mechanical work and fatigue The human body, like the bodies of other animals, is a complex system and therefore the results of simple mechanical considerations cannot be fully applied. Just two examples: 1. The tensioned spring does not do a work, but the tensioned muscle gets tired. When mechanical work should be done, it is necessary to move the body while applying force. A tensioned and fixed spring does not do any mechanical work, but just standing there (a person) is going. From the point of view of the release of mechanical energy,

46

Chapter One

in tis indifferent whether we are standing with tense legs or squatting. However, it tis more advantageous to use strong skeletal supports than to keep the gravity of the body by the tense of muscles. Walking downhill will get the person tired, even if the body´s center of gravity is constantly decreasing. 2. A muscle works more economically when it produces less power. With low power, muscle regeneration occurs in the muscle at the same time, so that a person is able to do more work with low power over a long period of time than with a large effort concentrated in a short time interval. For example, on a long climb, a slow steady pace is more advantageous than a fast ascent with frequent breaks.

Why doesn´t have a human wheel? Wheel rolling is a very efficient way to move. Low rolling friction and the ability to convert accumulated potential energy into kinetic energy (downhill riding) are the undeniable advantages of moving on a wheel (bicycle), so the wheel has become an absolutely dominant mode of land transport for technical civilization. Nevertheless, for the nature is a wheel unknown. Are good reasons why evolution did not create a wheel: a) Effective movement with wheels is possible only on a sufficiently flat surface and in a loose rugged environment it this completely unsuitable. b) The technical solution is very difficult, so it is no wonder that it has not been mastered by nature. For example, how would we solve the axis of rotation, the supply of blood to vessels and nerves to the rotating part, what muscle groups would drive the wheel, and how?

Does a human really have not wheels? The answer to this, at first glance an absurd question, may not be unambiguous if we compare the mechanics of walking with the rolling of a wheel:

Forensic Trasology

47

Table 1-5. Mechanics of walking Rolling of a wheel The center of gravity remains at a constant height. The contact area is at rest on the surface.

Walk The movement of the center of gravity is only slightly undulating. The contact area is at rest against the surface.

If we compare the energy consumption when walking with a sliding the body on a surface, we find that walking is equivalent to sliding two surfaces with a coefficient of friction f = 0.07. This value is lower than, for example the friction between two steel surfaces (f = 0.1). Our limbs therefore allow us not only to walk with small energy consumption in a difficult terrain, and to climb fences, climb trees, and swim. (Šteigler, 2001)

CHAPTER TWO BIOMECHANICS OF FALLS

Introduction The fall of the human body from the height is based principally on the physical nature of the body's body litter. It is a composite motion, consisting of moving in a horizontal direction (in the x-axis direction) and a free fall. It carries the body to which we assign the initial velocity in the horizontal direction. Trajectories of motion are part of the parabola with the top in the throwing spot. The length of the litter depends on the initial velocity v0 and the height h from which the body was thrown. In the case of biomechanical evaluations of falls from a height, it is necessary to strictly rely on the laws of physics. For objective assessment of factors affecting the course of the fall of the body and the impact position, it is necessary to take into account the conditions under which the body contact was lost at the starting point. The fall of the body is determined at the moment when body contact with the pad is lost. For forensic solutions to fall biomechanics, it is necessary to define the basic classification of falls and define some terminological problems of injury and traumas arising from falls from a height. Depending on the height of the fall, the falls can essentially be divided into three groups, namely, a drop from a stand, a fall from a height, and free fall. For Forensic Biomechanics, the most important are the drops from a height and the falls from standing. An objective solution to the question of height and type of fall is possible in principle in two ways. On the one hand, it is possible to create an optimal mathematical model and a theoretical simulation of the fall trajectory and body position at impact. Or maybe the second way, experimenting and simulating a fall with a suitable dummy that will meet the characteristics of the human body. This dummy can be dropped from a suitable height and to assess the conditions of their own fall and the impact conditions. For the gain of serious scientific knowledge, it is then the optimal comparison of

Biomechanics of Falls

49

theoretical simulations with experimental data on the fall of the biomechanical dummy. A study was published in the literature that dealt with 30 cases of death due to a fall from a height. Information on the injury, including the height of the fall and the location of the body from the base of the building (horizontal distance) was obtained from police investigation files. Further inquiries were made of relatives and interceptors. The height of the fall and the distance of the body's impact were confirmed by measuring personally at the crime scene, for each case being studied personally (Kiran Kumar, Srivastava 2013). Falls have been reported, for example, in one case, the father kept his baby in his arms on the balcony of his house when the child slipped out of his arms while trying to save it from falling from the balcony. In another case, a 10-year-old boy in a children's home slid on the railing along the staircase when he fell from a height of 5.1 meters. A thief climbed the eaves on the patio of the house and was revealed by a lady who slept on the terrace. When the shout started, the thief tried to hurry down the same way back and fell from a height of 14.4 meters. In most cases, the victim fell from a height of less than 10 meters (66.6 %). A fall of more than 20 meters was registered in just 5 cases (16.5 %). In most cases the victim fell near the building (76.6 %) and 1 m from the base of the building. Only in one case the body was found 8 meters from the building in which the thief jumped from the terrace (4th floor). To escape the police, he made a short-jump jump. The majority of fatal deaths occurred in adult men aged 21-50 years. Most of the falls were accidental from balconies or terraces. The most common cause of death after impact on the ground was craniocerebral head injury.

2.1 Biomechanical classification of falls In this chapter the biomechanical solution of falls will be discussed, both from a height and falls of a person from a standing position on a solid surface. The aim of the study is to analyze the size if a mechanical stress of the organism. We will also provide the calculation of the critical limit for fatal destruction of the organism, or determination of physical conditions of short-term survival, onset of unconsciousness, etc. To solve this question it

50

Chapter Two

is necessary to define the classification of falls from a height and describe the mechanism of injury arising from impact on the surface. From the point of view of biomechanics, it is possible to classify the fall of a person into a variety of categories. Only two of them. Theory distinguishes mainly from a standing down, a fall from a height and a free fall. This division is logically based on the specifics of the different factors that act on individual cases in the course of the fall. Stalling occurs when the body is tipped over a tilting edge formed by a line that passes through a flat footrest. The body then falls to the front or back. In these cases the air resistance values are absolutely marginal and the height of fall of individual parts of the body is different. The height from which the head falls on the head is naturally the largest when the upright body falls, the height from which the lower body parts fall, then decreases proportionately. The most common task of Forensic Biomechanics in relation to standing falls is to determine whether the fall was spontaneous or whether it was caused by stroke. A fall from a height occurs when the body is on a raised floor with respect to the plane, and when the body flips around the tipping edge and releases it from the pad and then falls. Depending on the presence of the applied forces and their size, the body moves either through the parabola, the vertical, in exceptional cases the general curve. These are falls from relatively small heights, i.e., heights up to 150 meters. Throughout the fall, motion of the body is evenly accelerated, depending on the gravity constant, while the air resistance can be neglected. As a fall, it is called the fall of a body from high heights over one hundred and fifty meters. The fall of the body corresponds to a certain point of the model of equally accelerated motion, so its speed is constantly increasing until it reaches its maximum. Experimentally, air resistance stabilizes the vertical velocity at falls from a height that is greater than 152 m. Then the air resistance equals the gravitational force FG = mg, and the speed of body movement is no longer increased. In addition, it is possible to classify falls according to whether the body is inactive or active in fall or fall. In passive falls the body is at rest before leaving the support, and its fall is essentially determined by gravitational acceleration only. In case of active falls, the body is in the fall and at the moment of detachment from the pad in motion and besides the gravitational acceleration it is further accelerated by other forces. These forces are created either by the person's own active activity or by the action of other subjects, most often by the other person. The course of the fall depends mainly on the

Biomechanics of Falls

51

action and orientation of the force-acting vector and on how its acceleration is added. The course of the fall is determined by the release of the body from the pad. From this point on, the body can take up either the vertical or horizontal position until the moment of impact, and it can also rotate during the fall. Rotation can occur in both passive and active falls. Its presence depends on various factors, but most often on the position of the center of gravity of the body when uncovering from the pad and on the size, direction and location of the force applied, especially if these forces act on or outside the center of gravity of the body. If there are other obstacle-forming bodies, such as parts of terrain, buildings, balconies, vehicles, etc., there is a so-called cascade collapse that causes the body to burst and change its path. The body is thus given secondary rotation. Impacted by fall, the fall may be slow. The body's impact on the ground mostly due to a strong impact on a certain part of the body occurs, depending on the height from which the body falls, to variously serious injuries. The first contact of the body after falling with the washer is called the primary impact. At the site of primary impact, the human body usually has a very high dynamic component of the force vector that is given by the impact velocity, body mass, and mechanical properties of the impact area. The subsequent impact of other parts of the body is referred to as secondary impact. With secondary impact, the lower impact force usually acts on the incident body part because the largest energy has already been absorbed at the primary impact site. The vertical position of the body during the fall occurs most often to the impact on the legs (especially the heels), the knee, buttocks or the head. If the body falls horizontally, the impact on the front, back or side of the body is considered. The extent of the injury depends on the impact force, which is given by the body speed at the time of impact, the contact surface of the body and the washer, the mechanical properties and the shape of the impact area, the angle of incidence and the nature of the tissues affected by the impact. The force of a blow to the destruction of the organism is, in addition to the factors mentioned above, such as body mass and its impact rate, also dependent on the length of the time period during which the body speed is zero. It follows that the harder the impact area, the greater the destructive effect will occur. According to the height of the fall, the falls can be divided into three groups:

52

Chapter Two

Figure 2-1. Classification of falls (Straus 2012)

1. A fall from a standing position occurs when the body is tilted around the tilting edge, which is formed by a straight line passing through the surface of the foot support. In these cases, the body falls on the abdomen or back, and for biomechanical analysis, the hit to the head and the associated consequences are dominant. 2. A fall from a height occurs when the body is on a raised surface, flips around the tipping edge, and a fall occurs. When falling, the center of gravity of the body moves along a parabola or vertically. The fall of the body is from such a height that throughout the fall, the body is accelerated by the weight constant and the resistance of the air can be practically neglected, its size is minimal. The movement of the body is a uniformly accelerated movement throughout the fall. These are most often falls from the windows of buildings. A fall from a height occurs when the body is on a raised floor with respect to the plane, and when the body flips around the tipping edge and releases it from the pad and then falls. Depending on the presence of the applied forces and their size, the body moves either through the parabola, the vertical, in exceptional cases the general curve. These are falls from relatively small heights, i.e., heights up to 150 m. Throughout the fall, motion of the body is evenly accelerated, depending on the gravity constant, while the air resistance can be neglected. 3. Free fall occurs when a person's body falls from a very high height. The body accelerates during the fall, and when it reaches its

Biomechanics of Falls

53

maximum speed, the air resistance increases to such an extent that it equalizes the force of gravity. Further moves the body at a constant speed. The movement of a falling body is at the beginning a uniformly accelerated movement and from a certain moment the movement is then at a constant speed. A typical example of free falls is falls in airplane disasters. Depending on whether the body before its own fall is at rest or in motion we distinguish: x passive falls - before the actual fall, the body is at rest, x active falls - at the moment of the fall, the body is in motion, the body is accelerated by the applied forces, Depending on whether the body rotates during a fall, we distinguish between falls: x with rotation, x without rotation

2.2 Injuries caused by falls As a result of the collapse, regardless of the type of fall, two categories of injury are created: local, primary or remote, otherwise referred to as secondary. Local (primary) injuries arise at places of immediate destructive force at the moment of impact on the mat. Secondary or distant injuries arise at sites of secondary impact of other parts of the body. On impact, the impact force is transmitted from the point of primary impact to the downstream part of the body even before the secondary impact of the other parts of the body occurs. This effect is noticeable, for example, in the vertical position where the body falls on the head or lower limbs. In both cases, the fall of the primary impact site, i.e., the head or the lower limbs, occurs in falls from height to devastating injury. However, due to the construction of the human support, the impact force is transferred to the spine, pelvic region and internal organs. When the body hits the surface when falling from the height, the body is injured both in the direct contact points of the body with the pad and on those parts of the body where the direct force impact was not directly directed. The traumatic damage to more distant parts of the body occurs when the fall is completed. After impact, a primary strong blow occurs on

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a certain part of the body, then the body can either bounce and fall with a secondary impact (in a case of free fall) or flip and fall on other parts of the body, usually on a larger area of the body. According to the body damage that is identificated, we distinguish the effects on: x primary impact of the body x secondary impact of the body The process of falling and the subsequent flight of a person (body) from a height is further limited by a number of laws and has several stages. In the case of a passive fall, the body is first tilted around the supporting edge without slipping, translation, and the movement of the body is rotated, namely rotation of the body and translation, followed by “cancellation” of body contact with the support and subsequent fall with or without rotation. If there are any obstacles in the trajectory of the next fall (e.g. parts of buildings, balconies) there will be an impact and the path of the falling body will change. In the case of active falls, the trajectory of the fall is influenced by the action and the orientation of the vector of the acting force (location in the center of gravity of the body or outside and also by the way of its acceleration. In the biomechanical assessment of the fall of the human body from a height, we consider quite often the fall of the body with the attached external force. The term applied force or force indicates the force that acts on the human body at the moment of detachment from the pad, and it can be developed by the person who falls, by his movement, or by another person acting on it. The vertical collapse or vertical is a designation for a convex line representing the perpendicular from the edge of the pad from which the body falls to the impact surface. If we are referring to a shift in the course of a fall, we mean the distance from which the center of gravity of the body moves from the starting position to the moment when the body leaves the support of the pad. Angle of tilt Į when leaving the support is the angle which forms the vertical of a fall with a line that is the joint of the edge of the pad from which the body falls and of the total body center of gravity. The angle of impact ȕ is the angle that is constrained by two lines when the body is at the water level. The first one passes through the point where the body first contacts the surface and is parallel to the perpendicular fall. The second line is a link between the total body center of gravity and the

Biomechanics of Falls

intersection of the water level plane and the first straight line.

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Figure 2-2. The impact of the body and the representation of the primary (triangle) and secondary injury (wheel) (Straus 2012).

For all subsequent considerations, suppose that the body acts as an open kinematic chain when it falls. The center of gravity of the body moves along the parabola in the fall. From the position to the point of contact loss (usually the horizontal position), the body moves along the circle. Only the forces that arose at the moment of reflection act on the body. The fall of the body is from a relatively small height, and therefore the strength of air resistance can be neglected. During the free fall a person who has started the fall in a certain position of the body can change the position through active activities of the limbs and the whole body. The position of the body during the fall can be changed with help of volition and the active doings of the falling person, the body can rotate around the center of gravity. From the moment of rebound or leaving the foothold until the moment of impact, the falling person can take several fundamental positions, namely: vertical – head down or feet down horizontal – forehead or back down or a position very close to this two one In a vertical body position (at the time of flight), the person may fall on

Biomechanics of Falls

x x x x

57

feet knee area head seating area

Figure 2-3. Evaluating the course of the fall from the height (Straus 2012)

With the body in a horizontal position, the body falls on the body area, namely x front surface x back surface x side part All types of impact can be combined, e.g., the impact on the knees and then the impact on the front surface of the torso. The extent of the body damage

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and of individual tissues depends on the velocity of the body at the moment of impact, the contact area of the body and the pad at the moment of impact, the nature and shape of the impact surface, the angle of impact and the nature of damaged issues. The force of the hit, which acts on the body at the moment of impact as a destructive force, depends primarily on the impact velocity and the weight of the body. Important is the time of destruction, the moment in which the velocity is zero. If a person is at rest until the beginning of the fall, then the velocity of his motion depends only on the height of the surface from the point of impact and the acceleration of gravity. The kinetic energy of the falling body, from which the force of the impact can be derived, is directly proportional to the weight of the body and the height of the fall. In the first second of the free fall the body has a velocity of 9.81 m˜s-1. It was experimentally found that in the 12th second the body has a velocity of 65 m˜s-1, i.e. 216 km˜h-1. Maximum velocities during free falls were measured during falls realized by athletes. In the low layers of the atmosphere, they reach velocities of 298 km˜h-1 (82.7 m˜s-1), at the highest altitudes measuring instruments fixed the velocity of 988 km˜h-1 (274 m˜s1). Deformation and destruction of the body at the moment of impact is not governed entirely by the laws of mechanics and physics, the human body is highly elastic, has varying degrees of flexibility, and these consequences, the force of impact and destruction is reduced. The reductions in destructive forces are also caused by the fact that at the moment of impact the limbs are folded and at the moment of impact the human body falls on two or more parts of the body. Irrespective of the type of fall, two kinds of injuries occur in principle: x local (primary) x distant (secondary). Local (primary, contact) injuries occur at the points of direct contact of the attached traumatic destructive forces at the moment of impact on the surface. Secondary (distant) injuries subsequently arise as secondary injuries away from the locus of primary injury. When the body falls in a vertical position and hits the head, primary injuries occur on the head, hand injuries are also widespread in these falls. The body turns around the head and hits the front, abdomen, or back part of the body. When a hit to the back occurs, secondary injuries to the knees, abdomen,

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and toes arise. Moreover, secondary injuries to the neck, seat (coccyx) and heels can be found. When the body falls in a vertical position and hits the feet, there are primary injuries in the area of the legs, feet, the secondary injury is again dependent on further tilting of the body. When the body is tilted forward, there are secondary injuries to the knees, elbows, and abdomen. When the body is tilted back, there are secondary injuries to the seat of the body, chest, and parietal part of the head. When falling with the impact and he knees, the primary injuries are located on the knees and the front area of the legs. For further biomechanical analyzes, the most important group are falls from a height these falls can be sub classified (for a subsequent biomechanical assessment) according the diagram.

Mathematical model of the trajectory of the center of gravity of the body during a free uncoordinated fall from a height An objective answer to the question concerning the height and the type of fall can be found in principle in two ways: 1. it is possible to create on optimal mathematical model and a theoretical simulation of the trajectory of the fall and the body emplacement on impact, 2. Providing experiments with a suitable dummy (with characteristics of a human body). This object can be dropped from a suitable height and the conditions of the fall and impact can be studied. In order to gain serious scientific knowledge, an optimal comparison of theoretical simulations with experimental data of the fall of a biomechanical dummy must be provided. When solving the task of the biomechanical analysis of the model of the fall, we rely mainly on theoretical methods. These methods are based on the synthesis of current knowledge and mathematical speculative procedures, based on the knowledge of mechanics applied to a living system. However, the human body has different mechanical properties than a rigid physical body. The general characteristics and physical laws will apply to the fall of the human body, but it is necessary to slightly correct them according to the biological characteristics of the human body. That is why it is necessary to schematize the situation, to simplify and to carry out model experiments with a biomechanical dummy whose weight ratios of the individual body

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segments will be the same as the living body. If we analyze in greater detail the whole situation of a person's fall from a height, then in a natural, uncoordinated fall, the body first pivots about an axis forward and falls only when the contact of the feet with the fall site is interrupted. The body (and therefore the center of gravity of the body) describes ideally a quarter circle, and when the body's longitudinal axis is in the horizontal position, the center of gravity trajectory turns into a dish.

Figure 2-4. Fall schedule (Modified by Kumar, Srivastava, 2013)

The initial speed can be calculated from the height and the horizontal movement in the event of a fall at various angles with the speculative formula: ‫ݒ‬଴ = ඨ

݃‫ ݔ‬ଶ ሺ‫ݔ‬sin ߠ െ ‫ݕ‬cos ߠሻcos ߠ 2

The solution to the question of biomechanics of falling from a height is very important and crucial for expert investigation in forensic biomechanics. If the response is to be serious, objective and appropriate to real conditions, it is necessary to have sufficient input information available for the subsequent biomechanical solution to the fall issue. This issue has not yet been satisfactorily solved, the current results correspond to the experiments in which a height drop with a training dummy was modeled, whose mass parameters, dimensions and location of the center of gravity were the same as the living person.

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For the experiments, a training dummy was used, used by wrestlers to practice technical-tactical shuffle actions in the game. The biomechanical dummy was constructed in such a way that the individual body segments corresponded to the live weight ratio of the man and the position of the manikin's overall center of gravity corresponded to the location of the center of gravity of the living person. The generalized results correspond to a sufficient number of experiments free uncoordinated falls from two different, fixed heights, 7 - 8 meters and 10 -11 meters, which corresponds to a fall from the second or third floor of a standard building. In the experiments, the aim was to determine and specify the free uncoordinated fall trajectory depending on the height of the fall, the starting position of the body at the onset of the fall, the site of the primary impact, the center of gravity of the body from the vertical to the point of origin of the fall, the position of the body at the secondary impact, the force, the place of the vector's field of external force. For the experiments and the fall modeling scenes, three starting positions were selected for the beginning of the fall: 1. Fall of the windowsill, the dummy was tilted from the vertical axis to 10° forward, followed by a free uncoordinated fall from the heights of 7.3-8.1 meters or 10.4 -11 meters. 2. A balcony from 10.4 to 11 meters high, with 10 kg (98.1 N) external force (stroke) attached. The external force vector was attached to the shoulders, the center of gravity, or the knees. This simulated a situation where a man is struck by a force of 10 kg in his shoulders, center of gravity or in his knees. 3. Fall from the "dream-on-hand" position from the balcony rail from a height of 10.4 -11 meters. In the first case, forward velocity v1= 1.37 m·s-1, in the latter case forward velocity v2 = 1.78 m·s-1. For these values, it is possible to express a linear relationship for calculating the probable forward velocity of the body's center of gravity during free uncoordinated fall. One form of a possible fall from a height is that at the moment of contact loss the outer body is attached to the body. This situation will occur in those cases where a person fights back. For free fall, there are physical laws that can be described by equations for motion evenly accelerated by gravitational acceleration (g). Considering that external forces are exerted on a person at a moment, then we will consider for the subsequent

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consideration that they must be reflected either from the spot or with the start, and thus the body is given an external force that causes the initial velocity of the v0. When jumping with the attached external force, the jumper is reflected upwards, the body's center of gravity trajectory (and the whole body) first flies up the parabolic curve upwards, and when it reaches the peak, it falls down. The maximum horizontal length of the jump can be affected by the size of the initial velocity vector and the angle D. The length of impact is deterministically determined by three factors, namely the height of the jump, the magnitude of the reflection velocity and the magnitude of the reflection angle. In principle, there may be two types of jumps, namely a long jump with a start and jump into the distance from the place (the so-called swimming jump). Initial jump distance with a start of 9.15 ± 0.11 m·s-1 and a jump of 2.70 ± 0.11 m·s-1, the angle of reflection was found to be 21° r 0.40° and to jump from 38° ± 1.33° (Shaw, Hsu 1998). Jumps from high heights are either suicidal jumps or unfortunate accidents when people want to overcome some distance. The reflection point, the reflection angle, the point of impact, and the height are the main determinants that can be used to determine the type of fall. The mathematical modeling of living systems has been recently considerably topical. Modeling is most often understood as a reproduction of some selected properties of the suited object constructed according to certain rules. The mathematical model is constructed according to the principle of mathematical modeling, which has a different nature than the object. Its behavior is described by a system of equations, which is identical to the system describing the examined aspect of the original. The construction of the model requires a certain analogy between the aspects and processes that take place in the model and the object. Modeling simplifies how to gain knowledge and how to analyze processes taking place in objective reality. The human body, when falling, behaves like an open kinematic chain. The movement of the body´s center of gravity is determined at the moment of rebound. If we consider falls from relatively small heights, then only those forces that were applied to the mass system at the moment of detachment

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63

Figure 2-5. Biomechanics of falls

from the ground are thought of. External forces can act on a falling body in those cases where the body falls from relatively high heights. The body then reaches a very high speed and, on the body, begins to act the air resistance force. For all subsequent considerations, assume the following mechanical conditions: x the body, when it falls, behaves like an open kinematic chain x when falling the center of gravity of the body moves along a parabola x from the standing position until the moment of loss of contact (usually a horizontal position), the body moves in a circle x only the forces that arose at the moment of rebound act on the body x the fall of the body is from a relatively small height, and therefore the force of air resistance can be neglected

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The biomechanics of a fall can be analyzed from two basic approaches: 1. It is possible to determine the main factors before the fall, whose evaluation can lead to determination of the type of fall. 2. It is possible to decode the fall information based on secured traces after the fall. The main factors of the fall, which determine the type of fall are the point of reflection, angle of reflection, point of impact and height of fall. The biomechanical analysis of falls allows to solve the following questions: 1. Was the person´s fall spontaneous, without the attached external forces, so did the person fall without any other fault, without being pushed out, or without his own reflection? 2. On the contrary, was the impact caused by external forces, i.e. did the person either bounce or were they pushed out? 3. Is it possible to calculate approximately the magnitude of the applied external force at the moment of loss of contact? 4. Does the distance of the body´s impact from the vertical of the fall correspond to the probable height of the fall? 5. If a person bounces, is it possible to calculate the size of the reflection velocity vector? 6. According to the mechanism of fall and impact, a suicidal jump or an unfortunate accident or intentional push by another person can be inferred. The information obtained from the crime scene investigation provides a necessary and unique basis for biomechanical analysis of the fall and determination of the original conditions at the moment of loss of contact with the rebound, i.e., assessing whether the person fell spontaneously without external forces or was pushed out or bounced at the rebound. Based on the literature analysis, the created mathematical model and several specific cases, we can state the requirements necessary for the objective biomechanical assessment of falls from height. To calculate the trajectory of the center of gravity, it is necessary to obtain the following information: 1. Body height and body weight. 2. Measuring the distance of the body´s impact from the vertical of the fall – the shortest distance, the longest distance. 3. Body position on impact – crouched, upright.

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65

4. Angle of the longitudinal axis of the body (torso axis) to the base of the building. 5. Assessment of the type of injury and intensity in the primary and secondary fall; assessed by a medical examiner and described in the autopsy report. 6. Departure of clothing components, especially shoes and headgear – whether the shoes flew off in the fall, where they were found, where, for example, hats were found, etc. 7. The height of the presumed fall, i.e., from where the victim approximately fell (for example the height of a window, windowsill) All the considerations, mathematical formulas and classification criteria presented here serve as an input for subsequent experiments, whose results will contribute to the objectification of expert evaluation of the biomechanics of falls from a height. Experimental data on the mechanical behavior of the human body when falling from a height are still lacking, and therefore research in these directions is very desirable and actual. In the process of investigating some crimes in which a fall from a height has occurred, the issues to clarify the circumstances of the fall itself are addressed, the position of the injured person´s body at the time of the fall, whether the fall occurred spontaneously without the other person´s contribution or with external forces. These questions can be answered seriously and objectively if there is sufficient evidence available to inspect the crime scene and a forensic medical report about wounds of the injured person. The results of research in modelling the conditions of a fall from different positions, different heights and with different applied forces were published in literature. (LebedČv 1986) The conditions were simulated using a dummy whose weight corresponded to a living body. Solution of the question of assessing the biomechanics of fall from a height is very important and crucial in crime scene investigation in the field of forensic mechanics. If the answer should be serious, unquestionable and in accordance with real conditions, it is necessary to have a sufficient amount of input information for the subsequent biomechanical solution of the fall question. Because this task has not been satisfactorily analyzed, we performed a series of experiments in which we modeled a fall from a height with a training dummy. The weight parameters, dimensions and location of the center of gravity of the body were identical to a living person. The dummy was a training manikin used by wrestler to practice technical a tactical grip actions during the match. The height of the

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dummy was 163 cm, the weight was 57 kg, and the center of gravity was in the height of 107 cm measured from the sole.

2.3 Analysis and experimental results Biomechanical studies were conducted by thirteen athletes through biomechanical measurement to test the running jump and standing jump (swimmer’s start jump) (Shaw, Hsu 1998). The initial velocity of the running jump and standing jump in normal athletes is 9.15 r 0.11 m·s-1 and 2.70 r 0.11 m·s-1 with jumping angles of 21 r 0.40 deg and 38 r 1.33 deg, respectively. The practical measurements of horizontal velocity of the running jump and swimmer’s start jump were 8.54 r 0.07 m·s-1 and 2.10 r 0.05 m·s-1, and vertical velocity, 3.88 r 0.12 m·s-1 and 1.59 r 0.07 m·s-1, respectively. These results suggest an initial velocity between 0 m·s-1 for the standing jump and 9.15 m·s-1 for the run-up and jump that may contribute to launch the fall from a height by a voluntary (suicidal) jump. The initial velocity of 9.15 m·s-1 can be defined as the maximal value of a normal individual engaging in a fall with a pre running acceleration before launch. Table 2-1. Biomechanical studies of standing (swimmer´s) jump and running (long) jump (Shaw, Hsu 1998). Biomechanical Maesurement Initial angle (deg) Initial velocity (m·s-1) Horizontal velocity (m·s-1) Vertical velocity (m·s-1)

Swimmer´s Jump (n = 9) 38.00 r 1.33 2.70 r 0.11 2.10 r 0.05 1.59 r 0.07

Long Jump (n = 30) 21.00 r 0.40 9.15 r 0.11 8.54 r 0.07 3.88 r 0.12

Standing Jump To present the typical standing jump, without adding any running activity, selective modes of the swimmer’s start jump provide unique jumping patterns that emulate the jumping activities through which the biomechanical measurements are obtained. Although many scholars have demonstrated how to find the initial velocity in sports that include a standing jump, the standing swimmer’s jump represents a distinctive pattern of jump from a height that can truly emulate the jump of falling from a height. A standard standing broad jump can generate up to 3.60 m·s-1 of initial

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velocity at an angle of 41.03 deg on the basis of the body gravity of normal athletic students. The swimmer’s start jumps, an ideal model to mimic the standing jump and falling from a height, makes it almost impossible to adjust the body position while the jumper has already left the jumping point, and thus permits us to measure the initial velocity and order related biomechanical parameters, including both horizontal and vertical velocity as well as jumping angle. Distinct body gravities may explain the lower value of the initial velocity of the standing jump while we compare the initial velocity of the standing broad jump with an adjustable gravity. A twohand push of a normal individual to other individuals (70 kg of body weight) can generate an initial velocity up to only 0.4 m·s-1 (Chen 1987). An initial velocity exceeding 2.70 m·s-1 or so becomes the criterion for the running jump that is distinguishable from being pushed or slipping before falling from a height. For distance, an initial velocity lower than 2.70 m·s-1 cannot be distinguished between suicide, homicide or accident (Shaw, Hsu 1998) Running Jump The running jump is a situation where is a running start to a jump from a height when an individual is really out of his mind or has convinced himself to jump from a height. This jump is preceded by a pre-running acceleration before launching to result in an intentional fall. When an individual actually launches at maximal force, the maximal horizontal movement can reach 42% of the height (42.21 m away from the jumping point while falling from a 100 m height at an angle of 11.44 deg with an initial velocity of 9.15 m·s1 ). A running jump initial velocity that reaches 9.15 m·s-1 reasonably explains the maximum capability of normal athletes. An initial velocity between 2.70 and 9.15 m·s-1 supports a jumping activity with pre-running assistance before the jump. Such data permit us to determine the pattern of fall and jump. Any initial velocity exceeding 9.15 m·s-1 should be carefully evaluated for other reasonable explanations, including wind factor, an inaccurate jumping point, a faulty impact point, launching machine assistance, etc. It is evident that falling after a running jump is a manner of intentional jump. Therefore, the decedent’s attempt to commit suicide should be considered. A falling fatality with an initial velocity exceeding 2.70 m·s-1 should not be mistaken for accidental or homicidal cause of death (Shaw, Hsu 1998).

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Figure 2-6. Falling patterns of standing jump and running jump above the jumping level: Running and standing jump are intimated at initial velocities of 9.15 and 2.70 m·s-1 at initial angles of 21 at 38 deg above the jumping level (Shaw, Hsu 1998).

Initial velocities from 2.70 to 9.15 m·s-1 may explain the running activity before jumping as well as the conviction of intentional running and jumping. Besides, it does become the standard criterion to characterize the voluntary jump as well as the suicidal fall. The initial velocities estimated from these experiments of standing and running jumps allow us to distinguish the jumping patterns of deaths caused by high falls. The difference between the standing and running jump can be recognized as the mental status of the jumper, including the determination or hesitation of the jumper’s thoughts. The results of biomechanical studies suggest that in initial velocity over 2.70 m·s-1 is a critical point for a voluntary jump while 9.15 m·s-1 is a cutoff point of maximal physical capability for an intentional jump. An initial velocity over 2.70 m·s-1 in a voluntary jump, with the help of prerunning acceleration before the jump, suggests that the attempt to commit suicide is considerable. The initial velocity can be derived from the height and horizontal distance of falling at various speculative angles by using eq. In conclusion, in every case, both the horizontal distance of movement and height should be used to estimate the initial velocity, to reconstruct the difference between the standing and running jump can be recognized as the mental status of the jumper, including the determination or hesitation of the jumper’s thoughts. The results of biomechanical studies suggest that in initial velocity over 2.70 m·s-1 the falling pattern, and to theorize on the manner of death so as to rule out the suicidal jump (Shaw, Hsu 1998).

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Table 2-2. Maximal horizontal movement and initial jumping angle varies with height at constant initial velocity of standing and running jump (Shaw, Hsu 1998). Height (m)

0.0 0.05 1.0 3.0 5.0 7.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0

Standing Jump v0 = 2.70 m·s-1 ------------------------------Dmax (deg )Horizontal Movement (m) 45 0.74 33.151.14 27.501.43 18.372.24 14.742.83 12.663.31 10.723.93 7.7 5.51 6.316.72 5.487.75 4.918.66 4.499.48 4.16 10.23 3.89 10.94 3.67 11.60 3.48 12.22

Running Jump v0 = 9.15 m·s-1 ------------------------------Dmax (deg)Horizontal movement (m) 45 8.54 43.42 9.03 41.99 9.49 37.4711.15 34.1712.59 31.6213.88 28.6815.62 22.7620.36 19.4524.20 17.2627.50 15.6730.45 14.4633.14 13.4935.62 12.6937.95 12.0240.13 11.4442.21

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Figure 2-7. Falling patterns intimated at various angles of jump at initial velocities of 2.70 m·s-1 (A) and 9.15 m·s-1 (B): Maximal horizontal movement can be achieved at about 40 deg; the angle at 50 deg or over starts to minimize the horizontal movement (Shaw, Hsu 1998).

Biomechanics of Falls

Figure 2-8. (A) - Falling patterns intimated at various angles of jump at initial velocities of 2.70 m·s-1 (A) and 9.15 m·s-1 (B), falling from height of 100 m (Shaw, Hsu 1998)

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Figure 2-9. (B) - Falling patterns intimated at various angles of jump at initial velocities of 2.0 m·s-1 (A) and 9.5 m·s-1 (B), falling from height of 100 m (Shaw, Hsu 1998).

Biomechanics of Falls

Figure 2-10. Range of maximal horizontal movement of standing jump and running jump at angles between 0 and 40 deg (Shaw, Hsu 1998).

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Figure 2-11. Body mass center trajectory comparison as relation of different kind of falls.

Biomechanics of Falls

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Biomechanics of Falls

Figure 2-12. Unprotected fall, v = 0.997 m˜s-1 ti = 40 ms

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2.4 Standing on a pad Introduction In the analysis of falls and head injuries in extreme dynamic loads of humans, a separate direction of investigation is formed by a group of falls that occur when the body is tipped around the tilting edge formed by the line passing through the flat surface of the feet. If there is no flexion in the knee joint (the person does not flex the knee) and there is no flexion in the hip joint, then the center of gravity of the body moves along a part of the circle. In the fall from a vertical standing position to a horizontal position, the body's longitudinal axis is tilted 90° and the center of gravity of the body moves along the quarter circle. In these cases, the body falls on the surface of the abdomen or the back, and the biomechanical analysis is the dominant blow to the head and the associated consequences.

Figure 2-13. Scheme of dropping the body from stand to pad (Zarubin 2003).

From the point of view of practice needs, the most common way is a fall from a standing position that causes a head injury, a fall back. The man falls from behind, falls on his back, and the greatest force strikes his head. In this type of fall, the person does not hold the head in the safe position with the neck muscles and, in the event of impact, strikes the head as a result of very strong dynamic forces. In the course of a movement, the falling person does not coordinate in the vast majority of cases, falls spontaneously, chaotically, and moves his back, curls his head, and in this case falls backwards on his head. The highest dynamic load then receives only the occipital portion of the head of the falling person. Exceptions may occur in the case of a very

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small group of specially trained athletes, especially junior sports (judo, wrestling, karate), who are specially trained on this type of fall and react reflexively, the fall damping by coordinated movements. They are perfectly capable of stunning, shock-absorbing, collapsing body when falling, and head-to-head contact does not come into contact with the right fallback technique. In the other considerations, we will not consider this type of fall, from a biomechanical analysis point of view we will be interested in the crisis variant of the fall, in which the person strikes the head. The essence of the biomechanical assessment is the assessment of the possible fall, head impact on the ground and the occurrence of the injury. The angular velocity of the falling body is

߱=

ସ.ଽଶ ξ௅

.

Or if we calculate the peripheral velocity of movement of the center of gravity of the head segment (vr), it is necessary to base the general relationship: vr = Z · ro If we know the distance between the center of gravity of the head and the rotation axis, it is possible to express the peripheral velocity of the head center of gravity movement during a spontaneous fall. According to biomechanical data (Korsakov 1991, Sažajeva 2008) the distance considered can be expressed as ro = 0.94 L Then you can enroll ‫ݒ‬௥ = Z × 0.94‫= ܮ‬

4.92 ξ‫ܮ‬

× 0.94 × ‫ܮ‬

After editing, we get (Korsakov 1991) ‫ݒ‬௥ = 4.62 ξ‫ ܮ‬or very precisely ‫ݒ‬௥ = 4.417 × ‫ܮ‬଴.ସଽ

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Mathematical modeling of a fall from a stand on a mat Mathematical modeling of the whole process and simulations of the human body by the mechanical model can express the magnitude of the forces that act at the moment of falling into the person's head. Calculation of the impact force is best suited to the theoretical modeling process from the empirically derived inputs, and compare the resulting computation with those literary data that were obtained, for example, by stroke. The experiments confirmed the expected and logical conclusion that the destruction time of the head varied depending on the surface hardness, it was found (Gromov 1979): a) For the hard surface it is ti = 0.006 – 0.007 s b) For the semi-hard surface ti = 0.007 – 0.009 s c) For a soft surface ti = 0.021 – 0.030 s From the known time of head destruction in the fall, it is possible to calculate the probable magnitude of the force that acts on the head of a person when falling back from the stand on a pad of varying quality of elasticity. The calculation of the force size depends on the weight of the person (G), resp. weight and body height (L) (Gromov 1979). a) For the hard surface it is F = (7.7 r 0.6) × ‫ × ܩ‬ξ‫ܮ‬ b) For the semi-hard surface F = (5.6 r 0.7 ) × ‫ × ܩ‬ξ‫ܮ‬. c) For a soft surface F = (1.6 r 0.3) × ‫ × ܩ‬ξ‫ܮ‬ Experimentally, these values, procedures, and formulas were verified by dropping the biomechanical dummy into a strain gauge plate that sensed the magnitude of the force generated by the head stroke of the fall. Differences between the calculation and the measured values were minimal, i.e., 50 kg, and the formula can therefore be accepted for forensic biomechanical analyzes. From the point of view of forensic biomechanical assessment of the fall from the stand on the pad, it is necessary to consider the case when the person is accelerated by the applied vector of force located above the center of gravity of the body. In practice, this is the case where a person is struck in the head, for example, by blowing his fists, kicking his foot, blowing open his palm, or some object. As a result of the strike, the head curves, the body bends downwards, and the impact on the pad faces the main blow to the head part of the head. The most common site of destruction of the skull is in the area of the lamb seam.

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Bend and fall from stand Falls caused by disruption of attitude or walking are a relatively frequent phenomenon in forensic biomechanics. In the Czech criminal area, 15% of cases are dealt with in forensic biomechanics. This issue is not used only in criminal cases, but also in civil cases, for example, in the fall caused by alleged slipping on the surface, in which a knee or hip injury occurs, when it is necessary to determine the mechanism of the fall, which often occurred without further witnesses. The case was also described when slipping on the head, and consequently the cause of the fall, which was underpinned by incomplete testimony, was extremely small, especially in comparison with established biomechanical models. Therefore, it is necessary to know the typical and appropriate features of individual disruptions. The following figures show the kinematic values of motion - the movement of the head and the body's body during fall.

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Figure 2-14. Typical course of head velocity and center of gravity over time, y axis: speed (m·s-1), axis x: time (s)

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Figure 2-15. Typical course of the angular velocity of the head and the angular velocity of the resting limb over time, in case of restored stability (walking speed: 6.9 km·h-1, response time: 0.08 s), y axis: angular velocity (rad·s), axis x: time (s).

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Figure 2-16. The course of the spontaneous fall of figurant

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2.5 Human reaction time Introduction Free reactions to the stimulus are much more complex than reflexes and require higher brain function. In case of free reactions, the signal from the eye or other sensory organ, or even several sensory organs at the same time, it is sent to the motoric centers of the brain that process it, determines the nature of the response, and transmits the given instruction to the muscles, which then perform the reaction, after a certain period of time. However, the response to a given stimulus does not react with a muscle reaction immediately but with some delay. The length of the reaction time is physiologically limited and, to a certain extent, influences the speed of the entire movement (in fact, the total duration of the movement), which is essential especially for short-duration movement movements of the order of seconds. Response rate is also essential in solving motor activity involving large muscle groups (Straus 2001). In forensic biomechanics, in recent years, the issue of addressing external and internal responses to reaction time has emerged as a very topical issue. As the current factor we consider the influence of alcohol on the decision time, i.e., the reaction time is a complex motor response.

Concept of reaction time The simplest is the reaction time (Danko 2013), the time that elapses from the beginning of the perception of the stimulus to the beginning of the response to the stimulus. The expanded concept of reaction capability was provided by Human Factors Design Handbook, defining a simple reaction time as the shortest possible time between the moment the senses detect the stimulus and the time at which the body begins to respond, while the complex reaction time additionally involves the process of human thinking. It is further characterized by the fact that the role of the complex reaction time is to create several stimuli with different modes of response. The distribution of simple reaction times and selective reaction times with a simple motor response is revealed by the fact that the visual information process is the most important part of the human reaction capacity. Additionally, the optional response time includes a decision-making process that logically causes delay, thus comparing with a simple reaction time, the overall reaction rate increases. Moreover, the time needed for the decision

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is the most variable component of the reaction rate. However, this difference provides an approximation of the determination of the decision time interval, according to specific conditions, respectively. The number and type of factors that will be further elaborated in this work. The most important factor here is the kind of incentive, because the need to make decisions based on a more or less standard incentive makes this component unstable compared to other components (Demirarslan 2008). The total reaction time can be expressed as the sum of the duration of the visual perception and the duration of decision-making that the motor response itself is immediately following. Visual perception includes the interval needed for the detection of the stimulus since it is detectable, while the decision time represents the time needed for selection and response decisions. Then the body starts the performance of the corresponding response. Above the definition of reaction time, the time required for muscle movement is built up, which nevertheless constitutes an unavoidable category, since exploring only the reaction speed without interest, motor responses would be lost to forensic biomechanics of practical significance. Expressing the reaction velocities in terms of these components is as follows: ‫ݐ‬୰୲ = ‫ݐ‬௣ + ‫ݐ‬୰ , trt… reaction time tp… the time required for perception tr… time needed for decision making.

Categorization of reaction times Donders in his publication (Donders 1969) first proposed a classification scheme in which experts continue to describe and distinguish between response rates: -

-

-

simple - consisting of the stimulus itself, to which the subject responds as quickly as possible, immediately after the discovery of the stimulus; recognition - consisting of two or more stimuli, but with only one response corresponding to one stimulus, while the rest can not respond; selective - consisting of two or more stimuli to which the subject must make different responses, if the subject must choose what

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signal was present and then make the response appropriate to that stimulus. The scheme concerns and continues to concern experimental psychology and closely related science disciplines. In a simplified way, this branch can be included in the reaction time, the essence of which is motor-friendly, and a terminologically slightly different scheme can be expressed in Fig. 2-17.

Figure 2-17. Categorization of reaction times

Response rate with complex motor response characterizes a situation where the subject engages in response to a large muscle group, unlike simple motor responses where it is absent.

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Recognition reaction time

Selective reaction time

Figure 2-18. The process of motor response formation for each type of reaction time, according to Donders (1969).

Components important to the duration of the action From the point of view of a relevant event, whether traffic accidents or conflict fighting analysis, it can create next to these components another important category of latency caused by the device. If a person performs a response by means of an instrument, then they form an inseparable system together with the human being, and the duration of a human reaction cannot be considered relevant. Most often, given these examples, it is certainly a means of transport or a firearm. The basis for the interpretation of the components includes, without question, the elucidation of the essence of perception, since perception is the basic process of man and the reaction of the initiation process in response to any stimulus. The most important types of perception are visual and auditory perceptions.

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Visual perception Visual perception is the most important in many situations. The entity obtains basic information about the situation. However, the eye has different areas of distinction. In this context, we talk about central and peripheral vision. The central frontal vision has a range of only a few degrees at the highest level of sharpness. For the optimal use of this vision, the subject needs to constantly change the direction of vision. Peripheral detection, general vision, on the other hand, captures the entire area outside the conical central vision. Visual perception is the most important for identifying information important for further decision making, which, as mentioned, plays a significant role. The general process of vision is as follows (Porada 2000): x the eye is oriented in the field of view with volatile micromovements; x the external stimulus attracts attention; x the visual receptor focuses and focuses on an interesting optical stimulus and, on the basis of the detected optical parameters of the optical situation (distance, brightness, etc.), is prepared for reception; x the stimulus processed by the optical system of the eye will hit the luminescent elements of the retina; x the transformation of the optical stimuli in the nerve impulses generates a response in the optic nerve, which leads to the brain center of vision where the sensation is generated; x synthesis generates a perception, on the basis of which the organism's response to the given stimulus is decided, so-called differentiation occurs; x the sensation can be lost or stored in memory or can be transformed into anxiety, spreading from the movement nerves to neuromuscular plaques; x in neuromuscular plaques, nerve impulses are transformed into nervous contractions; x during the process, the central nervous system is constantly informed about changes in the properties of the observed object and its surroundings; sends commands, controls the adaptive state smoothly.

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Theoretically, a role can also be played by perception within so-called "foveal vision," where the whole yellow spot does not come to the picture, but only in its part called the central well and only suppositories are filled. In this section, the highest quality display of items is displayed.

Hearing perception Hearing perception allows the subject to retrieve information that would be difficult to detect by sight, because it did not work, so he would not be able to handle it. Audio information, unlike optical, is perceived unconsciously inadvertently, without the intention of registering it. The hearing organ consists of three parts: the outer, the middle, and the inner ear. The outer ear consists of the bolt and the ear canal and ends with a drum. The outer ear captures the sound of the drum. This part of the auditory organ, along with the shadow of the head, influences the intensity of the stimuli coming to the drum from different directions, so it is important for the directional characteristics of the auditory organ. The sound is best received at the party and somewhat from the front. Directional effect occurs at high frequencies, while tones deep up to 200 Hz, perceive on all sides of the same volume. The middle ear has a transfer and protective function. The string of three auditory bones transmits and amplifies the vibration of the drum into the oval window of the inner ear. The sound energy is collected from a relatively large area of the drum, it concentrates on a small area of the oval window, and virtually no loss passes into the middle ear fluid. If a strong sound comes to the hearing organ, the two muscles will withdraw in a reflective fashion. This increases the tension of the drum and makes it difficult to transfer, especially deep tones. It happens at sound levels of 65 - 85 dB. Throughout the moment of stimulation, the perceptiveness of strong sounds is reduced, and the labyrinth is protected from damage. The reflex has a latency of 10 -150 ms. However, for sounds of impulse nature (duration up to 200 ms), this protective function of the middle ear is not actuated, so it is easier to damage the inner ear. The minimum sound level audible to the human ear is called the audible threshold, which corresponds to a sound pressure level of 10-5 Pa. If the intensity of acoustic waves on the ear increases, the perceived noise becomes louder and louder, when the hearing around 120 dB stops and changes into ticking, so that the tactile sensation also occurs in the hearing organ, which is referred to as the tactile threshold. However, if hearing sounds for long periods of time, the threshold of audibility is already in the

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first minutes. Adaptation is taking place and the noise is perceived at a lower volume. This adaptation phenomenon is followed by another storyline hearing fatigue that occurs already in the first minute and reaches its saturation in a period of 7 to 10 minutes. It also involves altered differentiation of frequencies, volume, and change of camouflage. It recedes in tens of minutes, hours and sometimes lasts all day.

Duration of action and its components The total duration of the action can be sequentially subdivided into separate sections for didactic purposes. These sections are the reaction time, the duration of the motor response, and facultative latency caused by the device. Clearly, this complex is represented by the following scheme:

Figure 2-19. Structure of the total duration of the action.

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Reaction time It represents a time that takes time from the moment the respondent registers the impulse that has occurred and decides on the response until the beginning of the response. This is the start-up phase of the whole process, consisting of the four subcategories listed below. Perception: the time that is required for sensor sensing by sensor sensors. The factors determining perception, detection and their actual influence on the reaction time value will be described extensively in the following chapter, however, there is a need to make a certain introduction to this topic. The character of the perception significantly affects the overall reaction time, the most important being the intensity of the stimulus, its complexity, and the circumstances in which the stimulus is perceived, as well as the person's readiness for the stimulus to occur. Recognition of perceptual nature: the time required to recognize the sense of perception. This component requires the application of information and experience from a person's memory to interpret the excitement coming from the sensory sensor. In some cases, there is an automatic answer, in this section is very short. In these cases, of course, there are simple reactions, including unconditional reflexes. In other cases, this is a controlled response, which represents a disproportionately significant time. Generally speaking, a new subject, a stimulant, unknown stimulus, slows the reaction time, a less intense signal, and the uncertainty, whether the source of the stimulus, the specific moment of appearance of the stimulus or its form, and, of course, surprise. Undoubtedly, there is a very close connection with the previous subcategory of perception, respectively. It is possible to conclude in many experiments the redundant character of subcategory recognition. However, its introduction brings a more complex theoretical basis to the problem of reaction time components. Finally, the results of the research justify the inclusion of this theoretical framework. Awareness: the time needed to recognize and interpret the nature of the environment, extract its meaning, and predict eventual development for the future. E.g., once the driver recognizes the pedestrian on the road and combines this perception with the knowledge of his own speed and distance, he will present a sequence of how and what will happen. As with the previous subcategory, the new stimulus slows down this phase, which is intelligently processed.

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Choice of response: the time needed to decide what kind of response will be needed. Selection of possible reactions slows the reaction time if a more diverse set of possible signals exists.

Time to move Once the response is selected, the subject must perform the required muscle movement. It is clear from the nature of the matter that the very beginning of the movement can be almost equal to the time of completion of the movement, especially in the simple reaction times. However, these cases are not very interesting for us. A more marked difference between the start of the reaction and the moment of completion of the reaction is observed for complex motor manifestations of behavior. For example, I can point out the situation in a confrontational struggle where the beginning of the reaction to an effective defense is totally irrelevant, as the defense itself becomes effective only after the transition to a certain stage of the technique. Of course, there are a number of factors influencing the time required to perform the movement on this stage. In general, the more complex movement is required, the higher the latency.

Meaning of reaction time components in confrontational combat The mandatory conditions of necessary defense in a clash, ie, the ability of the attacker to resist the attack by reacting, occur when the inequality of the success of the defensive action is fulfilled: ο‫ݐ‬ୢ < ο‫ݐ‬ୟ ο‫ݐ‬ୢ … duration of defensive action, ο‫ݐ‬ୟ … duration of the offensive action. The duration of the defensive action consists of two parts: ο‫ݐ‬ୢ = ο‫ݐ‬୰୲ + ο‫ݐ‬୫ ο‫ݐ‬୰୲ … the current response time of the defender,

ο‫ݐ‬୫ … duration of the defense movement.

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At the same time, the reaction time of the subject can be expressed by: ο‫ݐ‬୰୲ = ο‫ݐ‬୮ + ο‫ݐ‬୰ ο‫ݐ‬୮ … duration of perception, ο‫ݐ‬୰ … duration of the decision-making process. ο‫ݐ‬୮ + ο‫ݐ‬୰ + ο‫ݐ‬୫ < ο‫ݐ‬௔ There are several possibilities to increase the chances of effective defense: -

reducing the duration of the perception of the complaint, reducing the duration of the decision-making process, reducing the duration of the motor response.

Time requirements for individual components can be divided into three phases: visual perception, decision making, and muscle movement. Approximately 70% of the total reaction time is the time required for visual perception, while 30% requires a motor response. This ratio refers to the transport driver's motor response in the Demirarslana study 2008). Average division according to Bradáþ (1997) is 28.4 % for muscle movement, 71.6 % for visual perception, respectively 23.8 % to 76.2 %. The proportion of perceived speed response increased as the driver followed another object, either within a range not exceeding five degrees from the perpendicular to the relevant object, respectively. exceeding this value. Of course, in the case of a struggle, there are more complex motor responses, thus balancing the two components. Visual perception as a component of the reaction rate is influenced by the factors that will be discussed in the next chapter where the nature of the action will be explained. In general, the external environmental conditions, the spatial location of the subject towards the source, the direction in which the stimulus is exposed. The time required for decision making is the most variable component of the reaction time. The factors that act on it can be very difficult to categorize in some way. It is clearly determined by the subject itself caused by the psychic states of the infected person, by emotions, disturbance, inexperience in conflict, struggle, etc. Therefore, the reduction of the duration of this phase may be mainly the experience gained in these situations, the psychological resistance.

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In the motor, movement speed determines the time required to perform a particular movement act, due in particular to the training of the muscular apparatus and to the speed of muscle contraction of the involved muscles. Trained subjects are therefore better placed to reduce the duration of this phase. These people have reached a stage called stereotype motion stabilization, where the movements are carried out accurately, fluently, in a coordinated way, and economically. As a result, the time required for the motor response is greatly reduced, and the person is able to act precisely, thus increasing the chances of effective defense incomparably with the untrained. Another positive aspect in terms of effective defense is the fact that trained subjects generally gained the ability to perceive quickly in the trained area and, thanks to a stabilized dynamic stereotype, also reduce the time needed for decision-making. All these benefits of training contribute to a substantial reduction in the overall duration of action. However, many offensive actions can be made at short distances so quickly that they cannot be resisted. The attacker therefore detects the stimulus at maximum, but without a relevant motor response it has no meaning in terms of its effective defense. Therefore, it is desirable not to react to the impulse that has already occurred, because it makes an effective defense impossible. Regarding unarmed attacks, the easiest, fastest means to reach a criminal target in a violent way is to strike the limb, i.e., the stroke, and the kick. The velocity of the strike itself does not play a significant role because it achieves the desired effect in the event of an appropriate attitude and the optimum distance from the injured person. The effectiveness of the strike also affects the correct pronation, respectively forearm suppression and rotation and relocation of the hull. Similar motor operations are required when using a short cold weapon or heavy object strikes. However, the use of a short cold weapon is effective even with the movement of the limb itself. On the other hand, a stroke driven by just the movement of the limb would not be effective enough, but in both cases the initiation of motion and its detection as a stimulus for the injured would be less readable. Therefore, if the attack action takes a considerably shorter duration than the duration of the defensive action challenged, its defense as a reaction to it is unrealistic. In order for it to be possible, it is essential that the attacker responds not to the beginning of the offensive action of his opponent, but to something that has been sufficiently prevented and helped to identify the stimulus itself. The attacker then anticipates the future development of his

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opponent's behavior, which he then acts on. To anticipate the probable behavior, it also offers a solid opportunity to defend itself effectively. The determination of the components of the psychic and physical abilities of the subject was the subject of research by the already mentioned authors Olenika, Rožkova, Kargina (1984). We present the measured values according to the subject's preferred capabilities: Table 2-2. Average group indicators of the development of psychic properties of top wrestlers with different ways of fighting

Type

Player Stronger Tempaer

Simple reaction time (ms) 148.2 ± 10.2 157.7 ± 11.3 160.1 ± 11.1

Complex motion reaction (ms) 200.9 11.2 224.9 18.5 223.5 24.1

±

Response to a moving object (ms) 500 ± 190

±

610 ± 220

±

690 ± 250

Feel for time (s) 3.87 ± 1.86 4.93 ± 2.84 7.31 ± 4.20

Rationality of operative thinking (number of moves)

7.72 ± 0.41 8.32 ± 0.71 8.42 ± 0.66

For illustration, we also attach the results of the measurements of Novák, Skoupý, Špiþka (1991) concerning this narrow issue. From the reaction times mentioned, it is obvious that the experimental person responds to something that prevents the opponent's leg from moving away from the pad. These measurements were performed in the gym during normal evening lighting. Measurement has confirmed that the level of illumination and its location greatly affect the ability of the test person to respond. With good illumination in the right direction, the simple reaction time of a non-standard signal, whose substrate is offset followed by a kick, also gets negative values. The conventional start of the action takes the moment when the striker's leg has begun to move away from the pad. However, the last irreversible changes in the preparation of the trial person to carry out the follow-up action can be reliably identified under these conditions for 0.5-2 seconds before the determined start of the attacker's movement, which is sufficient for practical purposes.

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Table 2-3. Conventional simple reaction time before selected combat actions Response type after exposure to a standard visual signal Press the button Straight cast of the distant arm Hook aside External rotary key An arc kick from a far farther foot from a combat guard A circular kick from the bottom of the opponent's shin to the legs Loss of battle guard to trace forward Loss of combat prudence on track back Cover from the front of the arm from the battle guard Cover from top to front of arm from combat guard Reverse the head Bend your head aside

Conventional simple reaction time (ms) The shortest 153 211 229 228 220

Ordinary 180-200 270-330 270-330 260-290 240-280

300

300-380

226 210 203

260-300 260-280 220-250

211

230-250

211 201

230-260 230-280

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Table 2-4. Conventional simple reaction time before selected combat actions Type of combat action

Direct hit Hook aside External rotary stroke without strain The outer threshing of the distant arm from the combat guard Top down from the battle guard through the forward arm Direct the kick aside from the battle guard with the leg up to the knee The end kick from the bottom of the opponent's shin Arcing kick from the bottom of the battle guard to the far legs 90° The swinging knob aside from a combat guard close to the legs to the waist An outer kick from a combat guard over the legs to the waist Seoi-nage from the distance from the front of the arm to the front Placing over the calf (tai-otoši) External impact (o-soto-gari) Front thrust (uþi-mata) External cover (according to Šotokan school) Indoor cover (according to Šotokan school) Reverse the head Bend your head aside

Duration (ms) The shortest 91 120 181 139

of action

120-150 130-150 190-200 150-170

105

110-120

241

270-290

143

150-160

277

300-320

334

350-370

345

360-380

467

550-590

441 643 338 159 111 100 110

500-550 670-720 470-560 180-190 150-190 -

Ordinary

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Table 2-5. Duration of combat actions Experimental Person No. 2 performs:

Lift forward + arch to the waist height End kick from far farther leg The end kick is closer to the leg A kick from the bottom to the waist height A kick from the front to the waist Swing the kick aside from the front position to the waist height Lift forward + arc kick from bottom to waist height

Duration of simple Response Time of Experimental Person No. 1 (mean of measured values in ms) 48 62 93 88 37 6 115

Factors influencing the reaction time Response time determinants can be classified according to a number of criteria, including alcohol, drug-stimulating drugs, and therapies that are relevant for both theory and practice. drugs, age, training, fatigue, spatial orientation to the stimulus, warning of incoming, stimulus and tension. In the next, we were primarily interested in the question of changing the reaction time due to the level of alcohol. Alcohol reduces the speed of information processes, simple, selective, and recognition reaction times in experiments requiring a simple motor response in response. Finally, it also disrupts the cognitive abilities of the higher order, which is a prerequisite for the negative determination of complex motor responses.

Experimental part The main objective of the experiment was to find human reaction times in an experiment focused on complex reaction time selective to a complex motor response. In addition to this goal, we focused on quantifying and expressing the reaction time dependency on the amount of ingested alcohol, preparedness due to the distraction of the subject and the intensity of the

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auditory stimulus. Another task was to express the time duration of the stroke from the rest position, both in the free space and the rigid body. On the contrary, the aim was not to follow the analysis of simple reaction times, whether with a complex motor response or with a simple type of motor response. Likewise, it seemed desirable, given the goal set, to configure the experiment so that the stimulus would characterize its randomness caused by spatial and temporal uncertainty during exposure. Random stimulus signs for this experiment: an impulse from a defined set of stimuli with which the subject was informed before the experiment began, each of which was the only correct response, the most important of which is the complex motor response, unlike the typical patterns used in experimental psychology, there are no constant time intervals between stimuli, respectively. almost constant intervals (Experimental psychology uses time intervals between impulses whose duration is in the range of about 500-3500 ms, which inevitably, at least in some cases, decreases the reaction time due to the sequential effect), thus eliminating the so-called sequential effect; in this experiment, on the other hand, we worked with time frames ranging from tens of milliseconds to more than a minute upper limit. Furthermore, an important factor for the randomness of the stimulus is the fact that there was a change in the character of the stimulus, if there was an alternate exposure to the auditory stimulus (from the point of view of the complexity of the unimodal) with the audiovisual (in terms of bimodal complexity), and accidentally participated in an undefined impulse, which the subject did not react at all. There has also been an ongoing substitution of the spatial location of the source of the stimulus, again to maintain the variability with respect to the subject. The experiment was attended by 25 volunteers representing a group of very well-trained people. The practical part of the research was carried out at the police gym of the Police Academy of the Czech Republic. Experiments and measurements for all volunteers lasted roughly 60 minutes. Because of the nature of the experiment, only a complex and simple reaction time was present that required a complex motor response. The instructions were presented to the subjects before the start of the experiment. This was an outline of the focus of the experiment, with the focus on the research of reaction times for a random stimulus that requires complex motor reaction. In addition, the instructions consisted of defining the impulses, the kick, back, the pulling of the pistol, abdomen, sed, light,

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crank. It was explicitly stated that they should not respond to any further stimulus. Such instructions form the nature of a sample experiment - the subject responds to stimuli, for which they must choose the right response and, in addition, to distinguish undesirable stimuli. If simple types were present, this was the way the subject performed a "neutral reaction, a simple move", and only modified the process to a correct response during the movement. In this case, we determined the value of the simple reaction velocity, and then the latency, which determines the period from the start of the simple reaction to the actual reaction that is relevant to the given instruction. We are, of course, contemplating the overall reaction time, which we continue to work within the context of the analysis of addiction. Exposure to the sound of the conclusion was exclusively in the dorsal direction towards the subject. The criterion for selecting the suggestions made in the verbal expression was the requirement for a relatively equal duration of the instructions, which was also subject to the method of formulation of the assignment, the objective of which was not immediately obvious. Therefore, the subject received the appropriate instruction on the correct answers. In the case of more concise but significantly different signals in terms of length, it would be more likely to detect an undefined signal, thereby obtaining the conditions for a simple reaction time, and during its execution, the whole information would be expunged in the meantime, thereby "specifying" his / her response, i.e., he has made the required response. In other words, it could be said that the subject would be a signaling agent of the incoming stimulus, which is the exclusively positive determining factor of the reaction rate. The alcohol level in the blood was measured with a breath alcohol detector - Alcohol Tester, however, to eliminate alcohol in the breathing body for about a while. He spent 10 minutes exercising to remove alcohol from his breath and speed up the absorption of alcohol in the blood.

Methods and results of data analysis The methods of data analysis were that we extracted the unchanged soundtrack in the best quality from the video we obtained. We analyzed the video in VirtualDubMod 1.5.10.2 build 2540. The record was used to determine the moment when the subject responded, which meant 40 ms accuracy due to the methods used in the video. With the interleaving cancellation function, a sequence of 20 ms was finally created, meaning this limits errors in the output measurement. The input at the start of the stimulus

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exposure, the essence of which was the audio signal, was analyzed by Audacity 1.2.6, which was already working only with the audio track, for better accuracy and for the possibility of further analysis of the track and made it easy to work on the timeline with resolution less than 1 ms, this sensitivity to the circumstances was optimal. For each of the stimuli, we used a sound analysis that included the determination of the intensity of the auditory stimulus (expressed in dBFS units, the level of 0 dBFS corresponds to the maximum intensity), analysis of the frequency of the signal (frequency analysis) and its complete spectrum (spectrogram). Frequency analyzes and spectrograms do not, of course, be of primary importance in terms of the purpose of the work, but we consider their inclusion to be important in the complex processing of the given topic. The average response time of all subjects at zero alcohol level was 395.27 ms (ı = 113.37). This value represents the mean of all values without resolution. For a unimodal audible stimulus of 0 dBFS, the average of all subjects was 342.65 ms.

Figure 2-20. Dependence of reaction time on the intensity of the auditory stimulus.

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The nature of the dependence is obvious - faster reaction times reach the subject if the stimulus gets higher and vice versa. Of course, the curve created from our measured values does not apply to stimuli that have not reached such intensity that they are detected. Such incentives did not occur in our experiment. It is obvious from the very essence that the value of the reaction time would not increase, respectively. did not diminish indefinitely, if the theoretical impetus was infinitely small, respectively of great intensity. In the graph, such a circumstance would be represented by asymptotes, each of which would be parallel to the corresponding axis.

Figure 2-21. Reaction time dependencies on alcohol level - maximum alcohol level 0.6 ‰

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Figure 2-22. Reaction time dependencies on alcohol level - maximum alcohol level of 1.2 ‰ (Straus, Danko 2009).

Analysis of the experimentally determined values indicates the excitatory effect of alcohol for very low blood alcohol levels, namely, 0.17- 0.23 g/kg. The experiments and measurements carried out were performed only on a group of men, so it is not possible to say with certainty what specific values of reaction time women would achieve. Pilot research has shown the need to make the following steps in follow-up research: Performing multiple measurements from frontal positions vis-à-vis the subject, both with visual and audio-visual suggestions. The aim of this experiment would be to complete a set of reaction times for audio, visual, and audiovisual stimulation, which are the most important insights for expert research in the field. In all relevant areas of investigation, to obtain additional data to help clarify the dependencies and to clarify the reaction potential of the common population as well as to show the physiological boundary capabilities of highly trained subjects. To determine the effect of alcohol, it is necessary to further determine the influence of alcohol on the reaction capacity of other components of the reaction capability. In addition, we consider it necessary to obtain reaction time values for higher levels of alcohol in the blood than approx. 0.6 ‰. The hypothesis of reaction rate conditionality in response to stimulus intensity was confirmed. It was the only audio stimulus so far, but the sound stimulus responds most quickly to these types of stimuli. This has resulted in the best possible average response times, which will then be helpful in analyzing audiovisual stimulus responses, respectively. in the

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overall comprehensive assessment of human responsiveness from normal population or trained people. Analysis of the effect of distraction on readiness and thus on the value of the reaction time confirmed the expectations and formed a determinant of a non-negligible character. The data obtained again provide a solid basis for examining the readiness and its impact on reaction time. Interesting results have been obtained by analyzing the influence of alcohol on the reaction rate, where subjects were even excited on average at low levels, specifically at the level of alcohol in the blood, approx. 0.08 ‰. Subsequently, a negative determination occurred at approx. 0.4 ‰ and relatively high values. Analysis of the experimentally determined values indicates the excitatory effect of alcohol for very low levels of alcohol in the blood, for 0.17- 0.23 g/kg. The graphs in Figures 221, 2-22 show quite accurately the prediction of the reaction time for a random stimulus requiring a complex motor response depending on the level of alcohol in the blood.

Task - Measure the conduction velocity of the nerve excitation! Measure your reaction time to an optical or sound stimulus (we assume that the excitement reaches the brain immediately). Then measure the speed of your response to touch the end of the foot or hand. We assume that the length of the nerve transmitting this signal to the brain is small. The procedure for measuring the reaction time can be realized as follows: a) the man A holds a ruler between the fingers of man B, suddenly releases him and B as soon as he sees a falling ruler, catch him. From the length in which B catches the ruler and the free fall time, the ଶ௟

reaction time is determined ‫ = ݐ‬ට . ௚

b) Let something randomly appear on the computer monitor (change the background color, start a sound). At the moment when we register the signal, press the bottom keyboard (mouse). Let the computer record the reaction time. One can use the program http://www.happyhub.com/network/reflex/ We can use a ruler with a small spike or a piece of paper. We close our eyes and register a touch stimulus. The reaction time tn is comparable to the reaction time t when we were looking at the ruler. The time tn includes the

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time from the beginning of the ruler´s fall until the brain realizes that the ruler touched his hand. On t , on the other hand, the time of the beginning of the ruler´s falls to the moment when the brain realizes that the eye has seen the falling ruler. Therefore, it is an incorrect assumption that the optical (or sound) stimulus arrives in the brain immediately. The time it takes the stimulus to arrive in the brain can be estimated from the fact that the sampling frequency of the eye is about 20 Hz, so the sampling time is about 0.05 s.

CHAPTER THREE DACTYLOSCOPY

Introduction Dactyloscopy is one of the oldest identification methods of criminology, which deals with the identification of persons. The possibilities of identifying persons based on dactyloscopic principles were already known to ancient human culture, such as Chinese culture. It is known from history that already in ancient Chinese cultures various documents issued by the monarchs were provided with the fingerprint of the monarch's papillary finger lines to prove their authenticity. From the point of view of the history of criminology, dactyloscopy is the second forensic technical method used in practice to identify persons (the first method of identification was identification according to 11 anthropometric dimensions of the human body, co called bertillonage, introduced by Alphons Bertillon). Jan Evangelista PurkynČ, Czech scholar and physiologist, is an important Czech personality who is associated with the emergence of forensic dactyloscopy and its use for the identification of persons according to fingerprints. He studied the structure of human skin, its function, and shape characteristics of papillary lines. In most textbooks and professional forensic publications, he is mentioned as one of the founders of applications of dactyloscopy in forensics. A work of fundamental importance in the history of dactyloscopy appeared in a wide range of PurkynČ's works, and in 1823 he wrote "Comentatio de examine physiologico organi visus et systematis cutanei" (Debate on the Physiological Examination of the Organ of the Visual and Skin System). The credit of J. E. PurkynČ lies in the fact that he was the first to describe the basic patterns of papillary lines on the last sections of his fingers and to suggest their classification. His research was motivated solely by biological interests and he did not think about using papillary line drawing to identify people.

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Dactyloscopy of criminals has been carried out in our country since 1903, and since 1908 only dactyloscopic cards have been made for forensic identification purposes, bertillonage has been abolished. Dactyloscopy is a branch of forensic technology that examines the patterns of papillary lines on the inside of the last parts of the fingers, palms and toes and feet in terms of the laws of their origin, search, securing and research to identify a person. The importance of dactyloscopy lies in the fact that it makes it possible to identify a specific person under optimal conditions. By dactyloscopic traces, we mean all fingerprints and imprints of fingers, palms, and bare feet that reflect information about the surface structure reflecting parts of the human body and were caused by activities having a causal, local, temporal and other relationship to the clarified event. Papillary lines create complex and inherently unique patterns, the purpose of which has not yet been clearly elucidated. It has been clearly shown to be related to the sensitivity of the skin and its tactile properties. The papillary lines form continuously elevated reliefs, the height of which is 0.1 - 0.3 mm and the width of 0.3 - 0.6 mm. By crossing them, changing direction, branching, etc., various shapes are created in summary. The origin and existence of papillary line patterns are governed by the following generally accepted laws: 1) relative indelibility of papillary lines; 2) relative invariance of papillary lines; 3) relative individuality of papillary line drawing. Dactyloscopic traces are created in principle by a simple mechanism at the moment of direct action - at the moment of contact of two objects - a human and an object, which is able to accept the dactyloscopic trace and preserve the reflection of papillary lines for a certain time. The formation of one's own dactyloscopic trace can occur in several ways: 1) A mirror-inverted relief of the surface structure of the papillary lines is created, i.e., a volumetric (3D) dactyloscopic trace. This type of trace arises if the object that receives the trace is capable of plastic deformation (plasticine, wax, chocolate, cheese). Under suitable conditions (as long as the temperature does not exceed the melting point of the substance), the trace is preserved. The relief of the surface structure of the papillary lines is mirror inverted.

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2) The dactyloscopic trace is created by transferring the substance from the surface of the object to the papillary lines, thus disrupting the surface structure of the trace carrier. In places corresponding to the inter-papillary spaces, the original surface remains intact. The mechanism of these traces can be different: x Moisture dissolves a small amount of the substance, and the resulting solution has the ability to adhere to the top of the papillary lines - for example, water-soluble adhesives on postage stamps. x Substances that have their own adhesive properties adhere to the top (ridges) of the papillary lines - for example, fresh paint, dyes, glues, blood, etc. x At the top (ridges) of the papillary lines, a microscopic amount of a substance forming a continuous surface adheres to another, usually smooth object - for example, a fine layer of dust on furniture. This type of dactyloscopic trace usually occurs when the papillary lines of the fingers and palms are covered with a layer of sweat, which itself has adhesive properties. Because this mechanism of formation of this type of dactyloscopic trace always transfers a certain amount of substance from the continuous surface of the carrier to the papillary lines of the object forming the trace, they are referred to as layered traces. 3) The dactyloscopic trace is created by transferring a substance located on the surface of the papillary lines to a suitable carrier. The dactyloscopic traces formed in this way are referred to as layered dactyloscopic traces. Thus, the substance that previously adhered to them from the top of the papillary lines is transferred to various objects - for example, paint, blood, writing instruments, and dust. Depending on whether the resulting dactyloscopic traces are observable to the naked eye or not, they are divided into visible and invisible (latent). The vast majority of latent tracks are sweat, but this may not always be the rule. 4) Latent dactyloscopic traces are formed by sweat (weight approx. 10 micrograms, thickness 0.1 micrometer). Sweat is made up of more than 99 % water (some literature sources give somewhat different data - the fact will probably be due to a particular human individual), followed by lactic acid, sodium chloride, urea and amino acids. Lipids (fats), such as squalene, fatty acids, and mono-, di- and triglycerides, are also present. The composition of sweat is different in children and adults, in men and women, it is also affected by the type of food, health, and medication. In addition to time, the number of excreted substances in the trace is also due to the properties of

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the carrier on which the dactyloscopic trace is formed (to date, however, it has not been explained whether it is a physical or chemical process). The term latent trace is usually referred to as a trace formed by sweat, but it should be noted that daily contact with many objects of different nature can transmit to the fingers other chemicals (grease, cosmetics, etc.), which may affect the final composition, and thus possibly complicate the visibility of the dactyloscopic trace. The durability of latent dactyloscopic traces depends on several factors. Temperature, humidity, time from the formation of the dactyloscopic trace to secure solar radiation, etc. have a significant effect. Despite the above factors, it is not easy to determine the time after which the track would disappear. According to the literature, the oldest visible latent trace was 42 years old (visualized by ninhydrin). In practice, it is not a problem to make traces on the paper visible with ninhydrin several years old.

3.1 Searching, visibility and securing dactyloscopic traces Forensic technology has developed several methods that are used in the visibility of latent dactyloscopic traces, securing and documentation. When choosing a method, it is necessary to consider the type of track, the quality and character of its carrier the expected age of the track, and other factors. The choice of the method is based on the knowledge and experience of a police officer. The following methods can be used in principle: 1) Physical methods (argentorate, carborafin, iron filing, bronze powder, camphor, carbon black, graphite, fluorescent powder, magnetic powder, crystal violet, gentian violet, Sudan black). They are based on the principle of different adhesion of the individual components of sweat to solid, finely ground, and water-insoluble particles of the substance-induced substance. The adhesion of these particles decreases with the time (age) of the dactyloscopic trace, so their use is more suitable for relatively fresh dactyloscopic traces. There are currently a large number (several hundred) of dactyloscopic powders. The most used powder is argentorate, it is a silver-gray powder, a finely ground aluminum, which is applied with a fine brush (usually from hair or various animal hairs - currently glass fiber brushes are used, especially for their longevity), the imprint of which is visible in it has a silver-gray color.

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Good results are achieved especially on objects with a smooth and shiny surface, on glass, painted objects, etc. A visible print is provided on a black dactyloscopic foil. Not suitable for visualizing dactyloscopic traces on paper carriers. In some cases, other powders in large quantities are used to make dactyloscopic traces visible, for example, graphite (finely ground graphite), carborafin (finely ground charcoal), ultramarine, cinnabar. Reliable visibility of latent dactyloscopic traces on paper carriers is possible with the help of ferromagnetic powder (finely ground iron filings). The powder is applied with a so-called magnetic brush, which is a permanent magnet placed in a plastic case. Under favorable circumstances, dactyloscopic traces can also be developed and secured on textiles, using a fabric, which is a dark brown to black mixture of several substances. It can be used to develop dactyloscopic traces on fabrics with a smooth surface, such as nylon, damask, poplin, silk, etc. In contrast, with long pile fabrics or woven from thicker fibers with an uneven, rough surface, dactyloscopic traces cannot be provided by the fabric at all. The development is carried out by pouring the fabric on the fabric at the place of the presumed dactyloscopic imprint, the excess powder being poured onto the fabric by gentle, careful tapping of the fabric. The fabric adheres to the fabric in those places where sweat remains on the papillary lines when touched with a finger or palm. The traces thus developed are secured with a transparent dactyloscopic foil. They can also be documented photographically. The fabric does not damage the fabric to which it was applied and is easily dusted after securing the impression. It has not been used in practice in recent years. New agents used to induce dactyloscopic traces include higher intensity powders, either white or black. The powders show low adhesion to the track carrier. With these powders, traces of plastics, painted objects, weapons (white powder), hardened paper, etc. can be secured without any problems. Some mixtures were experimentally tested for visualization of latent dactyloscopic traces, which have elapsed since the creation of a long time. The most suitable are:

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x x x x x

ZnO (19 parts) + rosin (1 part), CuO (19 parts) + rosin (1 part), sodium salicylate (1 part) + starch (10 parts), carbon black (10 parts) + rosin (1 part), bismuth nitrate Bi (NO3) 3. 5 H2O (10 parts) + starch (1 part).

By using these mixtures, it is possible to make visible and ensure dactyloscopic traces up to 14 days old, even if they are exposed to weather conditions. In cases where the color of the developing powder coincides with the color of the trace carrier or in some cases conventional powders (e.g., argentorate) cannot be used, bronze powders can be used successfully, which, among other things, dust from traces and do not contaminate objects, such as argentorate. The use of bronze powders is recommended to make dactyloscopic traces visible, e.g., on formica, hard PVC, painted metals, etc. In recent years, many fluorescent powders have been produced and used, which fluoresce when illuminated with a suitable source and can also be used for visibility on light-reflecting surfaces, which causes problems in photographic security. Magnetic powders are composed of iron dust, sometimes mixed with copper or aluminum (discoloration) or also contain fluorescent dyes. They are applied with a magnetic brush, which prevents so-called brushing - insensitive application of classical powders with a brush and can cause smearing, in extreme cases even erasure of the impression. The use of carbon black burning camphor can also be included among the physical methods. The visibility of the trace is carried out by igniting the camphor and placing a dactyloscopic trace carrier over its flame (the flue gases contain a large proportion of very fine soot). After careful removal of excess soot (weakly flowing stream of water, pulp), a visible dactyloscopic trace in deep black can be observed. The use of camphor carbon black is especially suitable for the visibility of dactyloscopic traces on metal carriers (e.g., knives, scissors, firearms). 2) Chemical methods (ninhydrin, silver nitrate, 1,8-diazo-9-fluorene [DFO], osmium oxide, and ruthenium oxide). These methods are based on a chemical reaction between a sweat component and a chemical to form a colored compound. They are mainly used for developing dactyloscopic traces on paper carriers.

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3) Physic-chemical methods (iodine vapors, cyanoacrylate, laser methods). They are based on the adhesion of chemical compounds in places where the dactyloscopic trace is located. 4) Special methods (neutronography, autoradiography, 8hydroxychynoline, autoelectronography) that use special techniques or special procedures, such as X-rays, neutron flux, etc.

3.2 Examination of dactyloscopic traces Dactyloscopy allows in particular:

x identification of persons according to traces or footprints left (created) at the place of forensically relevant events;

x identification of corpses of unknown identity; x identification of persons who do not want or cannot prove their identity; x finding out whether the dactyloscopic trace has been created by a person who has already committed as yet unexplained criminal offenses; x In some cases, decide which finger or part of the skin covered by the papillary lines created the dactyloscopic trace. In practice, the most common comparisons are:

x Traces and prints obtained at the crime scene (criminally relevant events) with control (comparative) prints of selected, suspicious or domestic persons (a person who moved to the spot within another activity - apartment user, an employee of an organization, etc., but did not have a share of investigated events - these prints and comparative prints of these persons may not be used except in the given case and are destroyed after use); x traces obtained at the crime scene with comparative fingerprints of persons stored in dactyloscopic registrations; x Fingerprints of persons of unknown identity and corpses with fingerprints in registration. The actual dactyloscopic identification is performed based on the evaluation of dactyloscopic markers (individual features). The individuality of papillary line patterns is based on many markers that occur randomly and

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independently and ultimately form a unique structure. The individual markers, such as beginning, end, eyelet, bridge, forks, crossing, and others, differ from each other not only in the geometric shape, but also in the frequency of occurrence, which can also be used for further conclusions. The identification value of a sign (dactyloscopic marker) can be calculated according to the relation: I = - log n, where I - character identification value n - frequency of occurrence of the character on the area of 1 mm2. The significance of determining the identification value of individual dactyloscopic features lies in the fact that on their basis it is possible to determine the minimum number of features necessary to express a reliable categorical conclusion about the identity of the object. It is not the simple number of identical characters found that should be decisive for identification, but the total sum of their identification values. In the criminalistic practice of different countries, there are different opinions on this issue, and the required number of identification marks also differs. The number of dactyloscopic markers necessary for individual human identification in the Czech Republic is set at 10 markers or more. At present, the universally required number of dactyloscopic markers necessary for individual human identification has not been and has never been determined worldwide. In domestic forensic practice, a dactyloscopic trace with enough markers sufficient for individual human identification (i.e., a usable trace) is considered a dactyloscopic trace containing more than 10 markers. The partially usable track shows 7 to 9 markers. The unusable track then shows 6 or fewer markers. The division of dactyloscopic traces into partially usable and unusable is traditional in the domestic professional literature, but not very suitable. Both groups of dactyloscopic traces have a common forensic tactical use - they make it possible to exclude some of the suspects, but they do not allow their individual identification. The numbers of markers required for individual identification in other states are listed in the following table. Currently, the world literature discusses whether the condition of a minimum number of markers is necessary, or an individual evaluation of each case, where the frequency of individual markers can be considered. In addition to the use of dactyloscopic markers, the literature also mentions the use of the appearance and location of skin pores for identification

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purposes, especially by measuring their position, relative distances, size and more. This method is called poroscopy, it is almost not used in practice and in fact lies on the border of dactyloscopy. Comparative prints of the last links of the fingers, palm, or feet and toes are most often obtained with the help of dactyloscopic black, which is basically printed black. This black is applied in a thin layer on a solid and flat surface (glass, metal) and parts of the skin covered with papillary lines are pressed on this surface. For these purposes, it is also possible to use factory-produced plastic foils, on which the optimal layer of dactyloscopic black is already applied. The parts of the skin thus blackened are transferred to the marked part of the dactyloscopic card, in the case of the last finger joint by a rolling movement, in other cases by simple pressing. Before the actual dactyloscopy, the scanned part of the skin must be thoroughly washed, and the same is necessary even after dactyloscopy, because black adheres relatively firmly to the skin. Furthermore, the so-called "pure" dactyloscopy is used, which does not use dactyloscopic black, but a special, so-called wax paste, which is very lightly smeared on the pad and the impression is transferred to a specially treated paper. The chemical reaction between the paste and the paper components produces black prints. This method has been used and continues to be used in various modifications to obtain the prints of "domestic" persons. Comparative corpse prints are taken analogously as in the case of living persons. Again, dactyloscopic black is most often used, variously shaped dactyloscopic spoons are used, which enable a better fingerprinting with the help of an inserted piece of paper. If the corpse's skin is wrinkled, a suitable liquid can be injected under it with a syringe to turn it off. For some older corpses, it is more advantageous to dissect a part of the skin with papillary lines and obtain prints from them, or to photograph this skin. The process of dactyloscopic identification consists of three stages, which can be characterized as follows: x In the first stage, the expert examines the objects of identification in terms of their suitability for identification research. x In the second stage of dactyloscopic identification, the expert performs his own comparative examination, the principle of which is the evaluation of individual dactyloscopic identification features contained in both trace and comparative materials. The results are compared and conclusions are drawn about the similarity or

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difference of the examined (assessed) objects. At the same time, a reasoned explanation of the differences found in the ruling of one of the types of categories of courts is stated. x In the third stage, based on the quality and quantity of dactyloscopic features, the expert decides on the conformity or differences of the examined objects. Using formal and dialectical logic, he can pronounce four types of categorical judgments: - a categorically positive court (the trace from the crime scene and the comparative fingerprint were created by one person); - categorically negative court (the track and the comparative fingerprint were created by two persons); - partially positive and a partially negative court (differences were found in the footprint and the comparison sample, which the expert must explain before deciding on the overall conclusion of the investigation). In the past, there were (so far exist in some countries) dactyloscopic collections, in which the comparative fingerprints of the last fingerprints on the hands were classified according to established criteria and divided into groups and subgroups within the records. These records were used in the manual search for similar registered fingerprints according to dactyloscopically classifiable traces. Such systems were very diverse, common were singlefinger systems (monodactyloscopic), which allowed the identification of a person (especially the perpetrator) according to a single trace created by the last finger joint, and multi-finger systems (most often decadactyloscopic), which were used mainly to identify unknown persons. and corpses. These manual systems gradually became practically unusable due to the large number of registered persons, the evaluation time of individual dactyloscopic traces was unbearably long. Another disadvantage was the subjective evaluation, which to a large extent contributed to the low yield of these collections, and that some prints (literature reported up to tens of percent) could not be unambiguously classified according to the criteria and all had to be searched. In countries with large dactyloscopic records, the use of punched labels facilitated the search. With the further development of computer technology, automated dactyloscopic systems were constructed, which enable a very fast search for the most similar registered fingerprints on a trace secured at the crime scene. The main advantage of these systems is speed. Individual identification for the purposes of evidence is always a matter for the expert, computer systems do not have the primary role of identification, but elimination, is to offer the operator some of the most probable fingerprints, in which the same is then

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searched. In 1975, the use of the Automated Fingerprint Identification System (AFIS) began in the USA. At the end of the eighties, the first experiments with computer technology appeared in this area and in the country. In 1994, AFIS was installed at the Criminalistics Institute in Prague. It is currently saturated with more than 600,000 cards.

Other questions related to dactyloscopy Although dactyloscopy has already celebrated 100 years of success, some questions remain unanswered:

x Problems of visualization of latent dactyloscopic traces on the skin

x

x x

x x x x

of living persons or corpses - it has been experimentally proven that it is possible to visualize latent dactyloscopic traces on the skin of corpses and living persons, unfortunately to date there is no universally acceptable method. Composition of chemicals suitable for later DNA analyzes advances in forensic genetics allow the use of fewer and fewer biological materials to identify people by DNA, unfortunately most chemicals for visualizing latent fingerprints cause denaturation of its material nature and thus prevent later DNA analysis. Systematic testing of new methods for visualization of latent dactyloscopic traces. Characterization of the support - currently there is no simple method that would allow to characterize the surface properties of the support and its porosity, which would allow a better estimation of a suitable means of visibility. Sweat composition. Age of dactyloscopic traces - the moment of origin of the dactyloscopic trace has its tactical value, unfortunately currently there are no known methods that would allow this finding. Sequence of origin - a method that allows to distinguish (possibly gradually make visible) several overlapping tracks could provide additional information. Counterfeiting of dactyloscopic traces and their detection - at present there is no consensus on the risk of transmission of latent dactyloscopic traces or their creation (for example by means of castings) and the possibility of their detection. This issue will become increasingly topical, especially in connection with the increasing penetration of biometric methods into all areas

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(identification of persons, securing classified information and electronic transactions, etc.). x Number of markers necessary for individual identification - issues addressed especially in the USA for the last thirty years. In 1973, after three years of research, the International Association for Identification (IAI) stated: "There is no yardstick for deciding on the minimum number of markers that must be present on two fingerprints in order to establish a match - necessary to establish individual identification." The final decision is therefore up to the person making the comparison. This issue was reopened at an international conference in Israel (1995), known as the "Ne'urim Declaration", which merely repeated the IAI's 1973 conclusions. However, the question of uniqueness is far from over; most recently in 1999 (United States, v. Byron C. Mitchell, Pennsylvania). In this case, the defense stated that dactyloscopic traces had never been demonstrated to be unique in the sense of the definition of "Daubert" (the Daubert case - a decision of the US Supreme Court on the admissibility of forensic methods). After four days of hearing the defense and government experts, the court ruled that the fingerprints were unique and permanent.

Figure 3-1. Comparison of the secured track and the captured fingerprint

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Figure 3-2. Ridge characteristics (https://mrsblackmonsscienceblackboard.weebly.com/fingerprints.html)

Figure 3-3. Visibility of the dactyloscopic trace with finely ground ferric oxide

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Figure 3-4. Visibility of the dactyloscopic trace by reaction with ninhydride and by iodine vapor

Figure 3-5. Papilary lines

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3.3 Physics behind dactyloscopy 3.3.1 Optics A compound microscope has two (or more) lenses: the ocular lens (eyepiece) atop the body tube and the objective lens at the bottom of the tube. The magnification can be changed by using a different objective lens. A compound microscope

Figure 3-6. A compound microscope

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The objective has a short focal length, it is placed close to the object being examined. The objective lens produces a real, inverted image of the object 0 at I1. This is then viewed by the eye lens, and this gives a final virtual image at I. The magnifying power M of the instrument is given by the formula: Magnifying power M = ቀ

஽ ௙೐

௩

– 1 ቁ ቀ െ 1ቁ ௙೚

where fe is the focal length of the eye lens, fo that of the objective lens and v the distance of I1 from the objective lens.

Figure 3-7. Running rays (https://www.schoolphysics.co.uk/age1619/Optics/Optical%20instruments/text/Microscope_/index.html)

The magnifying power can be calculated also by the formula

M = mo x me where mo and me are the magnifications of the objective lens, and eyepiece, lens respectively.

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When viewing with two eyes, a single objective is employed in a binocular tube fitted with a matched pair of eyepieces. A beam splitting prism is used to divide the light beam from the image formed by the objective to each eye. When using the microscope, there are some geometric limitations resulting from the laws of geometric optics. 1. Resolution – the smallest distance at which two points can be seen as separate when viewed through the microscope 2. Resolving power of the microscope = O /2 n sin D Comparative microscope see Fig. 3-8:

Figure 3-8. Comparative microscope for forensics (http://www.mikro.cz/forenznikomparacni-mikroskopy)

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Figure 3-9. Ken – a – vision comparative microscope

A comparison microscope is an instrument used to compare two samples. The comparison microscope consists of two microscopes connected by an optical bridge, so the image in the eyepiece is divided into two parts, where you can see two separate objects that can be compared. Comparing images is simplified and the observer does not have to rely on his memory.

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Figure 3-10. Comparison microscope – image Tamasflex. Comparison microscope. Wikipedia: the free encyclopedia [online]. San Francisco (CA): Wikimedia Foundation, 2010 [cit. 2015-07-15]. https://upload.wikimedia.org/wikipedia/commons/5/5d/ComparisonMicroscope.pn g?uselang=cs

The inventor of the comparative microscope was the chemist Philip O. Gravelle, who was involved in the identification of the bullets. He worked with Calvin Gohhard. The T-19241C-230 comparison microscope is frequently used in forensic science. The microscope has two types of light according to its intensity. The more intensive can be used for microorganism imaging. I tis possible to display one field separately or both fields at once. The magnification of the microscope is 20-400times.

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3.3.2 Molecular physics – adhesion and cohesion Adhesion arises from the action of attractive forces that act between the particles located in the surface layer of various substances. If the two substances are water and the solid that the water wets, it is the so-called wettability (adhesion of the liquid). The force of wettability is the adhesive force. The adhesive force arises from the contact surface and the unevenness’s of both materials. A unique theory of adhesion does not exist. Basic theories or mechanisms of adhesion can be divided into the following groups: x Mechanical adhesion (e.g., Velcro, textile adhesives, sewing – the basis, blocking of surfaces, when cavities and gaps are filled when approaching surfaces) x Chemical adhesion (mediated by chemical bonding – covalent, ionic, or hydrogen bonding. For the formation of bonding, the two surfaces must be very close to each other (one nanometer), the bond is brittle. x Dispersion (van der Waals bonding) x Electrostatic adhesion (by the action of electrostatic forces touching the surface creates an analogy of a capacitor) x Diffusion (the connection is due to the diffusion of both materials, e.g. polymers) Practical example: Measurement of adhesive forces between a glass plate and water molecules in a surface layer: A glass plate is hung on a load cell and is placed on the water surface (the water is in a baker). Using the load cell, we are lifting the plate vertically upwards. By applying a force that is greater than the gravitational force of the plate, it is possible to “tear” the plate from the liquid surface. The underside of the plate is wet, it means that the water has turned off from the water and not from the glass.

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Figure 3-11. Measurement of adhesion forces http://pokusy.upol.cz/skolnipokusy/molekulova-fyzika-a-termika/kineticka-teorie-latek/prilnavost-27/

In practice, the problem of adhesion between car tires and road surface (or locomotive wheels and rails), is solved, so as in medicine, dentistry (denture attachment), construction, etc. Road grip depends on the quality of its surface and the surface of the tires. The highest adhesion occurs in the case of dry roads, the coefficient of adhesion is 0.6 – 1. Adhesion is affected by the temperature and structure of the tire surface. The coefficient of adhesion can then be more than doubled (see Formula 1 races – the warm-up lap). Cohesion arises from the existence of attractive forces (so-called cohesive forces) by which the particles of a substance act on each other. Cohesion occurs only in solids and liquids. All molecules in the surface layer with a depth of 10-9 m act on the whole liquid with a pressure called cohesive pressure. For water, the magnitude of the cohesive pressure is of the order of 109 Pa. Compared to the external

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pressure that can be applied to the liquid, this pressure is many times higher. This is also the explanation for the low compressibility of liquids. Surface layer of a liquid Molecular forces perpendicular to the surface of the liquid cause cohesive pressure. Molecular forces parallel to the surface of the liquid cause the liquid to try to change its surface – the so-called surface tension. Surface tension is expressed by the force acting in the direction of the tangent to the surface per length unit.

Figure 3-12. Molecular forces ௗி

Surface tension ߪ = , where dF is the increase in cohesive force acting ௗ௟ in the direction of the tangent to the liquid surface, dl is the length element. The unit of surface tension is N˜m-1. Let us study the contact of a liquid with a solid wall. There is an interaction of the attractive force between the liquid molecules (cohesive force) and between the surface molecules of the liquid and the wall (adhesion), which represents the mutual adhesion of two different substances (their surfaces). Two cases can occur: 1. Non-sticking liquid – adhesion is less than cohesion – the liquid does not wet the walls of the container, we observe the phenomenon depression 2. Adhering, wetting liquid – adhesion is greater than cohesion, we observe the phenomenon elevation.

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The surface tension decreases with temperature. Phenomena at the interface of three surfaces The liquid in the container represents the contact of three surfaces – the liquid, the walls of the container (solid), and the air (or steam). At a point A on the vessel wall, three surface layers meet: a liquid-solid layer, an air-solid layer, and an air-liquid layer. This corresponds to three surface tensions: V12 between air and liquid, V13 between air and solid, V23 between liquid and solid. In general, these surface tensions are not in equilibrium. At the wall of the vessel, the molecules can only move along this wall, so the equilibrium occurs when the components of surface tension parallel to the wall are compensated for each other. The following cases occur:

Figure 3-12. Surface tension – forces https://physics.mff.cuni.cz/kfpp/skripta/kurz_fyziky_pro_DS/display.php/molekul/ 9_4

a) If V13 ! V23, the molecules of the liquid move upwards along the wall of the container and the surface of the liquid forms an angle G, the so-called extreme angle The equilibrium requirement

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ߪଵଷ = ߪଶଷ + ߪଵଶ cos ߜ For the value of the extreme angle 0  G  S/2 the unevenness characterizes the case of perfect wetting of a solid body by a liquid and the elevation occurs. E.g., perfectly wets the clean glass surface with water or alcohol.

Figure 3-13. Liquid wetting the solid body

b) If V23 ! V13, the molecules move down along the wall of the container. The liquid forms an obtuse angle with the wall of the container and for the value S/2  G  S there is a case when the liquid does not wet the solid body – depression (e.g., glass-mercury).

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Figure 3-14. A liquid that does not wet the solid body

In the case of a drop of liquid on the surface of a solid body, a similar situation occurs.

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Figure 3-15. A drop of liquid

The equilibrium is ߪଵଷ = ߪଶଷ + ߪଵଶ cos ߜ. If V13 ! V23 + V12 (G = 0), the droplet spreads over the surface of the solid until a monomolecular layer is formed. I tis a perfect wetting of the solid body with a liquid. If the angle G is sharp, then an imperfect wetting occurs; if the angle G is obtuse, tension V23 and the projection of tension V12 cosG tend to give the drop a spherical shape. For the equilibrium is V13  V23+ V12 cosG . An extreme case can occur for small mercury droplets, when G = S and cos S = -1. A drop of liquid may be found on the surface of another liquid if the liquids do not mix with each other (e.g., a drop of oil with water).

Figure 3-16. Surface tension of a liquid drop on another liquid

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The component V13 tries to expand the drop and reduce the surface of the liquid (3), on the contrary components V12, V23 try to withdraw the drop and reduce its total surface. In order for the drop to maintain its shape, the requirement must be met

V13 = V12 + V23 If V13 ! V12 + V23 the droplet spreads over the surface of the liquid and forms a thin or monomolecular layer on the surface of the liquid (3). Capillarity Very closely related to the wettability are capillary phenomena. If we immerse a tube with a very small cross-section (capillary) in a liquid, then when the liquid comes into contact with a solid body, the surface of the liquid curves. A meniscus is formed in the capillary, which can be considered as part of a spherical surface with a radius R. When the capillary is immersed in a liquid that does not wet the walls of the capillary, a drop of liquid occurs, and the meniscus is convex.

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Figure 3-17. Capillary elevation and depression

When the liquid wets the surface of the capillary, the liquid in the capillary rises to a certain height and the meniscus is hollow. Due to the capillary pressure when the surface of the liquid is curved, two cases can occur:

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135

In the case of a non-wetting liquid, the capillary pressure is positive and causes an increase in the pressure in the liquid. The liquid in the capillary drops below the level of the surrounding liquid. The so-called capillary depression occurs. In the case of a wetting liquid, the capillary pressure is negative and causes a decrease in the pressure in the liquid. The liquid in the capillary rises higher than the level of the surrounding liquid and the socalled capillary elevation occurs. The liquid in the capillary rises to a height h at which the hydrostatic pressure of the liquid column is in equilibrium with the capillary pressure. The following applies 2ߪ = ݄ߩ݃ ܴ where U is the density of the liquid, R is the radius of curvature of the meniscus, g is the gravitational acceleration. From here, the height of the liquid column in the capillary can be expressed as

݄=

ଶఙ ோఘ௚

.

௥

From Fig. 3-17follows ܴ = and therefore for the height of the ascent ୡ୭ୱ ణ we can write ݄=

ଶఙ ୡ୭ୱ ణ ௥ఘ௚

.

It is obvious that the height of the liquid ascent in the capillary is inversely proportional to the radius of the capillary r. However, since the surface tension of the liquid depends on the temperature and decreases faster than the density of the liquid with increasing temperature, it also the capillary elevation, resp. depression decreases depending on the temperature. Capillarity plays an important role in nature science and technology (penetration of water into the roots of plants, separation of ore from rock, absorption of sweat by functional laundry, etc.) .

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Figure 3-18. Adhesion of dactyloscopic powder to a dactyloscopic brush

Figure 3-19. Adhesion of dactyloscopic powder to the solid surface

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Figure 3-20. Adhesion of dactyloscopic powder to a surface with a dactyloscopic trace

CHAPTER FOUR FORENSIC BALLISTICS

4.1 Forensic ballistics as a scientific discipline Forensic ballistics is a branch of science that studies the movement of a projectile in a weapon and the further movement of the projectile outside the weapon, until hitting the target. In addition, it deals with some other issues, in particular the assessment of weapons and ammunition in relation to their functionality and the investigation of postfire fumes. In solving its specific questions, forensic ballistics applies and creatively reworks the knowledge of military technical science in the issues of the mechanism of ballistic footprints, their form, and distribution. Of course, to cover all phenomena of criminalistic significance, criminological ballistics has required a substantial expansion compared to military ballistics, and at the same time the range of issues it examines has changed. Unlike military ballistics, which solves tasks related to the destruction of a given target, the accuracy of firing, and reliability of weapons, the main task of forensic ballistics is to identify the weapon according to the traces found in connection with a specific forensically relevant event. In addition, it also solves the issues of determining the shooter's location, shooting distance, weapon functionality, and other issues. Forensic ballistics is a scientific discipline that examines weapons, ammunition and its components, by-products of firing objects with traces of impact or effect of missiles, internal, transitional and external ballistics to determine group affiliation and perform individual identification under the current clarification of the causes and conditions under which the object was fired and damaged by gunfire. By forensic ballistics we mean: 1. The science of firearms of all kinds and types. 2. The science of ammunition and its components. 3. Identification of weapons by fired cartridge cases and projectiles and

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by-products of the shot. 4. The science of objects damaged by firing and the effects of missiles. 5. The science of internal, transitional, and external ballistics with application for forensic needs. The aim of professional research in the field of forensic ballistics is primarily to identify the weapon from which the projectile or cartridge was fired, to check the functionality of the weapon or ammunition, and to assess the distance and effects of firing on a particular object. The objects of forensic ballistics research are: x Especially small arms firearms of all kinds and types. To a much lesser extent, gas weapons and, exceptionally, mechanical weapons are the object of research. x Ammunition of all kinds and its individual components, cartridges, bullets, and cartridges as the main products of the shot, by-products of the shot as burned and unburned grains of gunpowder, smoke, flame and burn. x Items with traces of the impact and effect of bullets. One of the important fundamental issues that criminological ballistics deals with is the question of the mechanism of criminological traces during the actual course of the shot and after leaving the projectile from the weapon. From this point of view, ballistics is divided into: a) Internal ballistics - the doctrine of the laws by which the projectile is guided in its movement in the main weapon from the moment of initiation of the shot (ignition of the powder charge in the cartridge case or the provision of mechanical energy by another impulse) until the moment it leaves the barrel at a certain initial speed; internal ballistics is hardly concerned with mechanical weapons, b) Transitional ballistics - dealing with the study of the movement of the projectile immediately before the barrel, i.e. from the moment when the bottom of the projectile leaves the muzzle of the barrel and dust gases or air or carbon dioxide (used in gas weapons) still precede the projectile. This is a very limited section of the flight path of the projectile, its length is about 10-20% of the caliber of the weapon. The velocity of the projectile will increase by 1-2% for firearms at the end of transitional ballistics, and even less for other weapons, c) External ballistics - it is the part of ballistics that examines the movement of the projectile after leaving the barrel in the outer space.

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The result of solving the issue of external ballistics is the determination of the elements of the trajectory of a single or mass projectile (usually shots fired from a shotgun), especially its range and effective range, flight time, can’t and instantaneous speed and energy at certain points, d) Terminal ballistics - dealing with the study of movement and effects in the target (a special branch of terminal ballistics is wounded ballistics). Firearms are those weapons that are used to destroy a target at a distance by a projectile, which is set in motion by the immediate release of accumulated energy. The energy is used both to transport the projectile to the target and to destroy the target itself. According to the type of energy that was given to the projectile, we distinguish firearms into: 1. Mechanical weapons that usually use mechanical energy to fire (rubber, slingshot, bow, crossbow). 2. Gas weapons that use the pneumatic energy of air or other mechanically compressed gas to fire a projectile. The air can be precompressed in the pressure chamber or is compressed by a piston at the moment of firing (air rifle, gas gun, windbreakers). 3. Firearms in which the projectile is set in motion by the immediate release of the chemical energy of the gunpowder, or only match the composition itself. This is the most frequent type of weapon that forensic ballistics deals with. According to the controllability when shooting, all types of firearms can be divided into: 1. Small arms that can be operated with one or two hands. Small arms are divided into short and long. These weapons make up the dominant proportion of the weapons that forensic ballistics deals with. Mechanical weapons, as already mentioned, are of marginal importance. 2. Mounted weapons, which we hardly encounter in forensic ballistics (e.g., heavy machine guns, mortars, cannons, etc.). For practical reasons, firearms can be divided according to their purpose into: 1. Civil - hunting, sports, defense. 2. Military.

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3. Special - signal, insidious, alarm, home-made, industrial. However, this division is not of greater criminalistic significance - it is not decisive what kind of weapons, crime, or other forensically relevant events were committed. According to the bore, we can further divide small arms into weapons: 1. With a grooved or polygonal bore, which includes single-shot weapons. 2. With a smooth bore, which includes all weapons with mass bullet. 3. Combined, which are weapons having at least one barrel with a smooth bore and one barrel with a grooved bore (often hunting weapons). Firearms can be further divided according to the type of ammunition used into: 1. Firearms with a single projectile. 2. Firearms for mass bullet (shots). Firearms can also be divided according to the number of mains into: 1. Single barrel. 2. Multi-barrel. The degree of automation of the mechanism and construction of the weapon allows the division of firearms into: 1. 2. 3. 4.

One-shot. Repetitive. Self-charging (semi-automatic). Automatic.

It follows from the above division of firearms that a firearm is a modified object that allows the ignition and burning of a powder charge while generating a sufficient amount of gas, the pressure of which is used to transport the projectile to the target (and its subsequent destruction). One of the basic parts of any firearm is a barrel that meets the above conditions. It follows that from a forensic and technical point of view, a firearm does not always have to be only a classic weapon, but also only an adapted iron pipe, as is the case in some cases of home-made weapons, which can be

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encountered quite often in forensic ballistic research. As already mentioned, forensic ballistics, due to the nature of the use of weapons, mostly deals with the study of small arms. A single charge is used to fire current firearms. The single charge consists of a cartridge case, powder charge, projectile, and match. From a forensic technical point of view, under the firearm for small arms firearms, we mean rounds from the smallest caliber of 4 mm for target shooting to the most powerful rounds of 15.2 mm caliber intended for shooting from tropical rifles. The external dimensions of the shotgun charge correspond to the dimensions of the charge chamber. Depending on the application, it can be filled with either a mass or a single (but special) projectile. The cartridge case usually consists of a paper or plastic cartridge case, a metal bottom, and a powder quiver. The lower end of the cartridge case is closed by a bottom which is reinforced with metal fittings. A bed is created in the middle of the fitting for match setting. The caliber of the hub and the manufacturer's trademark are usually embossed on the lower part of the fitting. The dust quiver forms a stop for settling stoppers, supported by a dust disc and delimits the space for dust filling. For some species, a quiver is not used and the plug hits the dust directly. The bulk projectile consists of shots, which are made of almost pure metallurgical lead. To increase the hardness, antimony is added to the lead in an amount of 1 to 4%. Shots made of a lead-antimony alloy are called hard, shots made only of lead are called soft, and their main disadvantage is their high deformability. Single projectile round - used exclusively in weapons with a grooved bore. According to the location of the match composition, they are divided into: a) Ammunition with central ignition b) Ammunition with edge ignition c) Ammunition with needle (pin) ignition In the case of a cartridge with a central ignition, the bottom of the cartridge case is provided with a bed for a match, the cartridges with an edge ignition have a match composition located in their hollow edge. For a cartridge case with edge ignition, the diameter of the bottom is larger than the diameter of the cartridge case. The cartridge case with a central ignition has a circumferential groove (recess) at the bottom of the extractor claw. Needlefired cartridges are structurally obsolete and almost nonproduction. These

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ammunitions (unlike the two previous groups) must be placed in the weapon in only one possible position, otherwise they cannot be initiated. The shape of the cartridge case can be cylindrical, but less often conical, another shape is called bottled according to the fact that the cylindrical parts are provided with a neck. The cone to which the neck of the bottle cartridge is connected is called the abutment cone, the part between the abutment cone and the groove, or the edge is the shell of the cartridge case. In the case of cartridges without a neck, the casing of the cartridge case is the part between the groove or edge and its end, the so-called mouth. Bullets - at the time of using black gunpowder, the bullets were made of one piece of lead. The boreholes of the weapons were provided with a large number of grooves with considerable depth to ensure proper guidance of the soft lead bullet in the barrel (compact bullets). Smokeless powder allowed a higher initial velocity of the projectile than a less powerful black powder could give. The soft lead projectile did not withstand higher speeds, and therefore had to be fitted with a protective shell made of a stronger material, thus creating a shell projectile. The shell of the projectile is made of either brass, copper, and more recently deep-drawn steel. According to the construction, shell projectiles can be divided into all-shell, half-shell, fragmentation, and special. The all-shell projectile is covered over the entire surface with a shell. This gives the projectile considerable resistance to penetration by the opposing material of the target, whereby the projectile obtains very good penetrating effects. The half-shell projectile has the front part of the core exposed. This provides a prerequisite for easy deformation of the front of the projectile. The greater deformability of the projectile will cause a better transfer from energy of the projectile to the living target. The fragmentation projectiles have an even greater ability to deform, which have an expansion cavity formed in the front part, helping to deform (destroy) the projectile when hitting the target. The expansion cavity impairs the ballistic properties of the fragmentation projectile. After hitting the target, the projectile is destroyed and dismantled. Special missiles are used for various purposes and their construction is chosen due to the special use, they are, e.g., gas missiles, zinc missiles for slaughter, ammunition, missiles of military signal ammunition, some types of hunting ammunition (e.g., missiles for blasting harmful defense,

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explosive missiles), etc. The principle and essence of ballistic traces can be easily explained if we consider for this purpose the functional components of the weapon as general tools that, in contact with the material of the charge, create traces on its surface. The mechanism of formation of these traces is the same as for mechanoscopic traces. The principles for identifying weapons by cartridge and projectile are the same as the principles for identifying any instrument by its traces. In essence, it is a depiction of the typical and specific features of the individual functional components of the weapon in the relatively softer material of the cartridge case and projectile, with which they come into contact with each other during the shot. Properties of objects - individual parts of the weapon, which are displayed in its track and which are used for identification research, are called identification marks. Their essence lies in their originality, which is reflected in their rare occurrence in their specificity for a given part of the weapon in the fact that there is a possibility of finding and comparing them. Another feature of their essence is that they are relatively stable and can be used for comparative identification research even several years apart. The determination of group affiliation is performed in cases where the used weapon is not available. According to the cartridge case and projectiles secured at the crime scene, general features common to a particular type or group of weapons and typical of that type or group of weapons are examined and evaluated to facilitate the search for the weapon used in committing a crime. The basic method of weapon identification is to compare individual identification marks on cartridges from the crime scene with marks on cartridges obtained by experimental firing of a suspicious weapon. Individual identification is performed by microscopic comparison on a comparison microscope. The results of the research are documented photographically by macrophotography and microphotography. The identification of the weapon according to the projectiles fired (we mean a single projectile) is performed by mutual comparison of the identification features of the borehole, mainly on the projectile from the crime scene with the projectile experimentally fired from the suspect weapon. To obtain an experimental projectile with intact identification marks, it is fired into special projectile traps (in the country with a sufficiently thick

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layer of cotton wool, abroad also with water traps). The actual identification examination is performed by microscopic comparison, which compares the traces of the borehole fields, mainly on the projectiles fired. Another method of comparing two missiles is to compare photographs of optically developed missile shells, taken, for example, on a device such as a projectile or more modern on the already mentioned LUCIA system. The identification of the weapon according to the projectiles fired depends on the technical quality of the specific features of the field tracks and grooves that appeared on the surface of the projectile as it passed through the bore of the barrel. The caliber of a single projectile is usually smaller than the diameter of the bore mainly in the grooves, and larger than the diameter of the bore in the field. Therefore, the bore fields are cut on their surface into the shell of the projectile, forming a set of coiled tracks in the direction of the groove thread. Due to the production tolerances in the diameter of the bore in the grooves as well as the tolerance in the caliber of the projectiles, the entire bore can be displayed mainly in the shell of the projectile. In other cases, the bore of the barrel is not completely displayed. However, because the bullet in most cases closely fills the bore of the barrel, in the relatively soft shell of the bullet accurately displays a unique micro relief of the bore formed by a series of very fine protrusions and depressions. In the vast majority of cases, the traces of the fields on the projectile are deflected from the projectile axis by the same angle as the pitch of the thread in the grooves. When the projectile passes through the bore, especially at high speeds and under the action of large internal forces, the projectile does not move evenly through the bore, but its movement shows certain irregularities, which affect the formation of tracks. This is reflected in the fact that the angle of pitch of the field tracks or grooves on the projectile differs from the actual pitch of the thread of the grooved barrel. The final identification marks on the projectile arise only at the mouth of the bore of the barrel and their quality and quantity are affected by the shooting conditions, such as projectile speed, bore cleanliness, degree of corrosion, etc. Quality identification marks on the projectile arise if the weapon is carefully maintained and the movement of the projectile in the barrel is relatively slow. Ammunition is also examined for its ability to fire, and what is the effect of ammunition (projectiles) on hitting the target, e.g., to determine whether the weapon (taking into account the ammunition used and, if applicable, the shooting distance) can seriously injure or kill a person. By measurement or

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calculation, it is necessary to determine the impact velocity of the projectile at the point of anticipated impact (or already at the muzzle of the barrel), corresponding to the kinetic energy and load energy of the projectile crosssection. It was experimentally found that a projectile with an energy load of the cross-section (i.e., the ratio of kinetic energy to cross-sectional area of the projectile) greater than 50 J·cm-2 can cause death to a person (applied to projectile caliber 3 - 18 mm). In the range of 5 - 50 J·cm-2, such a consequence can be caused practically only by eye contact.

4.2 Physics behind the ballistic Energy Energy is a physical quantity that characterizes the form of motion (motion of matter). A single mass point (particle) has a mechanical energy if it moves with respect to a certain frame of reference (kinetic energy) or is located in the force field of other bodies (potential energy). It applies in all mechanical processes the potential energy changes into kinetic energy and vice versa, while the total energy of the isolated system bodies is constant throughout the process – the law of conservation of mechanical energy. Kinetic energy Kinetic energy is a physical quantity that characterizes the state of movement of the particle. The kinetic energy of a particle in a given reference frame is one-half its mass m multiplied by the square of its speed v measured in that reference frame 1 ‫ܧ‬௞ = ݉‫ ݒ‬ଶ 2 The unit is the newton-meter (N˜m). It is named joule (J). 1 J = 1 kg˜m2˜s-2 Kinetic energy is related to the work done by the forces acting on a particle. The force changes the velocity of the particle. This is the statement of the work-energy theorem for a particle – the work done on a particle equals the change of its kinetic energy: ‫ܧ‬୩ଵ =

ଵ ଶ

ଵ

݉‫ݒ‬଴ଶ , ‫࢑ܧ‬૛ = ݉‫ ݒ‬ଶ ଶ

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ܹ = ο‫ܧ‬୩ = ‫ܧ‬୩ଶ െ ‫ܧ‬୩ଵ In physics, work is defined as the force acting through the displacement 'x: In one-dimensional motion, the work is done by a constant force with the component Fx on a particle that undergoes a displacement 'x is the product of the force and displacement ܹ = ‫ ܨ‬ή ‫ ݏ‬ή cos ߙ where D is the angle between ‫ܨ‬Ԧ and ‫ݏ‬Ԧ. If ‫ܨ‬Ԧ A ‫ݒ‬Ԧ, then W = 0.

Work of variable force If the force acting on a body moves along a curve, and, moreover, this force is different in different places (it is a function of the path s), then the work that this force does on a certain section of the path (e.g., A, B) is given by ܹ = ‫ܨ‬ଵ ܿ‫ߙ ݏ݋‬ଵ ο‫ݏ‬ଵ + ‫ܨ‬ଶ ܿ‫ߙ ݏ݋‬ଶ ο‫ݏ‬ଶ + ‫ ڮ‬+ ‫ܨ‬௡ ܿ‫ߙ ݏ݋‬௡ ο‫ݏ‬௡ If the section of the trajectory between the points A, B is divided into many small sections, the resulting work will be equal to the sum of all elementary works. We can write ௦మ

ܹ = න ‫ߙ ݏ݋ܿ ݏ݀ܨ‬ ௦భ ௥ If the forces are conservative, then ܹ = ‫׬‬௥ మ ‫ܨ‬Ԧ ή d‫ݎ‬Ԧ , where d‫ݎ‬Ԧ is the భ displacement of the body.

Conservative forces are, for example, the force due to gravity, or the strength of elasticity. Proof of the connection between the change of kinetic energy and work A mass point of mass m is moved by the action of a force of magnitude F(x), which acts in the direction of the x-axis, from the initial position x1 to the final position x2. This force will do the work

Chapter Four

148 ௫మ

௫మ

ܹ = න ‫ܨ‬ሺ‫ݔ‬ሻ ݀‫ = ݔ‬න ݉ܽ݀‫ݔ‬ ௫భ

௫భ

(application of the 2nd Newton´s law) We can write ݉‫݉ = ݔ݀ݏ‬

݀‫ݒ‬ ݀‫ݔ‬ ݀‫ݐ‬

݀‫ݔ݀ ݒ‬ ݀‫ݒ‬ ݀‫ݒ‬ = = ‫ݒ‬ ݀‫ݐ݀ ݔ‬ ݀‫ݔ‬ ݀‫ݐ‬ ݉ܽ݀‫݉ = ݔ‬

݀‫ݒ‬ ‫ݒ݀ݒ݉ = ݔ݀ݒ‬ ݀‫ݔ‬

௩మ

ܹ = න ݉‫= ݒ݀ݒ‬ ௩భ

1 1 ݉‫ݒ‬ଶଶ െ ݉‫ݒ‬ଵଶ 2 2

Work in the gravitational field If we lift a body in a gravitational field, positive work is done. But the object does not gain kinetic energy. When lifted, the object acquires energy that depends on its position relative to the Earth – so called potential energy. Let’s have a particle of mass m, that is moving from the point A1 into the point A2 along the curve k in the gravity field of the Earth. The force of gravity ‫ܨ‬Ԧீ = ݉݃Ԧ is constantly acting on the particle. (Other forces are acting here – air resistance, influence of other particles etc.) The work done by the force ‫ܨ‬Ԧீ , is given ܹୋ = ݉݃ሺ݄ଵ െ ݄ଶ ሻ where h1, h2 are the heights of the points A1, A2 above an arbitrarily chosen horizontal plane H. The positional energy in the Earth´s gravitational field is ‫ܧ‬୮ = ݄݉݃

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i.e., when h = 0 then Ep = 0. The H plane is the level of zero gravitational energy.

Conservation of mechanical energy The mechanical energy of an isolated system (body) remains constant in time if the system is free of no-conservative forces, friction etc. The work done by the conservative force is independent on the path. Energy cannot be created or destroyed in an isolated system; energy can be only converted to another form of energy. ‫ܧ = ܧ‬୮ + ‫ܧ‬୩ ο‫ = ܧ‬ο‫ܧ‬୩ + ο‫ܧ‬୮ = 0 Consider a body in an isolated system that has a certain potential energy. We release this body, it starts to move, and thus its potential energy also starts to decrease. Thus, we can write െd‫ܧ‬୮ = ‫ܨ‬d‫ܽ݉ = ݏ‬d‫݉ = ݏ‬

d‫ݒ‬ d‫ݏ‬ d‫ݐ‬

d‫ݒ = ݏ‬d‫ݐ‬ and െd‫ܧ‬୮ = ݉

d‫ݒ‬ 1 ‫ ݒ‬d‫ݒ݉ = ݐ‬d‫ = ݒ‬d ൬ ݉‫ ݒ‬ଶ ൰ d‫ݐ‬ 2

So d(Ek + Ep) = 0, i.e. Ek + Ep = const. If the mechanical energy of the body (or the system) is preserved, we can compare the sum of the total kinetic and potential energy at different moments. It is not necessarily to consider the motion of the body (or of the system) in the interval between these moments and calculate the work of interaction forces of system particles.

Projectile energy In ballistics, we apply the calculation of energy in the case of determining the energy of the projectile impact related to the cross-sectional area of the

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projectile. It is the so called energy load of the projectile cross section. The greater this energy, the greater the force action on the material. It is also easier to overcome the resistance force (penetration below the surface of the material). E.g., a pressure greater than 100 kPa·cm-2 be applied to penetrate the projectile through the skin. ி

Pressure is a scalar physical quantity, defined by the relation ‫ = ݌‬, where ௌ F is the magnitude of the force acting perpendicular to a surface of size S. The basic unit of pressure is 1 Pa (Pascal). According to the formula for the calculation of kinetic energy, the energy of a projectile depends mainly on its velocity and its mass. This energy refers to the area corresponding to the cross section of the projectile. A small-caliber projectile that has a small cross section should therefore develop a higher energy load for a given amount of inertial (kinetic) energy. However, this is generally not the case. The cross-sectional load is determined on the basis of the following relationship: the mass of the projectile m a d3, cross-section S a d2, and the cross-sectional load m/S. This ration is decisive for the properties of the projectile. Compare: the volume of a cube with edge size a: V = a˜ a˜ a, the size of the area of this cube S = 6˜a˜a. If we increase the length of the edge to 2a, the volume is equal to 8times the original volume, cube area 4times. For 3a we have 27·V and 9·S. The ratios V : S are gradual a, 2a, 3a etc. As the body dimensions increase, the volume-to-surface ratio increases linearly. For the movement of the projectile it follows that small-caliber projectiles, even at high initial velocities due to the small cross-sectional load, quickly lose velocity (due to air resistance, wind etc.), and thus penetration. http://www.militaria.cz/cz/clanky/vojenska-technika/mytus-jmenemkwk.html

Forces acting on a single mass point in the Earth´s gravitational field The basic forces acting on a moving mass point are the gravitational force and the resistive force. Let us study the motion of a mass point in the Earth´s gravitational field, which will be affected only by gravity. We solve the following equations of motion:

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The movement takes place in the direction of the y-axis: ‫݉ = ୷ܽ݉ = ୷ܨ‬

d‫୷ݒ‬ dଶ ‫ݕ‬ = ݉ ଶ = െ݉݃ d‫ݐ‬ d‫ݐ‬

dଶ ‫ݕ‬ = െ݃ d‫ ݐ‬ଶ d‫ݕ‬ d‫ݒ‬ =‫Ÿݒ‬ = െ݃ d‫ݐ‬ d‫ݐ‬ Solution of this equation v = -gt + k1 We determine the integration constant k1 using the initial conditions: t = 0, v = vo Ÿ v = -gt + vo After inductance and further integration, we have: 1 d‫ݕ‬ = െ݃‫ ݐ‬+ ‫ݒ‬଴ ‫ = ݕ‬െ ݃‫ ݐ‬ଶ + ‫ݒ‬଴ ‫ ݐ‬+ ݇ଶ 2 d‫ݐ‬ We determine the integration constant k2 using the initial conditions: t = 0, yo = k2. The resulting equation for the motion of a mass point in the Earth´s gravity field is 1 ‫ = ݕ‬െ ݃‫ ݐ‬ଶ + ‫ݒ‬଴ ‫ ݐ‬+ ‫ݕ‬଴ 2 Resistive force: ଵ

‫ܨ‬௢ = ‫ ݒߩܵܥ‬ଶ , ଶ

where m is the mass of the object, C is the coefficient of air resistance (its value depends on the shape of the object, in ballistics it is the shape of the projectile – drag coefficient), S is the cross-sectional area of the object, U is the density of air, v is the velocity of the object.

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Air density The air density changes with the altitude and is temperature dependent. If we consider air as an ideal gas, the equation of state of an ideal gas can be used to describe air properties. We can write the Boyle-Marriote law (isothermal process) using pressure ௣ ௣ and density, then = ݇‫ݐݏ݊݋‬, ߩ = ߩ௢ ೌ , where U is the air density at a ఘ

௣ೌ೚

given altitude, Uo is the air density on the surface of the Earth, pa is the atmospheric pressure at a given altitude and pao is the pressure on the surface of the Earth. After substituting this relationship into the expression for calculating the change in air pressure written as dpa = Ugdy and integrating this equation, we obtain the barometric formula ‫݌‬௔௢ ˜ ݁ ିఘ೚௚௛Τ௣ೌ೚ This equation assumes that the air temperature is constant. However, this is not true, the temperature changes with altitude. The values of air pressure depending on the temperature can be found in meteorological tables. From the barometric equation it is possible to derive a relation about the dependence of air density according to the altitude. ߩ = ߩ௢ ˜ ݁ ିఘ೚௚௛Τ௣ೌ೚ It can be seen that the air pressure depends on the altitude, the magnitude of the gravitational acceleration, the temperature, the density of the atmosphere in a given place. Therefore, the so - called normal air pressure (normal atmospheric pressure) pn (also po) is introduced as the average value of air pressure at sea level at 45o at a temperature of 15 oC and gravitational acceleration gn = 9.80665 m˜s-2. It is defined by the exact value pn = 101 325 Pa = 1013.25 hPa. In general, air density is given as the ratio of the weight of air and the volume occupied by the air. It is usually expressed in kg˜m-3. The air density at a temperature of 0oC and a pressure of 1013.27 hPa is 1.293 kg˜m-3. If we compare humid and dry air, then under otherwise the same conditions, the density of humid air is always greater than the density of dry air. At constant pressure, the air density is inversely proportional to the air temperature.

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The non-uniform distribution of atmospheric pressure gives rise to a horizontal air flow – wind (horizontal component of the pressure gradient force). The air flow is also influenced by the Earth´s rotation – Coriolis force, friction, centrifugal force. Friction is applied in the boundary layer of the atmosphere – due to friction, the wind speed decreases and the wind direction turns (the Coriolis force decreases). Wind rotation (on the side of lower atmospheric pressure) is about 30° in our conditions.

Coriolis force Apparent, inertial force, the so-called Coriolis force, acts on bodies that move in a rotating non-inertial frame of reference. The force is perpendicular to the axis connecting the body to the center of rotation. This causes the body to deviate both sideways and in height. The trajectory of the body rotates against the direction of rotation of the object – it depends on the direction of movement of the object relative to the center of rotation. On Earth, in the northern hemisphere, the trajectory of the object, which moves in the direction of the meridians, turns to the right. In the case of an oblique throw of an object in the gravity field of the Earth (ballistics), the deviation depends on the position on the Earth (latitude), the speed of movement of the object, and the initial angle of the throw.

Figure 4-1. Coriolis force (https://windy.app/blog/what-is-the-coriolis-forcesimple-explanation.html)

We write the relation for the calculation of the Coriolis force in the form ‫ܨ‬Ԧେ = െ2݉߱ ሬԦ ൈ ‫ݒ‬Ԧ,

154

Chapter Four

where m is the mass of the object, ߱ ሬԦ is the angular velocity vector, ‫ݒ‬Ԧ is the velocity vector of the object in a given non-inertial frame of reference. The symbol „u“ denotes the vector product of both vectors, which implies that the magnitude of the Coriolis force can be determined by the relation ‫ܨ‬஼ = െ2݉‫߱ݒ‬sin ߠ, where ߠ denotes the angle between the two vectors. Further reading: http://fyzweb.cz/materialy/srazky_a_rotace/k27.php. The action of the Coriolis force is manifested, for example, in meteorology, in the case of ballistics in the calculation of the trajectory of missiles with a long range. See – World War I and so-called Paris Cannon http://www.guns-info.cz/modules.php?name=News&file=article &sid=2399)

Magnus effect As early as in the 17th century it was observed that the fired cannon ball deviated from the direction of flight. This phenomenon occurs when air rotates around the rotating object due to friction. When flowing around the object, a lateral force acts on the object, which arises due to the pressure difference on one and the other side of the object. Along with the rotating object, the so-called boundary layer of air also rotates. Because the law of conservation of energy applies, which can be expressed for fluid (air) using Bernoulli´s equation, there is a change in pressure – on one side of the object a negative pressure (air flow accelerates), on the other overpressure (air flow slows down). The pressure difference gives rise to a force that curves the trajectory of this object. Relation for calculating the magnitude of the Magnus force for a cylinder of radius r is ‫ = ܨ‬2ߨߩ‫ ݎ߱ݒ‬ଶ . For the case of a rotating sphere, the relation for calculating the Magnus force takes shape ଷ ሬሬሬԦ ‫ = ܨ‬2ߩ‫ݒ‬ ሬሬሬԦ × ߱ ሬԦ ߨܴଷ . ସ

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155

ሬሬሬሬሬԦ Figure 4-2. Air flow around the ball – force ‫ܨ‬ ெ goes from the area of lower pressure to the area of higher pressure (Reichl - Encyklopedie fyziky)

Bernoulli´s equation Bernoulli´s equation is a statement about the conservation of mechanical energy in a system. It is valid for an ideal fluid – for steady, incompressible, no viscous and irrational flow: ‫݌‬+

ଵ ଶ

ߩ‫ ݒ‬ଶ + ݄ߩ݃ = ܿ‫ݐݏ݊݋‬, ଵ

where p is the static pressure, ߩ‫ ݒ‬ଶ represents the dynamic pressure (it ଶ

corresponds to kinetic energy), and the product hUg is the hydrostatic pressure (h is the height above the Earth, U is the density of the fluid). If the fluid flows through a tube that is horizontal, then h = 0 and the hydrostatic pressure expression term falls out of the equation (it is zero).

Projectile motion Projectile motion is a two-dimensional motion of a particle thrown obliquely into the air.

Chapter Four

156

The motion is one of constant acceleration ݃Ԧ, ay = - g. The initial velocity is ሬሬሬሬԦ, ‫ݒ‬଴ the horizontal component is vx = vx0 + axt = v0 cos4, the vertical velocity (that of free fall) is ‫୷ݒ = ୷ݒ‬଴ + ܽ୷ ‫ݒ = ݐ‬଴ sin 4 െ ݃‫ݐ‬ The magnitude of resultant velocity vector at any instant is ‫ = ݒ‬ඥ‫ݒ‬௫ଶ + ‫ݒ‬௬ଶ . The range of the projectile (in general) ݀=

‫ݒ‬cos ߠ ቀ‫ݒ‬sin ߠ + ඥሺ‫ݒ‬sin ߠሻଶ + 2݃‫ݕ‬௢ ቁ ݃

In the case of initial conditions, where the initial height of the projectile is zero and the surface is horizontal, the flight duration can be calculated using the relation ݀=

‫ ݒ‬ଶ sinሺ2ߠሻ ݃

It follows from the given formula that the maximum flight range is reached for an elevation angle of 45°. This length is then equal to ݀ =

௩మ ௚

.

Flight time in general ‫=ݐ‬

݀ ‫ ߠ ݊݅ݏݒ‬+ ඥሺ‫ߠ ݊݅ݏݒ‬ሻଶ + 2݃‫ݕ‬௢ = ‫ߠ ݏ݋ܿݒ‬ ݃

For initial conditions, zero initial height and angle 45° we have ‫= ݐ‬

ξଶ௩ . ௚

If we want to calculate the elevation angle needed to reach the flight length d, we can write ‫݊݅ݏ‬ሺ2ߠሻ =

݃݀ ‫ݒ‬ଶ

1 ݃݀ ߠ = ܽ‫ ݊݅ݏܿݎ‬൬ ଶ ൰ 2 ‫ݒ‬ if the initial velocity v of the particle is known.

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Body height at distance x ‫ݕ = ݕ‬௢ + ‫ ߠ݃ݐݔ‬െ

݃‫ ݔ‬ଶ 2ሺ‫ݒ‬cosߠሻଶ

And the corresponding magnitude of the velocity at this point ଶ

௚௫

|‫ = |ݒ‬ට‫ ݒ‬ଶ െ 2݃‫ ߠ݃ݐݔ‬+ ቀ ቁ . ௩௖௢௦ఏ The magnitude of the resultant velocity vector is |‫ = |ݒ‬ඥ‫ݒ‬௫ଶ + ‫ݒ‬௬ଶ , where ‫ݒ‬௫ = ‫ ݒ‬cos ߠ, ‫ݒ‬௬ = ‫ ݒ‬sin ߠ െ ݃‫ݐ‬, ‫= ݐ‬

௫ ௩ୡ୭ୱ ఏ

and ‫ ݒ = ୷ݒ‬sinߠ െ

௚௫ ௩ ୡ୭ୱఏ

.

|‫ = |ݒ‬ඨሺ‫ ݒ‬cos ߠሻଶ + ቀ‫ ݒ‬sin ߠ െ

݃‫ ݔ‬ଶ ቁ ‫ ݒ‬cos ߠ

To reach a point with coordinates >x,y@ under given conditions, it is necessary to choose an elevation angle of magnitude ߠ = arctg ൭

‫ ݒ‬ଶ ± ඥ‫ ݒ‬ସ െ ݃ሺ݃‫ ݔ‬ଶ + 2‫ ݒݕ‬ଶ ሻ ൱ ݃‫ݔ‬

Solving the equation of motion of a thrown object (projectile) with the calculation of the resistance force (air resistance) is a complex tasks. Approximately, it can be solved using the following reasoning. We assume that the resistive force is proportional to v2, we use Newton´s formula in the form to express it ଵ

௩మ

ଶ

௞మ

‫ = ܨ‬െ ‫ ݒܵߩܥ‬ଶ = െ݉݃

,

Chapter Four

158

ଶ௠௚

where the symbol k is written as ݇ = ට

஼ௌఘ

. We will use this shape when

solving the equation of motion (excluding rotation) using the 2nd law of motion: ݉ ݉

ୢమ ௫ ୢ௧ మ

= ‫ܨ‬cos ߠ,

dଶ ‫ݕ‬ = ‫ܨ‬sin ߠ െ ݉݃ ݀‫ ݐ‬ଶ

If we denote the elementary length of the ballistic curve ds, the angle of the throw can be expressed as cos ߠ =

d‫ݔ‬ d‫ݕ‬ , sin ߠ = d‫ݏ‬ d‫ݏ‬

We obtain ୢమ ௫ ୢ௧ మ

= െ݃

௩ మ ୢ௫ ୢమ ௬

,

௞ మ ୢ௦ ୢ௧ మ

= െ݃ ቀ

௩ మ ୢ௬ ௞ మ ୢ௦

െ 1ቁ.

If the angle ߠ is small and the arc s can be replaced by the lenght x, the solution of the given equations is in the form ‫ݔ = ݕ‬tgߠ +

௞మ ଶ௩೚మ ୡ୭ୱమ ఏ

ቂ‫ ݔ‬െ

௞మ ଶ௚

൫݁ ൫ଶ௚/௞

మ ൯௫

െ 1൯ቃ.

This is the equation of a ballistic curve. If the air resistance is neglected (݇ ଶ ՜ f), the equation passes into the equation of the parabola. This solution does not include: the effect of rotation, Coriolis force, Magnus effect, the effect of air density. One of the computer programs can be used to calculate the ballistic curve, e.g. http://www.balistika.cz/vnejsi_program.html.

APPENDIX MATHEMATICS

Differential and vector calculus Differential calculus The ancient Greek mathematicians Eukleides of Alexandria, Archimedes of Syracuse and Apollonios of Perga solved some specific problems of differential calculus with specific methods. Many mathematicians in the 16th - 17th century systematically dealt with the issue of general methods for solving problems of this type, especially the determination of the tangent at any point of the plane curve, and relevant physical problems (determining the instantaneous velocity of a rectilinear motion of a material point and extremal problems). Let us mention, for example, the Italian physicist, mathematician and astronomer Galileo Galilei, the German mathematician, physicist and astronomer Johannes Kepler, Dutch mathematician, physicist and astronomer Christian Huygens, but also French mathematicians René Descartes and Pierre de Fermat. Isaac Newton was the first to lay the foundations of differential calculus. In his work "Method of Fluxes and Infinite Series", which he wrote in the years 1670–1671, he is based on the kinematic concept of the plane curve k as a trajectory of a continuously moving point. According to Descartes' ideas of analytic geometry, it assigns to this point in each (general) position its Cartesian coordinates [x; y], where y = f (x) from the equation of the curve k. Newton called the variables x, y fluents, i.e., "flowing quantities". He considered them dependent on time t as auxiliary variables. He denoted by o infinitely small increments of the variable t. The rates of change (increase or decrease) of fluents relative to the variable (time) t were denoted by ‫ݔ‬ሶ =

௫ሺ௧ା௢ሻି௫ሺ௧ሻ ௢

,

‫ݕ‬ሶ =

௬ ሺ௧ା௢ሻି௬ሺ௧ሻ ௢

Appendix

160

Newton expressed infinitely small (infinitesimal) increments of fluents by the expressions ‫݋ݔ‬ሶ, ‫݋ݕ‬ሶ and called them moments of fluxes. In Newton’s symbolism, an expression ௬ሶ ௫ሶ

=

௙ሺ௫ା ௫ሶ ௢ሻି௙ሺ௫ሻ ௫ሶ ௢

represented a slope of the tangent IJ to the curve k at the point [x; y] and ‫ = ݒ‬ට‫ ݔ‬ଶሶ + ‫ ݕ‬ଶሶ expressed the magnitude of the instantaneous velocity of the point [x; y], which moves along the curve k at time t, see Fig. A-1

Figure A-1. Magnitude of the instantaneous velocity of the point [x; y], which moves along the curve k at time t

Newton called the basic task of mathematical analysis (infinitesimal calculus) the following problem: The relation between fluents x, y is given, i.e. the equation of the plane curve k: y = f (x) ௬ሶ

Find the relationship between the respective fluxes ‫ݔ‬ሶ , ‫ݕ‬ሶ , i.e., the ratio . If ௫ሶ we convert this to a physical problem, we can express that as follows: The trajectory of a motion of a mass point is given as a dependence of its position on time t. Find its instantaneous velocity at any moment. Independently of Newton and only a few years later, G. W. Leibniz also laid the foundations of differential calculus. Leibniz labeled infinitesimal

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(infinitesimal) increments of the variables x, y with the symbols dx, dy and called them differentials. Although he did not define them exactly, what he understood by them follows from the fact that he calculates dy according to the formula dy = f(x + dx) – f(x) where in the expression f(x + dx) he neglects infinitesimal quantities of 2nd and higher orders. Furthermore, based on the ideas of B. Pascal from 1659, he introduced the concept of a characteristic triangle, and was the first to choose for it an infinitesimal right triangle with perpendiculars of lengths dx, dy and a hypotenuse of length d‫ = ݏ‬ඥd‫ ݔ‬ଶ + d‫ ݕ‬ଶ He further defined and used the differential ratio ݂ሺ‫ ݔ‬+ d‫ݔ‬ሻ െ ݂ሺ‫ݔ‬ሻ d‫ݕ‬ = d‫ݔ‬ d‫ݔ‬ expressing the slope of the tangent IJ to the curve k at the point P [x; y]. The basic concept of differential calculus is the concept of derivative. Derivatives describe the rate of rise or fall of a function at a given point, and thus determine how steep the function is in the neighborhood of a given point. To better understand this concept, we can approach the situation with the following physical problem of determining the instantaneous velocity: Problem: Determine the magnitude of the instantaneous velocity in nonlinear motion of a mass point at time to, given the functional dependence of its path s on the time t: s = s(t). Solution: The average velocity vp of a mass point movement in an interval ‫ݐۃ‬଴ ; ‫ ۄݐ‬is ‫ݒ‬௣ =

ο‫ݏ‬ ο‫ݐ‬

where ο‫ ݐ = ݐ‬െ ‫ݐ‬଴ and ο‫ݏ = ݏ‬ሺ‫ݐ‬଴ + ο‫ݐ‬ሻ െ ‫ݐ(ݏ‬଴ ). Its instantaneous speed in time ‫ݐ‬଴ is

Appendix

162

ο‫ݏ‬ ο௧՜଴ ο‫ݐ‬

‫ݐ(ݒ‬଴ ) = lim ‫ݒ‬୮ = lim ο௧՜଴

thus ‫ݐ(ݏ‬଴ + ο‫ )ݐ‬െ ‫ݐ(ݏ‬଴ ) ο௧՜଴ ο‫ݐ‬

‫ݐ(ݒ‬଴ ) = lim

The given problem is an interpretation of the limit of the differential ratio of the function f: y = f(x) at the point xo, i. e. ο௬ ο௫

for ¨x ĺ 0.

Due to its considerable importance and frequent use, it was given a special name for the derivative of the function f at the point xo. Definition: If there exists a limit of the differential ratio ο௬ ο௫

=

ο௬ ௛

for ݄ ՜ 0,

then this limit is called the derivative of the function f at the point x0 and is denoted by ݂ ´ (‫ݔ‬଴ ). The definition relation for the derivative ݂ ´ (‫ݔ‬଴ ) can be written in the following equivalent forms: ݂ ´ (‫ݔ‬଴ ) = lim

௛՜଴

௙(௫బ ା௛)ି௙(௫బ ) ௛

or ݂ ´ (‫ݔ‬଴ ) = lim

௫՜௫బ

௙(௫)ି௙(௫బ ) ௫ି௫బ

If the limit in these relations is proper, the number ݂ ´ (‫ݔ‬଴ ) is called proper derivative of the function f at the point x0. If this limit is an improper number, ݂ ´ (‫ݔ‬଴ ) is called the improper derivative of the function f at the point x0. In a similar way, one-sided derivatives of the function f are defined at the point x0, (right derivative ݂ା´ (‫ݔ‬଴ ) and left derivative ݂ି´ (‫ݔ‬଴ )): ݂±´ (‫ݔ‬଴ ) = lim

௛՜଴±

݂(‫ݔ‬଴ + ݄) െ ݂(‫ݔ‬଴ ) ݂(‫ )ݔ‬െ ݂(‫ݔ‬଴ ) = lim ௫՜௫బ± ݄ ‫ ݔ‬െ ‫ݔ‬଴

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Geometric meaning of the derivative of the function at the point: It follows from the definition of the derivative of the function f at the point x0 that the function f has its proper derivative ݂ ´ (‫ݔ‬଴ ) at the point x0 if and only if the graph of the function f at the point ܲ଴ ሾ‫ݔ‬଴ ; ݂(‫ݔ‬଴ )ሿ has a tangent t with the slope ݇௧ = ݂ ´ (‫ݔ‬଴ ). The equation of the tangent t is ‫ݔ(݂ = ݕ‬଴ ) + ݂ ´ (‫ݔ‬଴ ) (‫ ݔ‬െ ‫ݔ‬଴ )

The normal n (i.e., the line perpendicular to the tangent t) at point ܲ଴ has the slope kn; where kn kt = - 1, so the equation of the normal to the graph of the function at a given point is ‫ݔ(݂ = ݕ‬଴ ) െ

ଵ ௙´ (௫బ )

(‫ ݔ‬െ ‫ݔ‬଴ ), if ݂ ´ (‫ݔ‬଴ ) ് 0

and x = x0, if ݂ ´ (‫ݔ‬଴ ) = 0.

Figure A-2. Geometric meaning of the derivative (https://is.muni.cz/el/1431/podzim2012/C1480/um/web/ch02_s02_s01_s01.html)

If the function f is continuous at the point x0 and has an improper derivative in it, i. e. ݂ ´ (‫ݔ‬଴ ) = ±λ, then the tangent t to the graph of the function f at the point ܲ଴ ሾ‫ݔ‬଴ ; ݂(‫ݔ‬଴ )ሿ is parallel to the y-axis. The equation of the tangent t is x = x0 and the equation of the normal n is y = f (x0).

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164

ܽ) ݂ ´ (‫ݔ‬଴ ) ് 0, ݂ ´ (‫ݔ‬଴ ) ‫ܴ א‬

ܾ) ݂ ´ (‫ݔ‬଴ ) = 0 ܿ) ݂ ´ (‫ݔ‬଴ ) = +λ

Figure A-3. Tangent and normal of the graph of the function (http://cgi.math.muni.cz/kriz/analyza/kap5.html)

Physical meaning of the derivative of a function at a point: In physical problems, the functional variable (argument of functions) is often time t. In the case of non-uniform rectilinear motion of a mass point, its path is a function of time t, i. e. s = s(t) and the instantaneous velocity at time t0 is ‫ݐ(ݒ‬଴ ) =

d‫ݏ‬ (‫) ݐ‬ d‫ ݐ‬଴

This can be generalized: If the scalar physical quantity u is a function of time t, i. e. u = u(t), then its derivative according to t at time t0 represents the instantaneous amount of change of quantity u at time t0. Theorems about proper derivatives of a function in a point Theorem about the relationship between derivative and continuity of a function at a given point: If the function f at its point x0 has a proper derivative, then it is continuous at this point. (The continuity of a function at a point is a necessary condition for the existence of a proper derivative of a function at that point). It is important to note that this theorem holds only to the proper derivatives of the function. The reverse theorem does not hold: A function can be continuous at x0, but it does not have to have its proper derivative in it. (The continuity condition of a function at point x0 is not sufficient for the existence of the proper derivative of the function ݂ ´ (‫ݔ‬଴ )).

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If we consider the function f: y = | x |, then it is continuous at the point x0 = 0, but its one-sided derivatives at this point are different: ݂ା´ (0) = 1, ݂ି´ (0) = െ1 therefore, there is no derivative ݂ ´ (0). Theorems about derivatives of sum, difference, product and division of functions: If the functions f(x), g(x) have derivatives ݂ ´ (‫ݔ‬଴ ), ݃´ (‫ݔ‬଴ ), at the point x0, then the functions ݂(‫ )ݔ‬± ݃(‫)ݔ‬, ݂ܿ(‫( )ݔ‬for ܿ ‫)ܴ א‬, ݂(‫)ݔ‬. ݃(‫ )ݔ‬and for ௙(௫) have also derivatives and the following ݃(‫ݔ‬଴ ) ് 0 also the functions ௚(௫)

rules of derivation hold to them: ´

൫݂ܿ(‫)ݔ‬൯௫ୀ௫ = ݂ܿ´(‫ݔ‬௢ ), ೚

´

൫݂(‫ )ݔ‬± ݃(‫)ݔ‬൯௫ୀ௫ = f´(xo) ± g´(xo) ೚

´

൫݂(‫ )ݔ‬ή ݃(‫)ݔ‬൯௫ୀ௫ = f´(xo) g(xo) + f(xo) g´(xo) ೚

If ݃(‫ݔ‬௢ ) ് 0, then ´

ቆ

݂(‫)ݔ‬ ݂´(‫ݔ‬௢ )݃(‫ݔ‬௢ ) െ ݂(‫ݔ‬௢ )݃´(‫ݔ‬௢ ) ቇ = ݃(‫)ݔ‬ ݃ଶ (‫ݔ‬௢ )

Theorem on the derivation of a compound function: If a function g: u = g(x) has a derivative ݃´ (‫ݔ‬଴ ) at point x0 and the function f: y = f(u) has a derivative ݂ ´ (‫ݑ‬଴ ) at point u0 = g(x0), then a compound function h: y = h(x) = f(g(x)) has a derivative h´(x0) = f´(u0) g´(x0) at point x0. Symbolically, this formula is expressed in the form: ୢ௬ ୢ௫

=

ୢ௬ ୢ௨

.

ୢ௨ ୢ௫

,

166

Appendix

which is called a chain rule. It means that a derivation of a compound function y = f(g(x)), where f: y = f(u), g: u = g(x) is calculated as a product of the derivative of its external component (external function) f according to the variable u in point g(x0) and of the derivative of its internal component (internal function) g according to the variable x in point x0. If a function f has a proper derivative ݂ ´ (‫ )ݔ‬for all ‫ܦ א ݔ‬൫݂ ´ ൯ ‫)݂(ܦ ؿ‬ then f´: y = f´(x), ‫ܦ א ݔ‬൫݂ ´ ൯ is a new function called the derivative of the function f on the set ‫ܦ‬൫݂ ´ ൯. If (ܽ, ܾ) ‫)´݂(ܦ ؿ‬, we say that the function f has a derivative on an open interval (ܽ, ܾ). By using one-sided derivatives, this definition can be extended to a closed interval ‫ܽۃ‬, ܾ‫ۄ‬, possibly at semi-closed intervals (ܽ, ܾۧ and ‫ܽۦ‬, ܾ). The function f´ can also have a derivative at some point x0, which we call the second derivative of the function f at point x0 and we denote by f´´ (x0). The second proper derivative f´´: y = f´´ (x), ‫ܦ א ݔ‬൫݂ ´´ ൯ ‫)´݂(ܦ ؿ‬, represents a function on a set ‫ܦ‬൫݂ ´´ ൯. Similarly, higher order derivatives can be defined. Theorems on the mean value of differential calculus: The following sentences provide an important theoretical basis for investigating the course of functions. We will deal with the investigation of the function properties using derivatives. Fermat's theorem: If the function f acquires at a point c its highest values (maximum), or the smallest values (minimum) on the interval (ܽ, ܾ) ‫ؿ‬ ‫ )݂(ܦ‬and has a derivative at this point f´(c), then f´(c) = 0. Rolle's theorem: If the function f continuous on a closed interval ‫ܽۃ‬, ܾ‫ ۄ‬has at each inner point ‫ܽ( א ݔ‬, ܾ) the derivative f´(x) (proper or not) and f(a) = f(b), then there is at least one point ܿ ‫ܽ( א‬, ܾ) such that f´(c) = 0.

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Geometric interpretation of Rolle's theorem: If the assumptions of Rolle's theorem are fulfilled, there is at least one point ܶሾܿ, ݂(ܿ)ሿ, in which the tangent t to the function graph f is parallel to the axis x (Fig. A-4)

Figure A-4. Geometric interpretation of Rolle's theorem (https://math.fel.cvut.cz/mt/txtc/2/txc3ca2a.htm)

Lagrange‘s mean value theorem: If the function f continuous on a closed interval ‫ܽۃ‬, ܾ‫ ۄ‬has at each inner point ‫ܽ( א ݔ‬, ܾ) the derivative f´(x) (proper or not), then there is at least one point ܿ ‫ܽ( א‬, ܾ) such that ݂ ´ (ܿ) =

݂(ܾ) െ ݂(ܽ) ܾെܽ

Geometric interpretation of the Lagrange’s mean value theorem: If the assumptions of the Lagrange theorem are satisfied, there is at least one point ܶሾܿ, ݂(ܿ)ሿ, in which the tangent t is parallel to the line AB, where ‫ܣ‬ሾܽ, ݂(ܽ)ሿ, ‫ܤ‬ሾܾ, ݂(ܾ)ሿ, whose slope is ݂ ´ = ݇ = ‫= ߙ ݃ݐ‬

݂(ܾ) െ ݂(ܽ) ܾെܽ

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Figure A-5. Geometric interpretation of the Lagrange’s theorem (https://math.fel.cvut.cz/mt/txtc/2/txc3ca2a.htm)

It is worth noting that Lagrange's mean value theorem was proved in 1797 by the French mathematician Joseph-Louis Lagrange. Rolle's theorem was proved in 1691 by the French mathematician Michel Rolle, especially for polynomial functions. Physical meaning of Lagrange's theorem: If a quantity changes over time in a "smooth way", then at some point the instantaneous rate of change must be equal to the average rate. Physical meaning of Rolle's theorem: If a quantity changes over time in a "smooth way" so that it has the same magnitude at the beginning and end of the process, then at some point the instantaneous rate of change must be zero. Investigation of function properties using derivatives: The following theorem states the necessary and sufficient conditions for the monotonicity of the function f on the interval J. Let the function f have the derivative f´ on the open interval J. Then 1. The function f is non-decreasing (non-increasing) on J if and only if for each ‫ܬ א ݔ‬ ݂ ´ (‫ )ݔ‬൒ 0 ( ݂ ´ (‫ )ݔ‬൑ 0). 2. The function f is increasing (decreasing) on J if and only if for each ‫ܬ א ݔ‬ ݂ ´ (‫ )ݔ‬൒ 0 ( ݂ ´ (‫ )ݔ‬൑ 0).

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In practice, however, we often suffice with its following consequence, which expresses simple sufficient conditions of pure monotonicity of the function f. Theorem on sufficient conditions for pure monotonicity of a function: Let the function f have the derivative f´ on the open interval J. Then a) If the function ݂ ´ (‫ > )ݔ‬0 for each ‫ܬ א ݔ‬, function f is increasing on J. b) If the function ݂ ´ (‫ < )ݔ‬0 for each ‫ܬ א ݔ‬, function f is decreasing on J. Problem: Determine the monotonicity intervals of the function ݂: ‫= ݕ‬ ‫ ݔ‬ଷ െ 3‫ݔ‬. Solution: To derive a function f, we get ‫ = ´ ݕ‬3‫ ݔ‬ଶ െ 3. We will now solve the inequality ݂ ´ (‫ > )ݔ‬0, that is 3‫ ݔ‬ଶ െ 3 > 0. Consequently we get ‫ ݔ‬ଶ െ 1 > 0 ฻ |‫ > |ݔ‬1 ฻ ‫( א ݔ‬െλ; െ1) ‫( ׫‬1; +λ) Function f it is therefore increasing on intervals (െλ; െ1) and (1; +λ). Due to the continuity of the function in the set of all real numbers, it is increasing even on intervals (െλ; െ1ۧ, ‫ۦ‬1; +λ). Similarly, ݂ ´ (‫ < )ݔ‬0 yields 3‫ ݔ‬ଶ െ 3 < 0 and ‫ ݔ‬ଶ െ 1 < 0 ฻ |‫ < |ݔ‬1 ฻ ‫( א ݔ‬െ1; 1). Function f is decreasing on the interval (െ1; 1). Due to the continuity of the function on R is also decreasing on the interval ‫ۃ‬െ1; 1‫ۄ‬. In mathematical analysis, extreme function values of local and global type are distinguished and investigated: Local extremes of a function f are understood to be the largest, resp. the smallest values of the function f in some (unspecified) surroundings of the assessed point of the definition field D (f). They are local properties of the function f. Global extremes of function f means the extreme values (maxima, minima) of the function f on a certain (predetermined) set M, especially on M = D (f). They are global properties of the function f on the set M. Local extremes are defined as follows:

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We say that the function f has a local maximum at the point x0 (local minimum) if there is a neighborhood of the point ܷ(‫ݔ‬଴ ) so that for each ‫ݔ(ܷ א ݔ‬଴ ) ݂(‫ )ݔ‬൒ ݂(‫ݔ‬଴ ) (݂(‫ )ݔ‬൑ ݂(‫ݔ‬଴ )), strict local maximum (strict local minimum), if there exists a neighborhood ܷ(‫ݔ‬଴ ) such that for each ܷ(‫ݔ‬଴ ) െ ሼ‫ݔ‬଴ ሽ ݂(‫ݔ(݂ < )ݔ‬଴ ) (݂(‫ݔ(݂ > )ݔ‬଴ )).

Figure A-6. Local and global extremes of function

If we investigate local extrema of functions using their derivatives, it is necessary to use the following theorems: Theorem on the necessary condition of the existence of a local extreme of a function at the point where the function has a derivative: If the function f has a local extreme at the point x0 and if there exists a derivative f´´ (x0) at this point, then ݂´(‫ݔ‬଴ ) = 0 The given theorem is a direct consequence of Fermat's theorem. However, the reverse sentence does not hold. Example: Consider the function f: y = x3, which has the derivative f´´ (0) = 0 at the point x0 = 0 but has no local extreme at this point (it is increasing in the whole domain D(f)).

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Example: The function f: y = sin x has the derivative f´ (x) = cos x at each ଵ point of the interval ‫ۃ‬0; 2ߨ‫ۄ‬. At the point ‫ݔ‬ଵ = ߨ it has a local maximum and at the point ‫ݔ‬ଶ = points we get:

ଷ ଶ

ଶ

ߨ it has a local minimum. For the derivative at these 1 1 ݂ ´ ൬ ߨ൰ = cos ߨ = 0 2 2

and similarly 3 3 ݂ ´ ൬ ߨ൰ = cos ߨ = 0 2 2 Remark: The function does not have to have extremes, even if it is bounded. An example of such a function is a function ݂: ‫= ݕ‬ ௫మ

௫మ ௫ మ ାଵ

. This function is

bounded for every ‫ܴ א ݔ‬: 0 ൑ మ < 1, but has no maximum at any point ௫ ାଵ in its domain ‫ܴ = )݂(ܦ‬. It has only a local minimum at x0 = 0. Important points in the investigation of the course of the function are stationary points of the function f. These are such points ‫ݔ‬௞ ‫)݂(ܦ א‬, in which f´´ (xk) = 0. It does not necessarily follow that the function has a local extreme at point x0. Nevertheless, determining the 1st derivative of a function will be the first step to finding local extrema. Useful sufficient conditions are given in the following two sentences: Theorem on sufficient conditions for the existence of local extrema of a function expressed by its first derivative: Let the function f be continuous at x0 and have its proper derivative f´ (x) at some neighborhood ܷ(‫ݔ‬଴ ) for all ‫ݔ ് ݔ‬଴ . If at point x0 the signs of the derivative f´(x) change from f´(x) > 0 for x < x0 to f´(x) ݔ‬଴ , then the function f has a strict local maximum at point x0. If at point x0 the sign of the derivative f´(x) changes from f´(x) 0 for ‫> ݔ‬ ‫ݔ‬଴ , then the function f has a strict local minimum at point x0. Theorem on sufficient conditions for the existence of local extrema of a function expressed by its second derivative: Let ݂ ´ (‫ݔ‬଴ ) = 0 and ݂ ´´ (‫ݔ‬଴ ) ് 0. If ݂ ´´ (‫ݔ‬଴ ) > 0, then the function f at point x0 has a strict local minimum. If ݂ ´´ (‫ݔ‬଴ ) < 0, then the function f at point x0 has a strict local maximum.

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172

Problem: Determine all local extrema of the function ݂: ‫ ݔ = ݕ‬ଶ , ‫ܴ א ݔ‬. Solution: We start from the first derivative of the function f, ݂ ´ (‫ = )ݔ‬2‫ݔ‬, ݂ ´´ (‫ = )ݔ‬2, and therefore, for x = 0 ݂ ´ (0) = 0, ݂ ´´ (0) = 2 > 0 Thus ݂: ‫ ݔ = ݕ‬ଶ has a strict local minimum at x = 0. The function does not have other local extremes. Differential calculus can be used especially in physical applications such as kinematics. According to the shape of the trajectory, we divide the motion of the mass point into rectilinear and curvilinear. According to the size of the speed, we divide the movement into even and non-uniform. A special case of uneven movement is uniformly accelerated or uniformly decelerated movement. Problem: For the path along which a mass point moves, ‫ = )ݐ(ݏ‬0,5 ‫ ݐ‬+ 0,6‫ ݐ‬ଶ . Determine: a) the average velocity reached by the mass point from the 2nd second of the motion to the 4th second of the rectilinear motion; b) instantaneous velocity at time ‫ = ݐ‬2‫ݏ‬. Solution: a) for times ‫ݐ‬଴ = 2 s; ‫ = ݐ‬4 s we can calculate the path that the mass point travels during that time: ‫ݐ(ݏ‬଴ ) = ‫( ݏ‬2) = (0.5 ή 2 + 0.6 ή 2ଶ ) m = 3. 4 m ‫( ݏ = )ݐ(ݏ‬4) = (0.5 ή 4 + 0.6 ή 4ଶ ) m = 11. 6 m We calculate the average velocity according to the relation ‫ݒ‬୮ = ‫ݒ‬୮ =

௦ି ௦బ ௧ି ௧బ

, thus:

11. 6 െ 3. 4 m ή s ିଵ = 4. 1 m ή s ିଵ 4െ2

We calculate the instantaneous velocity by deriving the path according to the variable t, i.e.

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‫( = )ݐ( ´ ݏ = )ݐ(ݒ‬0. 5 ‫ ݐ‬+ 0. 6‫ ݐ‬ଶ )´ = 0. 5 + 1. 2‫ݐ‬ Now we substitute the time ‫ݐ‬଴ = 2 s into the relation for velocity, we get ‫(ݒ‬2) = (0.5 + 1. 2 ή 2) m ή s ିଵ = 2. 9 m ή s ିଵ Problem: Determine the acceleration at times ‫ݐ‬଴ = 0 s; ‫ݐ‬ଵ = 1 s; ‫ݐ‬ଶ = 2 s; ‫ݐ‬ଷ = 3 s, if ‫ = )ݐ(ݒ‬3 ‫ ݐ‬ଶ + 8 ‫ ݐ‬െ 2 holds for velocity. Solution: The magnitude of the instantaneous acceleration a at time t is calculated according to the relation ܽ(‫)ݐ( ´´ ݏ = )ݐ( ´ ݒ = )ݐ‬. So ܽ(‫( = )ݐ( ´ ݒ = )ݐ‬3‫ ݐ‬ଶ + 8‫ ݐ‬െ 2)´ = 6‫ ݐ‬+ 8. Then we calculate the value of the derivative of the function at a given point:ܽ(0) = (6 ή 0 + 8)m ή s ିଶ = 8 m ή s ିଶ ; ܽ(1) = (6 ή 1 + 8) m ή s ିଶ = 14 m ή s ିଶ ; ܽ(2) = (6 ή 2 + 8)m ή s ିଶ = 20 m ή s ିଶ ; ܽ(3) = (6 ή 3 + 8)m ή s ିଶ = 26 m ή s ିଶ ;

Vector calculus The physical concept of vectors has an ancient origin in the idea of force as a physical quantity of vector character. By the 4th century BC, Aristotle already had an intuitive idea of the rule of vector composition of forces. It was not until the 16th century that Simon Stevin, a Dutch military engineer, mathematician and physicist, began using it. For the first time, he worked with forces systematically as with vectors, which he represented with arrow lines. A little later, an explicit formulation of the rule on vector composition of forces can be found in the Italian astronomer and mathematician Galileo Galilei. However, the foundations of vector algebra in the plane were laid only at the end of the 18th century and the beginning of the 19th century in connection with the geometric interpretation of complex numbers. The Irish mathematician W. R. Hamilton, who in 1843 introduced the term scalar quantity (scalar), and in 1845 the term vector quantity (vector), deserved a generalized view of vector calculus. Hamilton not only developed the theory of quaternions but arrived at the basics of vector calculus in plane and space. He understood vectors as ordered pairs, resp. triples of real numbers. At the same time, he introduced the concepts of scalar and vector product of vectors.

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174

The axiomatic introduction of the term vector was credited in 1888 by the Italian mathematician Giuseppe Peano. G. Peano gave the basics of the theory of vector space, including the axiomatic definition of real vector space. Definition: By vector space (linear space) we mean the set M whose elements are called vectors, and on which vector operations of addition and multiplying vectors by real numbers are defined. The following axioms are satisfied: 1) ‫ݑ ׊‬, ‫ܯ א ݒ‬: ‫ ݑ‬+ ‫ ݒ = ݒ‬+ ‫ݑ‬, 2) ‫ݑ ׊‬, ‫ݒ‬, ‫ܯ א ݓ‬: (‫ ݑ‬+ ‫ )ݒ‬+ ‫ ݑ = ݓ‬+ (‫ ݒ‬+ ‫)ݓ‬, 3) ‫ܯ א ݋ ׌‬, ‫ܯ א ݑ ׊‬: ‫ ݑ‬+ ‫ݑ = ݋‬, 4) ‫ܯ א ݑ ׊‬, ‫ ׌‬െ ‫ܯ א ݑ‬: ‫ ݑ‬+ (െ ‫ = )ݑ‬0, 5) ‫ܯ א ݑ ׊‬, ‫ܴ א ܽ ׊‬: ܽ (‫ ݑ‬+ ‫ ݑܽ = )ݒ‬+ ܽ‫ݒ‬, 6)‫ܯ א ݑ ׊‬, ܽ, ܾ ‫ܴ א‬: (ܽ + ܾ)‫ ݑܽ = ݑ‬+ ܾ‫ݒ‬, 7) ‫ܯ א ݑ׊‬, ܽ, ܾ ‫ܴ א‬: ܽ(ܾ‫ݑ)ܾܽ( = )ݑ‬, 8) ‫ܯ א ݑ ׊‬: 1‫ݑ = ݑ‬. Many important physical quantities are given not only by a certain size, but also by the direction of action in space. Such quantities, e.g. speed, force, are called vector physical quantities. So-called oriented lines are used for their graphical representation. An oriented line is a line that has a specified start and end point, i.e., its two extreme points determine an ordered pair of points.

Figure A-7. Oriented line

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Most vector physical quantities are bound to a given point in space, which is usually called their point of action. Oriented lines, see Fig. A-7., that graphically represent them have a starting point at this point, i.e. they are bound to this point. If we then introduce for them the operation of addition and multiplication by real numbers in accordance with these operations for bound physical vectors, we get a physical model of the so-called bound geometric vectors, see Fig. A-8.

Figure A-8. Bound geometric vectors

The sum of bound geometric vectors u = PA and v = PB (where u  o, v  o) is the bound geometric vector u + v = PC, which we construct for vectors u, v not lying in one line as the diagonal of the vector parallelogram according to Fig. A-9 and for vectors u, v lying in one line according to Fig. A-10 in the same orientation, or according to Fig. A-11 in the non-matching orientation.

Figure A-9. Non-collinear vectors

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176

Figure A-10. Collinear vectors (same oriented)

Figure A-11. Collinear vectors (not same oriented)

It is further defined for ݇ ‫ ܴ א‬k-multiple of the bound geometric vector u = PA is a bound geometric vector ݇u = PA´ of size |࢛݇| = |݇||‫|´ۯ۾| = |ܝ‬, which has a concordant direction with the vector u, if ݇ > 0 Fig. A-12a, a non-concordant direction than the vector u if ݇ < 0 Fig. A-12b and is zero if ݇ = 0. ݇>0

݇ 0 and the work W is positive. In such cases, it is said

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Appendix

that the body acting by the force F does the work. If ߙ = 90°, then cos ߙ = 0, and therefore W = 0, i.e., the force F does not work. It is useful to express the work W in the form W = F  s, where F  s = F s cos Į is the so-called scalar product of vector quantities F, s. Based on this physical model, the scalar product of two vectors u, v is defined: The scalar product of the vectors u, v (denoted by u  v) is a real number assigned to the vectors u, v as follows: If both vectors u, v are nonzero and the magnitude of the angle of the vectors u, v is ij, then their scalar product u  v is given by ‫ ܝ‬ή ‫ |ܞ| ڄ |ܝ| = ܞ‬cos ߶. If at least one of the vectors u, v is zero, then their scalar product is equal to zero, i.e. for u = o or v = o is u  v = 0. In general, for the scalar product of nonzero vectors u, v we consider a vector ‫ |ܝ| = ୴ܝ‬cos ߶ ‫ܞ‬଴ , where ‫ܞ‬଴ =

‫ܞ‬ |‫|ܞ‬

,

which we call the rectangular projection of the vector u into the vector v. The real number |‫ |ܝ‬cos ) is often called by projecting the vector u into the vector v.

Forensics and Physics a) |‫ |ܝ‬cos ߶ > 0

181 b) |‫ |ܝ‬cos ߶ < 0

c)

Figure A-16. The scalar product of the vectors u, v

The rectangular projection of a vector into a vector has basic properties: for every three nonzero vectors u, v, w and every number ܿ ‫ ܴ א‬it holds (‫ ܝ‬+ ‫ ୵ܝ = ୵)ܞ‬+ ‫ ୵ܞ‬, (࢛ܿ)୴ = c‫୴ܝ‬ Theorem on the basic properties of the scalar product of vectors also holds: For every three vectors u, v, w (in the plane or in space) and every real number c it holds ‫ܝڄܞ=ܞڄܝ‬ (ܿ‫)ܞ ڄ ܝ( ܿ = ܞ ڄ )ܝ‬ ‫ ܞ( ڄ ܝ‬+ ‫ ܞ ڄ ܝ = )ܟ‬+ ‫ܟ ڄ ܝ‬ In conclusion, it is worth mentioning that the use of vectors can also be applied to harmonically time-varying quantities, i.e., quantities whose functional dependence on time t is expressed by the sine function. At each

182

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time t, a certain radius vector (so-called time vector) is assigned to these quantities, the initial value of which at time t = 0 is called the phasor.

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ABOUT THE AUTHORS

Renata Holubova RNDr. Renata Holubova, CSc. (1959) Palacky University in Olomouc graduate (MA: high school teacher - subjects mathematics – physics), PhD dissertation in Physics of condensed matter and acoustics defended at the Masaryk University in Brno. Currently working as a senior lecturer at the Department of Experimental Physics at the Faculty of Science in Olomouc. She is reading lectures in Thermodynamics, Didactics of Physics, Environmental Physics, Theory and Practice of Simple Experiments, Project based learning and teaching in Physics-Nature-Technique, Problems in Teaching Physics, Solving Tasks in Physics. She was thesis supervisor for more than 65 students. Research interest: Inquiry based science education, constructivism, low cost (simply) experiments in physics, motivation in science education, thermodynamics. The research activities were covered and supported by projects in the frame of the Operational Programme for Research and Education and the Czech Science Foundation (GAýR) – Constructivism and its application in integrated concept of science education, Improving quality of science teacher training in European Cooperation, Modules as a means of innovation in the integration of teaching modern physics and chemistry. In international Comenius+ projects and Erasmus+ projects, she is active as a co-researcher in projects Promote and Bridge2Teach. R. Holubova is an author and co-author of secondary school physics textbooks, university study texts, over 120 journal papers. She presents regularly outcomes of the activities at conferences GIREP, The Learner, at the Physics teachers´ invention fairs. Further reading about interdisciplinary relations criminology and science: x “Physics of non-Newtonian fluids and interdisciplinary relations (biology and criminology).” Published 14 December 2017 • © 2017 IOP Publishing Ltd, Physics Education, Volume 53, Number 2.

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x Holubova, R. and Straus, J. 2017. “Cross-Curricular Physics and Criminology Relationship on Example of Handwriting.” US-China Education Review A, July 2017, Vol. 7, No. 7, 323-335.

Jana Slezáková RNDr. Jana Slezáková, Ph.D. (1972) graduated from the Faculty of Science of Palacký University in Olomouc. She is currently an assistant professor at this faculty and also teaches at the Grammar School. She is professionally focused on the didactics of mathematics and the development of geometric imagination of pupils aged 11-16. At the same time, she deals with the issue of educational preparation of students of teaching science and mathematics for secondary schools. She is the co-investigator of many projects focused on the education of future mathematics teachers. She is the author of publications and articles about geometric imagination.

JiĜí Straus Prof. PhDr. JiĜí Straus, DrSc., is a leading European forensic scientist and forensic biomechanics. He is the Vice-Dean for Research and Publications of the Faculty of Legal and Administrative Studies and at the same time the guarantor of the study program Criminalistics and Forensic Disciplines at the University of Finance and Administration. He regularly co-organizes an international scientific conference on Criminal Law and Criminalistics Aspects of Evidence at the University. In the field of science, prof. Straus focuses on the field of forensic biomechanics in crime detection, in which he was appointed forensic expert.