Cambridge International AS & A Level Mathematics Mechanics Worked Solutions Manual with Cambridge Elevate Edition [Solution Manual ed.] 1108758924, 9781108758925

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Cambridge International AS & A Level Mathematics Mechanics Worked Solutions Manual with Cambridge Elevate Edition [Solution Manual ed.]
 1108758924, 9781108758925

Citation preview

Nick Hamshaw and Elaine Dorsett

Cambridge International AS & A Level Mathematics:

Mechanics Worked Solutions Manual

θ 11 180 − θ 8 180 − ϕ 12 ϕ

s (m) 30

0

10

t (s)

s (m) 30

0

20

t (s)

s (m)

0

t (s)

−10

s (m) 300

0

−200

30

50 t (s)

s (m) 200

0

10 t (s)

s (m) 200

0

10 t (s)

s (m) 175 100 75 0

5

15 t (s)

s (m)

1.25 0

−18.75

0.5 1

2.5 t (s)

s (m) 187.5

37.5 0

15 t (s)

5

s (m) 85 60 0

6

11 t (s)

s (m)

0

5

−20

7

t (s)

−60

s (m)

68 50 0

−88 −100

4

27

33 t (s)

s (m) 3000 0

t (s)       

  

s (m)

5000 0

   

40

t (s)

v (m s−1) 15

0

5

t (s)

v (m s−1) 5 2 0

6 t (s)

v (m s−1)

0

t (s)

−4

v (m s−1)

6 0

20 t (s)

v (m s−1)

20

0

4

5

−20 −21

v (m s−1)

0

0.5

5.5 t (s)

−5

v (m s−1)

0

−30

6

21 t (s)

t (s)

v (m s−1) 35

0

7

10.5

t (s)

v (m s−1) 10

0

2

t (s)

7

−8

s (m) 20

0

−20

2

7

t (s)

v (m s−1) 8 6

2 0

2.5

5 t (s)

s (m)

20 17.5

0

2.5

5 t (s)

v (m s−1)

4 0

17 t (s)

5

−6

s (m) 20 0 −16

5

9

17 t (s)

v (m s−1)

10 0

2

3

t (s)

−20

s (m) 20 5 0

2

3

t (s)

v (m s−1)

1.4 0 −1.75 −2.25

2.5

9.5 t (s)

s (m) 5 4.9

0

2.5

9.5 t (s)

v (m s−1) 30

15

0

5

t (s)

v (m s−1)

30

0

T

T

6

t (s)

s (m)

135

54 0

6

15

t (s)

v (m s−1) V

0

T1

T 1 + T2 552 t (s)

60

100

R

p+p+p+p

200

F

2200 F = combined force from sailors

a m s−2 150

200

3 m s−2 450

D

0.2 m s−2 R

1000

a m s−2 120

150

a m s−2 500 10000

a m s−2 1000

20 000

a m s−2 1800

21000

a m s−2 R

10

a m s−2 80 B

a m s−2 20 000

D

a m s−2 1000

D

0.01 a m s−2

0.05 × 10

240

a m s−2

12 × 10

2 a m s−2

0.4 × 10

0.8 a m s−2

0.5 × 10

R

−1.5 m s−2

0.01 × 10

a m s−2

0.3 × 10 + 0.15

0.15 a m s−2

0.3 × 10

2330 a m s−2

70 × 10

a m s−2

2 × 10 + R

R a m s−2

2 × 10

R a m s−2

70 × 10 = 700

R a = −3.5 m s−2

30 × 10 = 300

R

5.12 m s−2

40 × 10 = 400

R a = 2 m s−2

35 × 10 = 350

R + 1800 0 m s−2

200 × 10 = 2000

R a = 0.1 m s−2

0.7 × 10 = 7

R a = −2 m s−2

75 × 10 = 750

R a = 4 m s−2

45 × 10 = 450

R

a = −11 m s−2

4 × 10 = 40

a m s−2 40

R a m s−2

20 × 10 = 200

0.2 m s−2 R

60

0.4

a m s−2

0.3 × 10 = 3

a m s−2 400

4000

a m s−2 1000

4000

−0.015125 m s−2 R

50 a m s−2

60 × 10 = 600

2000 a m s−2

600

a m s−2 D

1200

27 s−2 − m 2

R + 0.1 × 10

0.35 a m s−2

0.1 × 10 = 1

a m s−2 200

a m s−2 200 + 2000 = 2200

7 a m s−2

0.3 × 10 = 3

am s−2 2

10

am s−2 2

310 am s−2

30 × 10 = 300

am s−2 R

R

5.5 m s−2

3 × 10 = 30

v(m s−1)

0 −10 −12

1

1.36 t (s)

15

15sin 40° (up)

40° 15cos 40 ° (Right)

12sin 55° (up)

12

55° 12cos 55° (left)

8cos20° (left) 20° 8sin20° (down)

8

21cos70° (right) 70°

21sin70° (down)

21

13sin35° (right)

13cos35° (up)

13 35°

65° 80cos65° (down)

80

80sin65° (right)

85° 8.5

8.5sin85° (left)

8.5cos85° (down)

120sin70° (up)

120

70°

110°

120cos70° (left)

F

Fy

20° Fx = 10

Fx

Fy = 8

F 25°

F

Fy = 8

θ Fx = 10

25 Fy θ 17

3.8

y = 3

F

θ F

x

R

30 10°

F

20

R 850 θ 700

0 750g

R F 15°

10

50

R

4000

20° T

5000g

F

θ

56

ϕ

G

5

3

17

15

ϕ

θ 4

8

R

1600sinθ

θ

F

400

6.5 144°

36° 6.5cos36°

3.8

19°

161°

3.8cos19°

42cos25°

42 25° 65°

12.3

12.3cos42° 42°

48°

4.5cos31°

4.5

31°

8

θ F 10

50°

R

F

15°

15°

3g

R

50

17° F 17°

8g

R F3 10° F2 15° 10° F1

425

20g

θ 11 180 − θ 8 180 − ϕ 12 ϕ

θ1

30

θ2

40 θ1

30

θ1

5g θ2

40

5g

θ

T θ

40°

20

T θ

7g 20 40° 7g

40 50° 130° F 180° − θ

35

θ

25

100°

150°

110°

T1

T2

T1 120° 90°

10 150°

T2

θ

ϕ

8

9 10

a R

20N 14°

5gN

a = 0.3 m s−2 R 6°

2g N

R 1.2

20 θ

10g

T

40

15° θ

30

3000 N 500 N

20° direction 10° of motion T

North T

2000 30°

0.1

θ

1000

T R

13° 200g N

20°

F

20° 70° 70°

20

20°

a

R

3 40°

2

0.8g

a

30

R

a

15°

10

25g

R a

10°

10° 3g

R 15

a

9° 5

9° 70g

R

a θ mg

θ

130

a

10° 15° T

N W

60° 30°

a

10°

5000

R a F 30°

5

9g

a

R

D

34° 100 34° 1200g

θ R

4g

20

R

θ

120° 200

500 60°

R θ 10° 20

35 110° 10°

60°

θ 15

4.019...

500

250

40° 350

θ 2.712... θ 82.7008...

θ R

θ 91.00122...

238.815...

Wsin50° + 150sin20°

150cos20 ° + Wcos50° 40°

D

F

θ

R

44

60°

θ F 30

a

R

15

40°

5

18g

N 400

360





R

2

T



5° 300g

32 24

θ

20°

45

R T

0.7

13°

13° 18g

R

a 12°

12° 8g

R

a

12° 20 12° 8g

a

R

8 35°

2.4

50g

65 R 16°

a

10°

10° 10g

R

a 10°

10g

R

a

180 × 4

600g

R

F

a θ

θ mg

R

a 30° 30°

mg

R

θ 55

48

v(m s−1)

4 3

0

5

30

40

t(s)

R

a

20°

20°

mg

29

70° 25

17

R 25 N 20°

F

R

F

15°

14g

15°

R

50

15° F

15°

14g

R

F

20 10° 15°

14g

15°

15 R 20°

15° F

15°

14g

R F

14°

14° 4g

P + F

R

25°

25° 3g

R P

25°

F 25°

3g

R + L 400

20° F 20°

100g

R

500 18°

F

200g

R F



200 5°

150g

R

F + 25

20°

20° 30g

100

40°

F

2g + R

200

40°

F

R + 2g

R1

T

60° F

R2 + 3g

R

F

a 15°

15°

3000g

R 50 30°

F

8g

R

a

15 5°

F

2g

2

R 20 10°

F

3g

R F + 50

22°

22° 80g

R 1000

a



F 6°

200g

a

R

F

250

50g

R

a

230

25°

F

50g

R 140

0

F

20°

150

120g

R a

F

150

120g

R

a

10°

F

10° 80g

R

a 15° F

15° 3g

R

20

15° F

15° 3g

R 13 000

F

35°

35° 1250g

R a

F

25°

25° 5g

R F

a 12°

12° 0.2g

R

a F

12°

12° 0.2g

R

F

a 29°

29° 3.5g

R

F

a

T

10° 18°

18° 4g

R

F1

θ F2

θ mg

µ

µ

µ µ

µ µ

µ

µ µ µ

µ µ µ

µ

µ µ µ

µ

R

F1

θ F2

θ mg

R 80

F

15°

17g

C

500 l θ

120g

P

C

l

12° 12° 20g

C

l

13°

4000



5° mg

C

T l θ

12g

µ

R

F2 25°

P + F1

25°

15g

R 40

a

13° F 13°

4.3g

R F a 5° 5°

5000g

R T θ

F

49g

5

3

θ 4

R F

16°

16°

1.3g

R F a 27°

27°

1.3g

R

26

24 39°



3g

R C 34° F

34°

34° 9g

20

R

a 20° F 20°

6g

R F

a 20°

20°

6g

R

F

15°

15° 8g

R

P + F1

25° F2 25°

0.6g

Whole system

Trailer R2

R 40N

400N

120000 N

Tractor R1 T

20000 N

40N T

400N

100000 N

Whole system

Box 2 R2

R

Box 1 R1

20N

250N

T

T

100N

150N

Whole system

Trailer R2

R 40N

300N

120N

40N

28000 N

300N

40000N

T

120N

300N

T 20000 N

Trailer R2

R 80N

Truck R1

8000 N

Whole system

120N

20N

80N

Truck R1 T

20000N

120N

300N

T 20000N

Whole system

Caravan R2

R 80N

D

20N

80N

Car R1 T

32000 N

12000 N

T1 Bucket 1

Whole system T1

T2

120N T2 240N

Bucket 2

120N

20N

D

T 20000 N

Whole system T1 T1

T1

T1 Bar T2

10N

50N

T2 Mass 40N

Whole system R 10 000 N

350 000 N

5 × 2000 N 2 000 000 N Last carriage R5

Next to last carriage R4 T5

2000 N

200 000 N

2000 N T5

T4

200 000 N

Whole system R

1

Caravan

0N 20

N 20 N 80

N 80

θ

θ

25 000 N

6000 N

Whole system

Caravan

0N 25 N 20 N 80

R

T

R1

N 80

R1

T θ

θ 25 000 N

6000 N

Whole system

N 20 N 80

R

DF q

24000N

Caravan

N 80

R1

0N −5 q

6000N

R T

F

T

40 N

30 N

R T1

T2

F

T1

T2 40 N

50 N

R

30 N

T T

F

30° 40N 30N

R2 R1

T

T

F 30 N

30° 40 N

Whole system T1

30N

2 kg box T1

20N T 2 T 3

0.4 kg box T2

4N

0.6 kg box T3

6N

R T T 30N

50N

R

T T

30° 40 N 10 N

µ

T

m kg particle

θ

5 kg particles T

T

θ

10m

50 N

km kg particle T

T 30°

m kg particles T

30°

10km

10m

3 kg mass

2 kg mass

4 kg mass

T2

R

T1

T2

T1

Fr 30 N

20 N

µ

40 N

Particle A R

Particle B T

T R

30° 3N

60° 3N

Cylinder T T

Mass T

40 N

30 N

Overall system T1

10 N

T1

T1

2 N

L ower shelf T1

10 N

T2

T2

T2

10 N

T2

Whole system

Lift

Crate

5000N

5000N

R

10m R

200 N

10m 200 N

Whole system

Lift

Crate

T

T

R

3000N

200N

3000N

R

200N

Whole system

Lift

Crate

T

T

R

4600N

400N

4600N

R

Whole system 4620 N

10m

200 N 100 N 50 N

400N

Third box

Second box

First box

R3

R2

R1

50N

R3 100N

R2

200N

Man

Whole system

820 N

11 070 N

800 N

10m

800 N

Whole system 4992 N

750 N 30 N

20 N

4000 N

Box R

F

Ball T

T

30° 80N

120N

µ

Whole system R 10 N 30 N

Car R1

DF

24 000 N

T 30 N

DF

18 000 N

Crate R

Platform 10 N

150 N

50 N

R

R

A

B T

T

30° 5N

5N

A

B T

R F

T

5 N

µ

25 N

Crate

Ball

T

T

Second pulley T

T θ

θ

200 N

300 N 4 m 2 m θ

10m

µ

Particle A R

F

30°

µ

5N

T

Particle B T

25N

µ

µ

Particle A R1

Particle B R2

T F1 T

30°

20N

F2 20N

v 10 5

0 −5 −10 −15

1

2

3

4

5

6

7

8

9 10 11 12 t

Particle Q

Particle P R F

T T

30N

50N

Whole system R

Engine

Carriage

R1

R2

4N F T

4N 10N BF 100N

T

10N

40N

60N

Travelling up wards R

F 25° 40 N

R F

20° 1N

before

after

5 ms−1

0 ms−1

0.5 ms−1

2.75 ms−1

25 kg

m kg

25 kg

m kg

before −1

12 m s

m1 kg

after −1

8 m s

m2 kg

−1

10 m s

m1 kg

10 m s−1

m2 kg

before 5 m s−1

0 m s−1

0 m s−1

4 kg

1 kg

2 kg

u m s−1

5 kg

after first impact 0 m s−1

2 kg after second impact

v m s−1

5 m s−1

5 kg

2 kg

before

after

4 m s−1

v m s−1

5 m s−1

66 kg

6 kg

60 kg

before

after

u m s−1

2 m s−1

2.3 m s−1

60 kg

40 kg

100 kg

before

after

2 m s−1

1 m s−1

v m s−1

(v + 1) m s−1

0.3 kg

0.5 kg

0.3 kg

0.5 kg

before

after

2 m s−1

1 m s−1

v m s−1

(v + 1) m s−1

0.3 kg

0.5 kg

0.3 kg

0.5 kg

before

after

2 m s−1

1 m s−1

v m s−1

(v + 1) m s−1

0.3 kg

0.5 kg

0.3 kg

0.5 kg

before

after

2 m s−1

1 m s−1

v m s−1

(v + 1) m s−1

0.3 kg

0.5 kg

0.3 kg

0.5 kg

before

after

4 m s−1

4 m s−1

u m s−1

3 kg

2 kg

5 kg

before 5 m s−1

after 7 m s−1

u m s−1

(40000 + 4(50000)) kg (40000 + 2(50000)) kg 2(50000) kg

before −1

after −1

−1

3 m s

0 m s

v m s

4 kg

6 kg

4 kg

4v m s−1

6 kg

before −1

after −1

−1

3 m s

1 m s

v m s

2 kg

5 kg

2 kg

before

v m s−1

5 kg

after

5 m s−1

2 m s−1

3 m s−1

m kg

0.2 kg

(m + 0.2) kg

before

after

4 m s−1

4 m s−1

0.4 m s−1

v m s−1

3 kg

2 kg

3 kg

2 kg

0.03 kg

0 m s−1

sx 30 m sy 20 m s−1

0.06 kg

before

after

15 m s−1

5 m s−1

u m s−1

0.03 kg

0.01 kg

0.04 kg

before −1

4 m s

0.4 kg

1 m s−1

0.4 kg

1 m s−1

0.4 kg

0 m s−1

0.8 m s−1

m kg

0.25 kg

after first impact u m s−1

m kg

after second impact v m s−1

m kg

0.8 m s−1

0.25 kg

1 m s−1

0.25 kg

1 m s−1

0.4 kg

after second impact v m s−1

m kg

1 m s−1

0.25 kg

before

after

−1

−1

0.51 m s

−1

0 m s

0.03 kg

0.06 kg

0.03 kg

before −1

0.01 m s

v m s−1

0.01 m s

0.06 kg

after −1

−1

0.125 m s

0.03 kg

0.15 m s

0.06 kg

0.03 kg

before

v m s−1

0.06 kg

after X and Y collide

1 m s−1

0 m s−1

0 m s−1

v m s−1

3v m s−1

0 m s−1

2 kg

1 kg

2 kg

2 kg

1 kg

2 kg

after Y and Z collide v m s−1

v m s−1

2v m s−1

2 kg

1 kg

2 kg

after X and Y collide again v 4

−1

m s

2 kg

v 2

m s−1

1 kg

2v m s−1

2 kg

R

5N

15° 250 N

R

N 1005 N 15° 250 N

R

1.5

60° 5N

N

R F

15° 50 N

µ

before

after

uA m s−1

uB m s−1

vA m s−1

vB m s−1

m kg

m kg

m kg

m kg

before

after

1 m s−1

0 m s−1

0.4 m s−1

v m s−1

2 kg

1 kg

2 kg

1 kg

before

after

1.2 m s−1

0 m s−1

0.4 m s−1

v m s−1

1 kg

2 kg

1 kg

2 kg

R 0.03R 500 N

µ

R F

45° 600 N

µ

µ

µ µ

µ

R

P 40 N a 200N

R

800 a

DF

N 160 000 N

R

F

DF a 160 000 N

µ

R

200

θ 1000 N

N

µ

v

50° 12 cos 45°

θ 1 m

1 m

h speed = √10 ms−1

0.4 m θ

0.4 m

R F

30° 100 N

µ

µ

µ µ µ

µ

µ

µ

R

DF

00 N 20 10°

18 000 N

R

DF

N 263

8° 12 000 N

R

DF

N 400 2°

10 m

y 3 m

T X

x 1.2 m

20 N

R

T

T

F 40 N

20° 20 N

µ

R

16

DF

θ 16 000 N

00

N

R T 30°

F

250 N

µ  

µ

µ

before 10 m s−1

3 kg

after first collision

0 m s−1

0 m s−1

6 kg

17 kg

2 m s−1

v m s−1

3 kg

6 kg

0 m s−1

17 kg

µ

µ

before

after

5 m s−1

0 m s−1

u m s−1

4 m s−1

4 kg

3 kg

4 kg

3 kg

R

3200N

3° 240 000N

DF

50N R

10°

250N



30N

before

after

3 m s−1

5 m s−1

1.5 m s−1

2.5 m s−1

1.2 kg

m kg

1.2 kg

m kg

velocity

v

100 m 20

30

35

time

R T

F 4N

T

1N

µ