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
−0.015125m s−2
R
50
a m s−2
60 × 10 = 600
2000
a m s−2
600
a m s−2
D
1200
− 27 m s−2
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 = 0.3m 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
8°
5°
R
2
T
5°
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
5°
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
6°
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°
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°
9°
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
2N
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
5N
µ
25 N
Crate
Ball
T
T
Second pulley
T
T
θ
θ
200 N
300 N
4m
2m
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
ParticleQ
ParticleP
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
5
ms−1
after
0
ms−1
m kg
25 kg
0.5
ms−1
12 m s
m1 kg
m kg
25 kg
before
−1
2.75 ms−1
after
−1
8 ms
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 ms
0 ms
v ms
4 kg
6 kg
4 kg
4v m s−1
6 kg
before
−1
after
−1
−1
3 ms
1 ms
v ms
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 ms
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
0.25kg
after first impact
u m s−1
m kg
after second impact
v m s−1
m kg
0.8 m s−1
0.25kg
1 m s−1
0.25kg
1 m s−1
0.4 kg
after second impact
v m s−1
m kg
1 m s−1
0.25kg
before
after
−1
−1
0.51 m s
−1
0 ms
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
0.125m 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 andZ collide
v m s−1
v m s−1
2v m s−1
2 kg
1 kg
2 kg
after X andY collide again
v
4
−1
ms
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
1m
1m
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
3m
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
2m
s−1
3 kg
v m s−1
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
5°
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