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A series of physics problems involving motion, forces, and calculus. The problems cover various topics such as projectile motion, harmonic motion, collisions, and friction. Students can use this document as study notes, summaries, or schemes and mind maps to prepare for exams or assignments.
Typology: Exams
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1. (a) A particle falls from rest from a point P. When it has fallen a distance 19·6 m a second particle is projected vertically downwards from P with initial velocity 39·2 m s−^1. The particles collide at a distance d from P.
Find the value of d.
1 2 2 1 2 2
1 2 2
1 2 2
1 2 1 2 2 2 2 2
1 2 2
2 s
44·1 m
s ut ft gt t
d g t
d t gt
g t t gt t t t t
t
d g t
1. (b) A car, starts from rest at A , and accelerates uniformly at 1 m s−^2 along a
straight level road towards B , where AB = 1914 m. When the car reaches its maximum speed of 32 m s−^1 , it continues at this speed for the rest of the journey.
At the same time as the car starts from A a bus passes B travelling towards A with a constant speed of 36 m s−^1. Twelve seconds later the bus starts to decelerate uniformly at 0·75 m s−^2.
(i) The car and the bus meet after t seconds. Find the value of t.
(ii) Find the distance between the car and the bus after 48 seconds.
{ ( )} { ( ) ( )( )}
40 s
bus
(i) car 2
2
2 4 3 2 1
2 2 1
2
2 2
t
t t
t t t
s s
s
s ut ft
t s
t s
v u ft v u fs
c b
1 2 2
1 2 2 3 1 3 4 2 4
(ii) car 32 8 0 256
bus 36 40 12 15 8 64
s ut ft
v u ft s ut ft
distance 256 96 352 m
v = s =
2 (b) At noon ship A is 50 km north of ship B.
Ship A is travelling southwest at 24 2 kmh−^1. Ship B is travelling due west at 17 kmh−^1. (i) Find the magnitude and direction of the velocity of B relative to A.
A and B can exchange signals when they are not more than 20 km apart.
(ii) At what time can they begin to exchange signals? (iii) How long can they continue to exchange signals?
( ) ( )
− 24
25 tan
(i)
BA
BA B A
i j
i j i j
2 2
(ii) 50sin 50 14 25 20 14 2 51 1· 50cos 14· 24 50 14·2829 33· 25
time 13 : 21
(iii) 2 28·
BC
CE
t
t
time = 1 h 9 min
θ B
The range of the particle is 940·8 m. Find
(ii) the difference between the two times of flight.
2
(i) 98cos. 940· 9· cos
98sin. 4·9 0 9· 98sin 4·9 0 cos
sin 2 0· 2 73·74 , 106·
t
t
t t
1
2
2 1
(ii) cos 3· 12·
cos 53· 16·
4 s.
t
t
t t
4. (a) Two particles A and B each of mass m are connected by a light inextensible string passing over a light, smooth, fixed pulley. Particle A rests on a rough plane inclined
tan. 12
Particle B hangs vertically 1 m above the ground. The coefficient of friction between A and the inclined plane is
The system is released from rest. (i) Find the speed with which B strikes the ground. (ii) How far will A travel after B strikes the ground?
1 2
2 2
2 2
(i) sin cos
(ii) 2 2 0
mg T mf T mg mg mf
mg mg mg mf
g f
v u fs g
g v
v u fs g
m. 11
g s
s
1 m
α
m g
μR
mgcosα
mgsinα
4 (b) Two particles of mass m kg and 2 m kg lie at rest on horizontal rough tables. The coefficient of friction between each
particle and the table it lies on is
μ μ
The particles are connected by a light inextensible string which passes under a smooth movable pulley of mass 4 m kg. The system is released from rest. (i) Find, in terms of m and (^) μ , the tension in the string.
(ii) If the acceleration of the m kg mass is f , find the acceleration of the 2 m kg mass in terms of f.
1 2
(i) 2 2 4 2 4 2 2
(ii) 4 1 5 4 5 5
T mg mf T mg mp mg T m f p mf mp
mg T T mg T mg mg T
mf T mg mg mg
g g f
mp
T mg mg mp mg
g g p
p f g
m
4 m
2 m
5 (b) A smooth sphere P collides with an identical smooth sphere Q which is at rest. The velocity of P before impact makes an
where 0 ° ≤ α< 90 .°
The velocity of P is deflected through an angle θ by the collision. The coefficient of restitution between the spheres is
Show that (^2) 2 tan tan. 1 3 tan
1 3 tan
2 tan tan
tan tan 3 tan 3 tan tan
3 tan
cos
3 sin 1 tan tan
tan tan
sin tan
cos
cos 0 3
PCM cos 0
2
2
1
1
1 2
1 2
u
u
v
u
u v
v v u
m u m mv mv 5
P^ θ Q
6. (a) A particle of mass 0·5 kg is suspended from a fixed point P by a spring which executes simple harmonic motion with amplitude 0·2 m. The period of the motion is 2 seconds.
Find (i) the maximum acceleration of the particle
(ii) the greatest force exerted by the spring correct to one place of decimals.
2 2 2
2
2
i
a A
ii F m a
T mg
g T
7. (a) A uniform wire ABC is bent at right angles at B. When it is suspended from B the parts AB and BC make angles of 30 and 60 respectively with the vertical.
If the mass per unit length of the wire is m and AB = hBC find the value of h.
1 2 1 2
2
sin 30 sin 60
mgh BC h BC mg BC BC
h
h
7. (b) Two rough rings of equal weight W are a distance d apart on a horizontal rod. A light smooth inelastic string of length 2 connects the rings. Another ring of weight 2 W slides on the string. The coefficient of friction between the rough rings and the rod is μ.
Show that the system remains at rest if (^2)
d
μ μ
( )
1
2 2 1 1
2 2
2 2
2 2 2 2 2 2 2 2 2
2
2
2 cos 2 cos
cos 2
d d d R W R
d d d W W W
d d
d d d
d
d
d
μR 1 W
d
μR 2
θ
θ θ
T
8. (b) A string is wrapped around a smooth pulley wheel of radius r. A particle of mass m is attached to the string.
The axis of rotation of the wheel is horizontal, perpendicular to the wheel, and passes through the centre of the wheel.
The moment of inertia of the wheel about the axis is I. The particle is released from rest and moves vertically downwards. (i) Find, in terms of I , m and r , the tension in the string.
(ii) If the acceleration of the particle is , 5
g find the mass of the pulley
wheel in terms of m.
M m
mr Mr
mr I mr
g I mr
mgr f
I mr
mgI T
I mr
mr mg
T mg mf
mg T mf
I mr
mgr
m r
mg f
m mg r
f
m fh mgh r
fh I
fh
v u fs
mv mgh r
v I
I mv mgh
(ii)
(i)
2 2
(^21)
2 2
2
2
2
2
2
2
2
2
2
2
1 (^22)
1
2 2
2 2 1 2
2 2 1
2 2
(^21) 2 1
m
9. (a) Stainless steel is an alloy of iron, chromium and nickel. A piece of stainless steel consists of 70% iron, 20% chromium and 10% nickel by volume. The relative densities of iron, chromium and nickel are 7·8, 7·2 and 8· respectively. Find the relative density of stainless steel.
m I mC mN mSS
s
