1. This question is about conservation of momentum and conservation of energy.
(a) State Newton’s third law.
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(1)
(b) State the law of conservation of momentum.
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(2)
The diagram below shows two identical balls A and B on a horizontal surface. Ball B is at rest
and ball A is moving with speed V along a line joining the centres of the balls. The mass of each
ball is M.
A B
v
B e f o r e c o l l i s i o n
During the collision of the balls, the magnitude of the force that ball A exerts on ball B is FAB
and the magnitude of the force that ball B exerts on ball A is FBA.
(c) On the diagram below, add labelled arrows to represent the magnitude and direction of the
forces FAB and FBA.
A B
D u r i n g t h e c o l l i s i o n
(3)
The balls are in contact for a time Δt. After the collision, the speed of ball A is +vA and the
speed of ball B is +vB in the directions shown.
A B
A f t e r t h e c o l l i s i o n
vA vB
(d) Use Newton’s second law of motion to deduce an expression relating the forces acting during the collision to the change in momentum of
(i) ball B.
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(2)
(ii) ball A.
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(2)
(e) Apply Newton’s third law and your answers to (d), to deduce that the change in
momentum of the system (ball A and ball B) as a result of this collision, is zero. ... ... ... ... ...
(4)
(f) Deduce, that if kinetic energy is conserved in the collision, then after the collision, ball A
will come to rest and ball B will move with speed V.
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(3) (Total 17 marks)
2. Momentum
(b) An ice hockey puck collides with the wall of an ice rink. The puck is sliding along a line
that makes an angle of 45 to the wall.
w a l l
i c e r i n k
d i r e c t i o n o f p u c k
b e f o r e c o l l i s i o n d i r e c t i o n o f p u c ka f t e c o l l i s i o nr 4 5 4 5
The collision between the wall and the puck is perfectly elastic.
(i) State what is meant by an elastic collision.
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(1)
(ii) Discuss how the law of conservation of momentum applies to this situation.
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(2)
(c) The diagram below is a scale diagram that shows the vector representing the momentum
of the puck before collision. Scale: 1.0 cm = 0.10 N s
By adding appropriate vectors to the diagram, deduce that the magnitude of the change in momentum of the puck as a result of the collision is 0.71 N s.
(d) The sketch-graph below shows the variation with time t of the force F exerted by the wall on the puck.
F
0
0 t
The total contact time is 12 ms. Estimate, explaining your reasoning, the maximum force exerted by the wall on the puck.
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(3) (Total 12 marks)
3. Collisions
A large metal ball is hung from a crane by means of a cable of length 5.8 m as shown below.
c r a n e c a b l e
5 . 8 m
w a l l
In order to knock down a wall, the metal ball of mass 350 kg is pulled away from the wall and
then released. The crane does not move. The graph below shows the variation with time t of the
speed v of the ball after release.
v
/
m
s
–1
0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4
t / s 3 . 0
2 . 0
1 . 0
0 . 0
The ball makes contact with the wall when the cable from the crane is vertical.
(a) For the ball just before it hits the wall,
(i) state why the tension in the cable is not equal to the weight of the ball;
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(1)
(ii) by reference to the graph, estimate the tension in the cable. The acceleration of free
fall is 9.8 m s–2.
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(b) Use the graph to determine the distance moved by the ball after coming into contact with the wall.
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(2)
(c) For the collision between the ball and the wall, calculate
(i) the total change in momentum of the ball;
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(2)
(ii) the average force exerted by the ball on the wall.
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(2)
(d) (i) State the law of conservation of momentum.
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(2)
(ii) The metal ball has lost momentum. Discuss whether the law applies to this
situation.
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(e) During the impact of the ball with the wall, 12 of the total kinetic energy of the ball is converted into thermal energy in the ball. The metal of the ball has specific heat capacity
450 J kg–1 K–1. Determine the average rise in temperature of the ball as a result of
colliding with the wall.
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(4) (Total 18 marks)
4. This question is about momentum and energy.
(a) Define impulse of a force and state the relation between impulse and momentum.
definition
... ... relation
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(2)
(b) By applying Newton’s laws of motion to the collision of two particles, deduce that
momentum is conserved in the collision.
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(c) In an experiment to measure the speed of a bullet, the bullet is fired into a piece of plasticine suspended from a rigid support by a light thread.
2 4 c m
s p e e d V
b u l l e t
p l a s t i c i n e
The speed of the bullet on impact with the plasticine is V. As a result of the impact, the
bullet embeds itself in the plasticine and the plasticine is displaced vertically through a
height of 24 cm. The mass of the bullet is 5.2×10–3 kg and the mass of the plasticine is
0.38 kg.
(i) Ignoring the mass of the bullet, calculate the speed of the plasticine immediately
after the impact.
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(2)
(ii) Deduce that the speed V with which the bullet strikes the plasticine is about
160 m s–1.
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(iii) Estimate the kinetic energy lost in the impact.
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(3)
(d) Another bullet is fired from a different gun into a large block of wood. The block remains
stationary after impact and the bullet melts completely. The temperature rise of the block is negligible. Use the data to estimate the minimum impact speed of the bullet.
mass of bullet = 5.3×10–3 kg
specific heat capacity of the material of the bullet = 130 J kg–1 K–1
latent heat of fusion of the material of the bullet = 870 J kg–1
melting point of the material of the bullet = 330°C
initial temperature of bullet = 30°C
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5. This question is about momentum.
(a) Define
(i) linear momentum.
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(1)
(ii) impulse.
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(1)
(b) In a ride in a pleasure park, a carriage of mass 450 kg is travelling horizontally at a speed
of 18 m s–1. It passes through a shallow tank containing stationary water. The tank is of
length 9.3 m. The carriage leaves the tank at a speed of 13 m s–1.
1 8 m s c a r r i a g e , m a s s 4 5 0 k g
9 . 3 m
w a t e r - t a n k 1 3 m s– 1 – 1
As the carriage passes through the tank, the carriage loses momentum and causes some
water to be pushed forwards with a speed of 19 m s–1 in the direction of motion of the
carriage.
(i) For the carriage passing through the water-tank, deduce that the magnitude of its
total change in momentum is 2250N s.
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(1)
(ii) Use the answer in (b)(i) to deduce that the mass of water moved in the direction of
motion of the carriage is approximately 120 kg.
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(iii) Calculate the mean value of the magnitude of the acceleration of the carriage in the water.
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(3)
(c) For the carriage in (b) passing through the water-tank, determine
(i) its total loss in kinetic energy.
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(3)
(ii) the gain in kinetic energy of the water that is moved in the direction of motion of
the carriage.
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(1)
(d) By reference to the principles of conservation of momentum and of energy, explain your
answers in (c).
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6. This question is about linear motion.
A car moves along a straight road. At time t = 0 the car starts to move from rest and oil begins to
drip from the engine of the car. One drop of oil is produced every 0.80 s. Oil drops are left on the road. The position of the oil drops are drawn to scale on the grid below such that 1.0 cm
represents 4.0 m. The grid starts at time t = 0.
d i r e c t i o n o f m o t i o n
1 . 0 c m
(a) (i) State the feature of the diagram above which indicates that, initially, the car is
accelerating.
...
(1)
(ii) On the grid above, draw further dots to show where oil would have dripped if the
drops had been produced from the time when the car had started to move.
(2)
(iii) Determine the distance moved by the car during the first 5.6 s of its motion. ... ...
(1)
(b) Using information from the grid above, determine for the car,
(i) the final constant speed.
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(2)
(ii) the initial acceleration.
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7. This question is about the collision between two railway trucks (carts).
(a) Define linear momentum.
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(1)
In the diagram below, railway truck A is moving along a horizontal track. It collides with a stationary truck B and on collision, the two join together. Immediately before the collision, truck
A is moving with speed 5.0 ms–1. Immediately after collision, the speed of the trucks is v.
B A
5 . 0 m s– 1
I m m e d i a t e l y b e f o r e c o l l i s i o n
I m m e d i a t e l y a f t e r c o l l i s i o n B A
v
The mass of truck A is 800 kg and the mass of truck B is 1200 kg.
(b) (i) Calculate the speed v immediately after the collision.
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(3)
(ii) Calculate the total kinetic energy lost during the collision.
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(c) Suggest what has happened to the lost kinetic energy.
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(2) (Total 8 marks)
8. This question is about driving a metal bar into the ground.
Large metal bars can be driven into the ground using a heavy falling object.
o b j e c t
m a s s = 2 . 0 × 1 0 k g3
b a r
m a s s = 4 0 0 k g
In the situation shown, the object has a mass 2.0 × 103 kg and the metal bar has a mass of
400 kg.
The object strikes the bar at a speed of 6.0 m s–1. It comes to rest on the bar without bouncing.
As a result of the collision, the bar is driven into the ground to a depth of 0.75 m.
(a) Determine the speed of the bar immediately after the object strikes it.
(b) Determine the average frictional force exerted by the ground on the bar.
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(3) (Total 7 marks)
9. This question is about momentum and the kinematics of a proposed journey to Jupiter.
(a) State the law of conservation of momentum.
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(2)
A solar propulsion engine uses solar power to ionize atoms of xenon and to accelerate them. As a result of the acceleration process, the ions are ejected from the spaceship with a speed of
3.0 × 104 m s–1.
x e n o n i o n s
s p e e d = 3 . 0 × 1 0 m s4 – 1 s p a c e s h i p 2
m a s s = 5 . 4 × 1 0 k g
(b) The mass (nucleon) number of the xenon used is 131. Deduce that the mass of one ion of
xenon is 2.2 × 10–25 kg.
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(c) The original mass of the fuel is 81 kg. Deduce that, if the engine ejects 77 × 1018 xenon
ions every second, the fuel will last for 1.5 years. (1 year = 3.2 × 107 s)
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(2)
(d) The mass of the spaceship is 5.4 × 102 kg. Deduce that the initial acceleration of the
spaceship is 8.2 × 10–5 m s–2.
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(5)
The graph below shows the variation with time t of the acceleration a of the spaceship. The solar
propulsion engine is switched on at time t = 0 when the speed of the spaceship is 1.2 × 103 m s–
1.
a / × 1 0 m s– 5 – 2
1 0 . 0
9 . 5
9 . 0
(e) Explain why the acceleration of the spaceship is increasing with time.
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(2)
(f) Using data from the graph, calculate the speed of the spaceship at the time when the
xenon fuel has all been used.
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(4)
(g) The distance of the spaceship from Earth when the solar propulsion engine is switched on
is very small compared to the distance from Earth to Jupiter. The fuel runs out when the
spaceship is a distance of 4.7 × 10–11 m from Jupiter. Estimate the total time that it would
take the spaceship to travel from Earth to Jupiter.
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(2) (Total 19 marks)
10. Momentum
(a) State the law of conservation of linear momentum.
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(b) A toy rocket of mass 0.12 kg contains 0.59 kg of water as shown in the diagram below. h i g h - p r e s s u r e a i r
n o z z l e , r a d i u s 1 . 4 m m
w a t e r
The space above the water contains high-pressure air. The nozzle of the rocket has a circular cross-section of radius 1.4 mm. When the nozzle is opened, water emerges from
the nozzle at a constant speed of 18 m s–1. The density of water is 1000 kg m–3.
(i) Deduce that the volume of water ejected per second through the nozzle is 1.1 10–4
m3.
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(2)
(ii) Deduce that the upward force that the ejected water exerts on the rocket is
approximately 2.0 N. Explain your working by reference to Newton’s laws of motion.
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(4)
(iii) Calculate the time delay between opening the nozzle and the rocket achieving lift-off.
