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1
Name: ______________________________
Date: ______________________________
Topic 2 Review
2.1 Kinematics
1. A ball is held at rest in air. The ball is then released. Which one of the following graphs best
shows the variation with time t of the distance d fallen by the ball?
A.
C.
B.
D.
d
d
d
d
t
t
t
t
0
0
0
0
0
0
0
0 (1)
2. The diagram below shows the variation with time t of the velocity v of an object.
v
t0
0
The area between the line of the graph and the time-axis represents
A. the average velocity of the object.
B. the displacement of the object.
C. the impulse acting on the object.
D. the work done on the object. (1)
2
3. The graph shows the variation with time t of the velocity v of an object.
v
t
Which one of the following graphs best represents the variation with time t of the acceleration a
of the object?
0 0
0 0
0 0
0 0
a a
a a
t t
t t
A. B.
C. D.
(1)
4. An object is dropped from rest from a point several hundred metres above the surface of the
Earth at time t = 0. The object strikes the ground at t = T and air resistance is not negligible.
Which of the following sketch graphs best shows the variation with time t, of the speed v of the
object?
B.
D.
A.
C.
T t
TT
T
tt
t
v
vv
v
0
0
0
0
0
000
(1)
3
5. A ball is thrown vertically upwards at time t = 0. Air resistance is not negligible and the
acceleration of free fall is g. The ball reaches a maximum height at time t = T and then
descends, reaching a terminal speed.
Which graph best shows the variation with time t of the acceleration a of the ball?
A. a
+g
0 0 T t
B. a
+g
0 0 T t
C. a
+g
0
–g
0 T t
D. a
+g
0 0 T t
–g –g
–g
(1)
6. Two identical metal spheres are held above the ground as shown.
spheres
(not to scale)
ground
The separation between them is small compared to their distance above the ground. When the
spheres are released, the separation of the spheres will
A. remain constant.
B. decrease continuously.
C. increase continuously.
D. increase initially and then remain constant. (1)
4
7. A car accelerates uniformly from rest. It then continues at constant speed before the brakes are
applied, bringing the car to rest.Which of the following graphs best shows the variation with
time t of the acceleration a of the car?
a
a
a
a
0
0
0
0
0
0
0
0
t
t
t
t
A.
B.
C.
D.
(1)
8. The graph shows the variation with time t of the acceleration a of an object.
0
20
15
10
5
a / ms–2
0 1 2 3 4 5 6 7 8 9 10
t / s
The object is at rest at time t = 0.
5
Which of the following is the velocity of the object at time t = 6.0 s?
A. 0.50 m s–1
.
B. 2.0 m s–1
.
C. 36 m s–1
.
D. 72 m s–1
. (1)
9. A ball, initially at rest, takes time t to fall through a vertical distance h. If air resistance is
ignored, the time taken for the ball to fall from rest through a vertical distance 9h is
A. 3t.
B. 5t.
C. 9t.
D. 10t. (1)
10. The diagram below shows the path of a projectile in the absence of air resistance.
Verticalposition
Horizontal position
Which one of the following diagrams best represents the path of the projectile under the same
initial conditions when the air resistance is taken into account? (The path in absence of air
resistance is shown for comparison as a dotted line.)
Verticalposition
Verticalposition
Verticalposition
Verticalposition
Horizontal position Horizontal position
Horizontal position Horizontal position
A. B.
C. D.
(1)
6
11. A stone is thrown at an angle to the horizontal. Ignoring air resistance, the horizontal component
of the initial velocity of the stone determines the value of
A. range only.
B. maximum height only.
C. range and maximum height.
D. range and time of flight. (1)
12. A stone X is thrown vertically upwards with speed v from the top of a building. At the same
time, a second stone Y is thrown vertically downwards with the same speed v as shown.
Building
X Y
v
v
Air resistance is negligible. Which one of the following statements is true about the speeds with
which the stones hit the ground at the base of the building?
A. The speed of stone X is greater than that of stone Y.
B. The speed of stone Y is greater than that of stone X.
C. The speed of stone X is equal to that of stone Y.
D. Any statement about the speeds depends on the height of the building. (1)
13. Two forces P and Q act at a point X. The individual forces are represented in magnitude and
direction in the diagram below.
Which of the following diagrams best shows the value of S, where S = (P – Q)?
7
P
P P
PQ
Q
Q
Q
S
S S
S
A. B.
C. D.
(1)
14. Two identical metal spheres X and Y are released at the same time from the same height above
the horizontal ground. Sphere X falls vertically from rest. Sphere Y is projected horizontally as
shown below.
ground
X Y
Air resistance is negligible.
Which of the following statements is correct?
A. Sphere X hits the ground before sphere Y because it travels a shorter distance.
B. Sphere Y hits the ground before sphere X because its initial velocity is greater.
C. The spheres hit the ground at the same time because horizontal motion does not affect
vertical motion.
D. The spheres hit the ground at the same time because they have equal weights. (1)
8
15. A stone is thrown with speed v from the top of a cliff of height H, as shown below.
h
v
cliff
H
sea
The stone is thrown at an angle to the horizontal so that it rises to a height h above the top of the
cliff before falling into the sea. The acceleration of free fall is g. Air resistance is negligible.
Which one of the following expressions gives correctly the speed of the stone as it hits the sea?
A. ( )ghv 2+
B. ( )gHv 2+
C. { }( )Hhg +2
D. ( )gHv 22
+
(1)
16. A projectile is fired from the ground at time t = 0. It lands back on the ground at time t = T.
Which of the following sketch graphs best shows the variation with time t of the vertical speed
VV and horizontal speed VH of the projectile? Air resistance is negligible.
speed
speed
speed
speed
T
T
T
T
t
t
t
t
0
0
0
0
0
0
0
0
A. B.
C. D.
VH
VH
VHVH
VVVV
VV
VV
(1)
9
17. A sailing boat is moving with constant velocity v to the right parallel to the dock.
Sailor Hulot, up on the mast, drops his telescope at the moment he is opposite Lucie who is
standing on the dock. Which one of the following best shows the path of the falling telescope as
seen by Lucie?
A. B.
C. D.
(1)
18. The graph below shows how a quantity y varies with time t for a falling object.
y
0 0 t
Which one of the following quantities could be represented by y?
A. Speed when air resistance is negligible
B. Speed when air resistance is not negligible
C. Distance moved from rest when air resistance is negligible
D. Distance moved from rest when air resistance is not negligible (1)
10
19. An archer shoots an arrow at an angle to the horizontal. Air resistance is negligible. Which of
the following graphs best represents the variation with time of the horizontal component of the
arrow’s velocity from the time it is launched to the time just before it hits the ground?
A. velocity
0
B. velocity
00 time
0 time
C. velocity
0
D. velocity
00 time 0 time
(1)
20. A ball is dropped from rest at time t = 0 on to a horizontal surface from which it rebounds.
Which one of the following graphs best shows the variation of speed v of the ball with time t
from the time t = 0 to the time that the ball leaves the surface?
0
0
0
0
v
v
v
v
0
0
0
0
t
t
t
t
A.
C.
B.
D.
(1)
11
21. The graph below shows the variation with time t of the velocity v of an object moving on a
straight-line.
Which of the graphs below best represents the variation with time t of the acceleration a of the
object?
(1)
22. The graph below shows the variation with time t of the acceleration a of an object from t = 0 to
t = T.
a
00 T t
The shaded area under the graph represents change in
A. displacement.
B. velocity.
C. momentum.
D. kinetic energy. (1)
12
23. Which one of the following is a true statement concerning the vertical component of the
velocity and the acceleration of a projectile when it is at its maximum height? (The acceleration
of free fall is g.)
Vertical component of velocity Acceleration
A. maximum zero
B. maximum g
C. zero zero
D. zero g
(1)
24. This question is about linear motion.
A police car P is stationary by the side of a road. A car S, exceeding the speed limit, passes the
police car P at a constant speed of 18 m s–1
. The police car P sets off to catch car S just as car S
passes the police car P. Car P accelerates at 4.5 m s–2
for a time of 6.0 s and then continues at
constant speed. Car P takes a time t seconds to draw level with car S.
(a) (i) State an expression, in terms of t, for the distance car S travels in t seconds.
........................................................................................................................... (1)
(ii) Calculate the distance travelled by the police car P during the first 6.0 seconds of
its motion.
...........................................................................................................................
........................................................................................................................... (1)
(iii) Calculate the speed of the police car P after it has completed its acceleration.
...........................................................................................................................
........................................................................................................................... (1)
(iv) State an expression, in terms of t, for the distance travelled by the police car P
during the time that it is travelling at constant speed.
........................................................................................................................... (1)
(b) Using your answers to (a), determine the total time t taken for the police car P to draw
level with car S.
13
.....................................................................................................................................
.....................................................................................................................................
..................................................................................................................................... (2)
(Total 6 marks)
25. This question is about earthquake waves.
(a) (i) Light is emitted from a candle flame. Explain why, in this situation, it is correct to
refer to the “speed of the emitted light”, rather than its velocity.
...........................................................................................................................
...........................................................................................................................
........................................................................................................................... (2)
(ii) By reference to displacement, describe the difference between a longitudinal wave
and a transverse wave.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
........................................................................................................................... (3)
The centre of an earthquake produces both longitudinal waves (P waves) and transverse waves
(S waves). The graph below shows the variation with time t of the distance d moved by the two
types of wave.
d / km
t / s
P wave S wave1200
800
400
0
0 25 50 75 100 125 150 175 200 225
14
(b) Use the graph to determine the speed of
(i) the P waves.
...........................................................................................................................
...........................................................................................................................
........................................................................................................................... (1)
(ii) the S waves.
...........................................................................................................................
...........................................................................................................................
........................................................................................................................... (1)
The waves from an earthquake close to the Earth’s surface are detected at three laboratories L1,
L2 and L3. The laboratories are at the corners of a triangle so that each is separated from the
others by a distance of 900 km, as shown in the diagram below.
L L
L
1 2
3
900 km
The records of the variation with time of the vibrations produced by the earthquake as detected
at the three laboratories are shown below. All three records were started at the same time.
time
start of trace
L
L
L
1
2
3
On each record, one pulse is made by the S wave and the other by the P wave. The separation of
the two pulses is referred to as the S-P interval.
15
(c) (i) On the trace produced by laboratory L2, identify, by reference to your answers in
(b), the pulse due to the P wave (label the pulse P). (1)
(ii) Using evidence from the records of the earthquake, state which laboratory was
closest to the site of the earthquake.
........................................................................................................................... (1)
(iii) State three separate pieces of evidence for your statement in (c)(ii).
1 .................................................................................................................
.................................................................................................................
2 .................................................................................................................
.................................................................................................................
3 .................................................................................................................
................................................................................................................. (3)
(iv) The S-P intervals are 68 s, 42 s and 27 s for laboratories L1, L2 and L3 respectively.
Use the graph, or otherwise, to determine the distance of the earthquake from each
laboratory. Explain your working.
Distance from L1 = ......................km
...........................................................................................................................
Distance from L2 = ......................km
...........................................................................................................................
Distance from L3 = ......................km
........................................................................................................................... (4)
(v) Mark on the diagram a possible site of the earthquake. (1)
16
There is a tall building near to the site of the earthquake, as illustrated below.
building
ground
direction of vibrations
The base of the building vibrates horizontally due to the earthquake.
(d) (i) On the diagram, draw the fundamental mode of vibration of the building caused by
these vibrations. (1)
The building is of height 280 m and the mean speed of waves in the structure of the building is
3.4 × 103 ms
–1.
(ii) Explain quantitatively why earthquake waves of frequency about 6 Hz are likely to
be very destructive.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
........................................................................................................................... (3)
(Total 21 marks)
17
26. This question is about projectile motion.
A small steel ball is projected horizontally from the edge of a bench. Flash photographs of the
ball are taken at 0.10 s intervals. The resulting images are shown against a scale as in the
diagram below.
00
20
20
40
40
60
60
80
80
100
100
120
140
distance / cm
distance / cm
(a) Use the diagram to determine
(i) the constant horizontal speed of the ball.
...........................................................................................................................
........................................................................................................................... (2)
(ii) the acceleration of free fall.
...........................................................................................................................
...........................................................................................................................
18
(b) Mark on the diagram the position of the ball 0.50 s after projection.
In the space below, you should carry out any calculations so that you can accurately
position the ball.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
..................................................................................................................................... (3)
(c) A second ball is projected from the bench at the same speed as the original ball. The ball
has small mass so that air resistance cannot be neglected. Draw on the diagram the
approximate shape of the path you would expect the ball to take. (3)
(Total 10 marks)
27. This question is about projectile motion.
A ball is projected from ground level with a speed of 28 m s–1
at an angle of 30° to the
horizontal as shown below.
wall h
16m
30°
There is a wall of height h at a distance of 16 m from the point of projection of the ball. Air
resistance is negligible.
(a) Calculate the initial magnitudes of
(i) the horizontal velocity of the ball;
.........................................................................................................................
.........................................................................................................................
......................................................................................................................... (1)
(ii) the vertical velocity of the ball.
.........................................................................................................................
.........................................................................................................................
......................................................................................................................... (1)
19
(b) The ball just passes over the wall. Determine the maximum height of the wall.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
................................................................................................................................... (3)
(Total 5 marks)
28. This question is about projectile motion.
A stone of mass 0.44 kg is thrown horizontally from the top of a cliff with a speed of 22 m s–1
as shown below.
22 m s–1
cliff32 m
sea level
The cliff is 32 m high.
(a) Calculate the total kinetic energy of the stone at sea level assuming air resistance is
negligible.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
................................................................................................................................... (3)
(b) In practice, air resistance is not negligible. During the motion of the stone from the top of
the cliff to sea level, 34% of the total energy of the stone is transferred due to air
resistance. Determine the speed at which the stone reaches sea level.
...................................................................................................................................
20
...................................................................................................................................
................................................................................................................................... (2)
(Total 5 marks)
29. Motion of a ball
A ball of mass 0.25 kg is projected vertically upwards from the ground with an initial velocity
of 30 m s–1
. The acceleration of free fall is 10 m s–2
, but air resistance cannot be neglected.
The graph below shows the variation with time t of the velocity v of this ball for the upward part
of the motion.
30.0
25.0
20.0
15.0
10.0
5.0
0.0
3.02.52.01.51.00.50.0
v / ms–1
t/s
(a) State what the area under the graph represents.
................................................................................................................................... (1)
(b) Estimate the maximum height reached by the ball.
...................................................................................................................................
................................................................................................................................... (1)
21
(c) Determine, for the ball at t = 1.0 s,
(i) the acceleration;
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
......................................................................................................................... (3)
(ii) the magnitude of the force of air resistance.
.........................................................................................................................
.........................................................................................................................
......................................................................................................................... (2)
(d) Use the graph to explain, without any further calculations, that the force of air resistance
is decreasing in magnitude as the ball moves upward.
...................................................................................................................................
...................................................................................................................................
................................................................................................................................... (2)
(e) The diagram below is a sketch graph of the upward motion of the ball.
Draw a line to indicate the downward motion of the ball. The line should indicate the
motion from the maximum height of the ball until just before it hits the ground.
30
20
10
0.0
–10
–20
–30
4.02.00.0 t / s
v / ms–1
(2)
22
(f) State and explain, by reference to energy transformations, whether the speed with which
the ball hits the ground is equal to 30 m s–1
.
...................................................................................................................................
...................................................................................................................................
................................................................................................................................... (2)
(g) Use your answer in (f) to state and explain whether the ball takes 2.0 s to move from its
maximum height to the ground.
...................................................................................................................................
...................................................................................................................................
................................................................................................................................... (2)
(Total 15 marks)
1
Name: ______________________________
Date: ______________________________
Topic 2 Review
2.2 Forces and Dynamics
1. Which one of the following is the condition necessary for an object to be in translational
equilibrium?
A. The lines of action of all the forces acting on the object must pass through a single point.
B. Every force must be balanced by another force that is equal in magnitude but opposite in
direction.
C. The resultant of all the forces acting on the object in any direction must be zero.
D. The total upward force on the object must be equal to the total downward force. (1)
2. A uniform metal bar XY of weight W is hung from a horizontal support at point P by two wires
of negligible mass.
P
TT
X Y
W
Each wire makes an angle θ with the vertical.
Which of the following is equal to the tension T in one of the wires?
A. θ
Wcos
B. θ
Wcos2
C. θ
Wsin
D. θ
Wsin2
(1)
2
3. A bird of weight W lands at the midpoint of a horizontal wire stretched between two poles. The
magnitude of the force exerted by each pole on the wire is F.
F F
W
The bird will be in equilibrium if
A. 2F > W.
B. 2F = W.
C. 2F < W.
D. F = W. (1)
4. A small electrically charged sphere is suspended vertically from a thread. An oppositely charged
rod is brought close to the sphere such that the sphere is in equilibrium when displaced from the
vertical by an angle of 45°.
+ –
Which one of the following best represents the free body diagram for the sphere?
A.
C.
B.
D.
(1)
3
5. A trolley of mass 1.5 kg is pulled along a horizontal table by a force of 5.0N.
force 5.0N
The frictional force acting on the trolley is 0.50N.
The acceleration of the trolley is
A. 0.30 m s–2
.
B. 0.33 m s–2
.
C. 3.0 m s–2
.
D. 3.3 m s–2
. (1)
6. A general expression for Newton’s second law of motion is
F = t
p
∆
∆
What condition is applied so that the law may be expressed in the form F = ma?
A. The mass m is constant.
B. The acceleration a is constant.
C. The force F is constant.
D. The direction of the force F is constant. (1)
7. A block of mass m is pulled along a horizontal, frictionless surface by a force of magnitude F.
The force makes an angle θ with the vertical.
F
block
The magnitude of the acceleration of the block in the horizontal direction produced by the force
F is
A. .m
F
B. .sin
m
θF
C. .cos
m
θF
4
D. .tan
m
θF
(1)
8. A light inextensible string has a mass attached to each end and passes over a frictionless pulley
as shown.
pulley
string
mass m
mass M
The masses are of magnitudes M and m, where m < M. The acceleration of free fall is g. The
downward acceleration of the mass M is
A. ( )( )mM
gmM
+
−.
B. ( )
M
gmM −.
C. ( )( )mM
gmM
−
+.
D. ( )mM
Mg
+.
(1)
9. An elevator (lift) is used to either raise or lower sacks of potatoes. In the diagram, a sack of
potatoes of mass 10 kg is resting on a scale that is resting on the floor of an accelerating
elevator. The scale reads 12 kg.
10 kgelevator
scale
The best estimate for the acceleration of the elevator is
5
A. 2.0 m s–2
downwards.
B. 2.0 m s–2
upwards.
C. 1.2 m s–2
downwards.
D. 1.2 m s–2
upwards. (1)
10. Mandy stands on a weighing scale inside a lift (elevator) that accelerates vertically upwards as
shown in the diagram below. The forces on Mandy are her weight W and the reaction force
from the scale R.
lift
scale
acceleration
The reading of the scale is
A. R + W.
B. W.
C. R.
D. R – W. (1)
11. An object of mass m is initially at rest. An impulse I acts on the object. The change in kinetic
energy of the object is
A. .2
2
m
I
B. .2
m
I
C. I2m.
D. 2I2m.
(1)
6
12. A rocket is fired vertically. At its highest point, it explodes. Which one of the following
describes what happens to its total momentum and total kinetic energy as a result of the
explosion?
Total momentum Total kinetic energy
A. unchanged increased
B. unchanged unchanged
C. increased increased
D. increased unchanged
(1)
13. The engine of a rocket ejects gas at high speed, as shown below.
high speed
gas
rocket
direction of
motion of rocket
The rocket accelerates forwards because
A. the momentum of the gas is equal but opposite in direction to the momentum of the
rocket.
B. the gas pushes on the air at the back of the rocket.
C. the change in momentum of the gas gives rise to a force on the rocket.
D. the ejected gas creates a region of high pressure behind the rocket. (1)
14. This question is about Newton’s laws of motion, the dynamics of a model helicopter and the
engine that powers it.
(a) Explain how Newton’s third law leads to the concept of conservation of momentum in the
collision between two objects in an isolated system.
...................................................................................................................................
...................................................................................................................................
................................................................................................................................... (4)
7
(b) The diagram illustrates a model helicopter that is hovering in a stationary position.
rotatingblades
downward motion of air
0.70 m0.70 m
The rotating blades of the helicopter force a column of air to move downwards. Explain
how this may enable the helicopter to remain stationary.
...................................................................................................................................
...................................................................................................................................
................................................................................................................................... (3)
(c) The length of each blade of the helicopter in (b) is 0.70 m. Deduce that the area that the
blades sweep out as they rotate is 1.5 m2. (Area of a circle = πr
2)
...................................................................................................................................
................................................................................................................................... (1)
(d) For the hovering helicopter in (b), it is assumed that all the air beneath the blades is
pushed vertically downwards with the same speed of 4.0 m s–1
. No other air is disturbed.
The density of the air is 1.2 kg m–3
.
Calculate, for the air moved downwards by the rotating blades,
(i) the mass per second;
.........................................................................................................................
......................................................................................................................... (2)
(ii) the rate of change of momentum.
.........................................................................................................................
......................................................................................................................... (1)
8
(e) State the magnitude of the force that the air beneath the blades exerts on the blades.
................................................................................................................................... (1)
(f) Calculate the mass of the helicopter and its load.
...................................................................................................................................
................................................................................................................................... (2)
(g) In order to move forward, the helicopter blades are made to incline at an angle θ to the
horizontal as shown schematically below.
While moving forward, the helicopter does not move vertically up or down. In the space
provided below draw a free body force diagram that shows the forces acting on the
helicopter blades at the moment that the helicopter starts to move forward. On your
diagram, label the angleθ. (4)
(h) Use your diagram in (g) to explain why a forward force F now acts on the helicopter and
deduce that the initial acceleration a of the helicopter is given by
a = g tanθ
where g is the acceleration of free fall.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
................................................................................................................................... (5)
(i) The helicopter is driven by an engine that has a useful power output of 9.0 × 102 W. The
engine makes 300 revolutions per second. Deduce that the work done in one cycle is 3.0
J.
...................................................................................................................................
................................................................................................................................... (1)
9
(j) The diagram below shows the relation between the pressure and the volume of the air in
the engine for one cycle of operation of the engine.
A
B C
D
pressure
volume
(i) State the name given to the type of process represented by D→A.
......................................................................................................................... (1)
(ii) During one cycle of the engine, the gas absorbs Q1 units of thermal energy and Q2
units of thermal energy are transferred from the gas. On the diagram above, draw
labelled arrows to show these energy transfers. (2)
(iii) The efficiency of the engine is 60%. Using your answer to question (i), calculate
the values of Q1 and Q2.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
......................................................................................................................... (3)
(Total 30 marks)
10
15. This question is about the collision between two railway trucks (carts).
(a) Define linear momentum.
.....................................................................................................................................
..................................................................................................................................... (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.
BA
5.0 ms–1
Immediately before collision
Immediately after collision
BA
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.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
........................................................................................................................... (3)
11
(ii) Calculate the total kinetic energy lost during the collision.
...........................................................................................................................
........................................................................................................................... (2)
(c) Suggest what has happened to the lost kinetic energy.
.....................................................................................................................................
..................................................................................................................................... (2)
(Total 8 marks)
16. This question is about units and momentum.
(a) Distinguish between fundamental units and derived units.
.....................................................................................................................................
..................................................................................................................................... (1)
(b) The rate of change of momentum R of an object moving at speed v in a stationary fluid of
constant density is given by the expression
R = kv2
where k is a constant.
(i) State the derived units of speed v.
........................................................................................................................... (1)
(ii) Determine the derived units of R.
...........................................................................................................................
...........................................................................................................................
........................................................................................................................... (2)
(iii) Use the expression and your answers in (b)(i) and (b)(ii) to determine the derived
units of k.
...........................................................................................................................
........................................................................................................................... (1)
12
(c) Define
(i) linear momentum.
...........................................................................................................................
........................................................................................................................... (1)
(ii) impulse.
...........................................................................................................................
........................................................................................................................... (1)
(d) 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
.
18 m s
carriage, mass 450 kg
9.3m
water-tank 13 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.
...........................................................................................................................
........................................................................................................................... (1)
(ii) Use the answer in (d)(i) to deduce that the mass of water moved in the direction of
motion of the carriage is approximately 120 kg.
...........................................................................................................................
...........................................................................................................................
........................................................................................................................... (2)
13
(iii) Calculate the mean value of the magnitude of the acceleration of the carriage in the
water.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
........................................................................................................................... (3)
(e) For the carriage in (d) passing through the water-tank, determine
(i) its total loss in kinetic energy.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
........................................................................................................................... (3)
(ii) the gain in kinetic energy of the water that is moved in the direction of motion of
the carriage.
...........................................................................................................................
........................................................................................................................... (1)
(f) By reference to the principles of conservation of momentum and of energy, explain your
answers in (e).
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
..................................................................................................................................... (3)
(Total 20 marks)
17. This question is about momentum and energy.
(a) Define impulse of a force and state the relation between impulse and momentum.
definition
.....................................................................................................................................
.....................................................................................................................................
14
relation
.....................................................................................................................................
..................................................................................................................................... (2)
(b) By applying Newton’s laws of motion to the collision of two particles, deduce that
momentum is conserved in the collision.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
..................................................................................................................................... (5)
(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.
24cm
speed Vbullet
plasticine
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.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
........................................................................................................................... (2)
15
(ii) Deduce that the speed V with which the bullet strikes the plasticine is about
160 m s–1
.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
........................................................................................................................... (2)
(iii) Estimate the kinetic energy lost in the impact.
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................ (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
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
..................................................................................................................................... (5)
(Total 19 marks)
18. This question is about momentum and the kinematics of a proposed journey to Jupiter.
(a) State the law of conservation of momentum.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
16
(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.
xenon ionsspeed = 3.0×10 m s
4 2–1spaceshipmass = 5.4×10 kg
(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.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
..................................................................................................................................... (2)
(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)
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
..................................................................................................................................... (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
.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
..................................................................................................................................... (5)
17
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 / ×10 m s– 5
7
– 2
t / ×10 s
10.0
9.5
9.0
8.5
8.00.0 1.0 2.0 3.0 4.0 5.0 6.0
(e) Explain why the acceleration of the spaceship is increasing with time.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
..................................................................................................................................... (2)
(f) Using data from the graph, calculate the speed of the spaceship at the time when the
xenon fuel has all been used.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
..................................................................................................................................... (4)
18
(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.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
..................................................................................................................................... (2)
(Total 19 marks)
1
Name: ______________________________
Date: ______________________________
Topic 2 Review
2.3 Work, Energy, and Power
1. A force stretches a wire that is fixed at one end. The value of this force increases from zero to a
maximum value and then returns to zero. The graph below shows the variation with force F of
the extension x of the wire.
x
0
area R
area Q
area P
0 F
Which area, or areas, represents the net work done on the wire by the force?
A. Area P
B. Area Q
C. Area R
D. Area Q and area R (1)
2. An object of weight 50 N is dragged up an inclined plane at constant speed, through a vertical
height of 12 m. The total work done is 1500 J.
The work done against friction is
A. 2100 J.
B. 1500 J.
C. 900 J.
D. 50 J. (1)
3. A box of mass m is moved horizontally against a constant frictional force f through a distance s
at constant speed v. The work done on the box is
A. 0.
B. mgs.
C. 2
1mv
2.
D. fs.
2
(1)
4. The graph below shows the variation with displacement x of the force F acting on an object. The
force F always acts in the same direction as the displacement.
F
F
x x
Q
00 Q
Q
At point Q, the displacement is xQ and the force is FQ.
Which of the following gives the work done by the force on the body as the displacement
increases from zero to xQ and then returns to zero?
A. Zero
B. QQ21 xF
C. FQ xQ
D. 2FQ xQ
(1)
5. A spring is compressed by a force F.
F
For a compression e, the force F is given by F = ke. When the compression force is removed,
the spring returns to its original length in time t. The best estimate for the power developed by
the spring during its expansion is
A. .2t
ke
B. .t
ke
C. .2
2
t
ke
D. .3
t
ke
(1)
3
6. An object of mass m falls from rest in a vacuum. As the object falls it loses an amount E of
gravitational potential energy. The speed of the object is then
A. m
E2.
B. E
m
2.
C. m
E2.
D. E
m
2.
(1)
7. An electric motor has an input power of 160W. In raising a load, 120W of power is dissipated.
The best estimate for the efficiency of the motor is
A. 25%.
B. 33%.
C. 57%.
D. 75%. (1)
8. Water flows out from a tank down a pipe, as shown below.
tank
water
water flow
pipe
The pipe is always full of water.
Which of the following gives the change in the kinetic energy and in the gravitational potential
energy of the water as the water flows down the pipe?
kinetic energy gravitational potential energy
A. constant decreases
B. constant increases
C. increases decreases
D. increases increases
(1)
4
9. An electric motor is used to raise a weight of 2.0 N. When connected to a 4.0 V supply, the
current in the motor is 1.5 A. Assuming no energy losses, the best estimate for the maximum
steady speed at which the weight can be raised is
A. 0.3 m s–1
.
B. 3.0 m s–1
.
C. 9.0 m s–1
.
D. 12.0 m s–1
. (1)
10. A machine lifts an object of weight W at constant speed through a vertical distance h. The
efficiency of the machine is 25%. The total input energy to the machine is
A. 0.25Wh.
B. 0.75Wh.
C. 2.5Wh.
D. 4.0Wh. (1)
11. A petrol engine P is used to drive a generator G. The energy-flow diagram for this system is
shown below.
input energy
E
energy input to
generator E
electrical energy
generated EOGP
energy losses energy losses
engineP
generatorG
What is the efficiency of this electrical energy generating system?
A. P
G
E
E
B. P
O
E
E
C. G
O
E
E
D. ( )
P
GO
E
EE +
(1)
5
12. This question is about mechanical power and heat engines.
Mechanical power
(a) Define power.
...................................................................................................................................
................................................................................................................................... (1)
(b) A car is travelling with constant speed v along a horizontal straight road. There is a total
resistive force F acting on the car.
Deduce that the power P to overcome the force F is
P = Fv.
...................................................................................................................................
................................................................................................................................... (2)
(c) A car drives up a straight incline that is 4.80 km long. The total height of the incline is
0.30 km.
4.80 km
0.30 km
The car moves up the incline at a steady speed of 16 m s−1
. During the climb, the average
resistive force acting on the car is 5.0 × 102 N. The total weight of the car and the driver
is 1.2 × 104 N.
(i) Determine the time it takes the car to travel from the bottom to the top of the
incline.
.........................................................................................................................
.........................................................................................................................
......................................................................................................................... (2)
(ii) Determine the work done against the gravitational force in travelling from the
bottom to the top of the incline.
......................................................................................................................... (1)
6
(iii) Using your answers to (i) and (ii), calculate a value for the minimum power output
of the car engine needed to move the car from the bottom to the top of the incline.
.........................................................................................................................
.........................................................................................................................
......................................................................................................................... (4)
(d) From the top of the incline, the road continues downwards in a straight-line. At the point
where the incline starts to go downwards, the driver of the car in (c) stops the car to look
at the view. In continuing his journey, the driver decides to save fuel. He switches off the
engine and allows the car to move freely down the incline. The car descends a height of
0.30 km in a distance of 6.40 km before levelling out.
6.40 km
0.30 km
The average resistive force acting on the car is 5.0 × 102 N.
Estimate
(i) the acceleration of the car down the incline;
.........................................................................................................................
.........................................................................................................................
......................................................................................................................... (5)
(ii) the speed of the car at the bottom of the incline.
.........................................................................................................................
......................................................................................................................... (2)
(e) In fact, for the last few hundred metres of its journey down the incline, the car travels at
constant speed. State the value of the frictional force acting on the car whilst it is moving
at constant speed.
................................................................................................................................... (1)
7
The heat engine
(f) The diagram below shows the idealised pressure-volume (P-V) diagram for one cycle of
the gases in an engine similar to that used in the car.
pressure P
B C
D
A
volume V
The changes A → B and C → D are adiabatic changes.
(i) Explain what is meant by an adiabatic change.
.........................................................................................................................
.........................................................................................................................
......................................................................................................................... (2)
(ii) State the name given to the change B → C.
......................................................................................................................... (1)
(g) The useful power output of the engine is 20 kW and the overall efficiency of the engine is
32%. The car engine completes 50 cycles every second. Deduce that QH = 1.3 kJ.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
................................................................................................................................... (3)
(Total 24 marks)
8
13. This question is about power and an ideal gas.
(a) Define power.
.....................................................................................................................................
..................................................................................................................................... (1)
(b) A constant force of magnitude F moves an object at constant speed v in the direction of
the force. Deduce that the power P required to maintain constant speed is given by the
expression
P = Fv
.....................................................................................................................................
.....................................................................................................................................
..................................................................................................................................... (2)
(c) Sand falls vertically on to a horizontal conveyor belt at a rate of 60 kg s–1
.
sand
60 kg s
2.0 m s -1
-1
The conveyor belt that is driven by an engine, moves with speed 2.0 m s–1
.
When the sand hits the conveyor belt, its horizontal speed is zero.
(i) Identify the force F that accelerates the sand to the speed of the conveyor belt.
........................................................................................................................... (1)
(ii) Determine the magnitude of the force F.
...........................................................................................................................
...........................................................................................................................
........................................................................................................................... (2)
9
(iii) Calculate the power P required to move the conveyor belt at constant speed.
...........................................................................................................................
........................................................................................................................... (1)
(iv) Determine the rate of change of kinetic energy K of the sand.
...........................................................................................................................
...........................................................................................................................
........................................................................................................................... (2)
(v) Explain why P and K are not equal.
...........................................................................................................................
...........................................................................................................................
........................................................................................................................... (2)
(d) The engine that drives the conveyor belt in (c) operates in a cycle. In part of this cycle, air
is compressed in a cylinder of the engine such that the pressure and the temperature of the
air increases. Assuming that the air in the cylinder behaves as an ideal gas, outline how
the kinetic model of an ideal gas accounts for this increase in temperature and pressure.
temperature:
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
pressure:
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
..................................................................................................................................... (7)
(Total 18 marks)
10
14. This question is about work, energy and power.
(a) Define the work done by a force.
.....................................................................................................................................
..................................................................................................................................... (2)
A body of mass m is in a gravitational field of strength g. The body is moved through a distance
h at constant speed v in the opposite direction to the field.
(b) Derive an expression in terms of
(i) m, g and h, for the work done on the body;
...........................................................................................................................
...........................................................................................................................
........................................................................................................................... (2)
(ii) m, g and v, for the power required to move the body.
...........................................................................................................................
...........................................................................................................................
........................................................................................................................... (2)
(c) A mass falls near the Earth’s surface at constant speed in still air. Discuss the energy
changes, if any, that occur in the gravitational potential energy and in the kinetic energy
of the mass.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
..................................................................................................................................... (3)
A sample of an ideal gas is contained in a cylinder fitted with a piston, as shown below.
Ideal gasPiston
Cylinder
11
(d) (i) Explain, in terms of molecules, what is meant by the internal energy of the gas.
...........................................................................................................................
........................................................................................................................... (2)
(ii) The piston is suddenly moved inwards, decreasing the volume of the gas. By
considering the speeds of molecules, suggest why the temperature of the gas
changes.
...........................................................................................................................
...........................................................................................................................
........................................................................................................................... (5)
(iii) The gas now expands at constant pressure p so that the volume increases by an
amount ∆V. Derive an expression for the work done by the gas.
...........................................................................................................................
...........................................................................................................................
........................................................................................................................... (4)
An engine operates by using an isolated mass of an ideal gas. The gas is compressed
adiabatically and then it is heated at constant volume. The gas gains 310 J of energy during the
heating process. The gas then expands adiabatically. Finally, the gas is cooled so that it returns
to its original state. During the cooling process, 100 J of energy is extracted. The cycle is shown
below.
100 J
300 K
D
A
B
C
6.1×10
1.0×10
0.32×10 6.0×10–4 –4
3
6
5
pressure / Pa
volume / m
310 J
12
(e) (i) Mark, on the diagram, arrows to show the direction of operation of the stages of the
cycle. (1)
(ii) Using data for point A, calculate the number of moles of gas.
...........................................................................................................................
...........................................................................................................................
........................................................................................................................... (2)
(iii) Determine the temperature of the gas at point B in the cycle.
...........................................................................................................................
...........................................................................................................................
........................................................................................................................... (2)
(iv) State what is represented by the area ABCD on the diagram and give the value of
this quantity.
...........................................................................................................................
........................................................................................................................... (2)
(v) Calculate the efficiency of the engine.
...........................................................................................................................
...........................................................................................................................
........................................................................................................................... (3)
(Total 30 marks)
13
15. This question is about driving a metal bar into the ground and the engine used in the process.
Large metal bars can be driven into the ground using a heavy falling object.
object mass = 2.0×10 kg
bar mass = 400kg
3
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.
.....................................................................................................................................
.....................................................................................................................................
..................................................................................................................................... (4)
(b) Determine the average frictional force exerted by the ground on the bar.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
..................................................................................................................................... (3)
14
(c) The object is raised by a diesel engine that has a useful power output of 7.2 kW.
In order that the falling object strikes the bar at a speed of 6.0 m s–1
, it must be raised to a
certain height above the bar. Assuming that there are no energy losses due to friction,
calculate how long it takes the engine to raise the object to this height.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
..................................................................................................................................... (4)
The diagram below shows the relation between the pressure and the volume of the air in the
diesel engine for one cycle of operation of the engine. During the cycle there are two adiabatic
processes, an isochoric process and an isobaric process.
volume
pressure
thermal energy
B C
D
A
(d) Explain what is meant by
(i) an adiabatic process;
...........................................................................................................................
...........................................................................................................................
........................................................................................................................... (2)
(ii) an isochoric process;
...........................................................................................................................
........................................................................................................................... (1)
15
(iii) an isobaric process.
...........................................................................................................................
........................................................................................................................... (1)
(e) Identify, from the diagram, the following processes.
(i) Adiabatic processes
........................................................................................................................... (1)
(ii) Isochoric process
........................................................................................................................... (1)
(iii) Isobaric process
........................................................................................................................... (1)
During the process B → C thermal energy is absorbed.
The diesel engine has a total power output of 8.4 kW and an efficiency of 40%. The cycle of
operation is repeated 40 times every second.
(f) State what quantity is represented on the diagram by the area ABCD.
..................................................................................................................................... (1)
(g) Determine the value of the quantity that is represented by the area ABCD.
.....................................................................................................................................
..................................................................................................................................... (1)
(h) Determine the thermal energy absorbed during the process B → C.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
..................................................................................................................................... (2)
(Total 22 marks)
1
Name: ______________________________
Date: ______________________________
Topic 2 Review
2.4 Uniform Circular Motion
1. A point mass is moving in a horizontal circle with a velocity of constant magnitude v. At one
particular time, the mass is at P. A short time later, the mass is at Q, as shown below.
P
Q
v
v
Which vector diagram correctly shows the change in velocity ∆v of the mass during this time?
A.
C.
B.
D.
∆
∆ ∆
∆v
v v
v
v
v v
v
v
v v
v
(1)
2. The centripetal force that causes a car to go round a bend in the road is provided by
A. the force produced by the car engine acting on the wheels.
B. the friction between the tyres and the road.
C. the weight of the car.
D. the force exerted by the driver on the steering wheel. (1)
2
3. Points P and Q are at distances R and 2R respectively from the centre X of a disc, as shown
below.
Q
P 2R
R
X
The disc is rotating about an axis through X, normal to the plane of the disc. Point P has linear
speed v and centripetal acceleration a. Which one of the following is correct for point Q?
Linear speed Centripetal acceleration
A. v a
B. v 2a
C. 2v 2a
D. 2v 4a
(1)
4. In a fairground ride, a car of mass M travels on rails around a vertical loop of effective radius R.
At the top of the loop, the speed of the car is v. The car stays in contact with the rails, as shown
below.
v
R
The acceleration of free fall is g.
Which of the following is the correct expression for the force that the rails exert on the car?
A. MgR
Mv −2
B. R
Mv2
C. Mg
D. MgR
Mv +2
(1)
3
5. A brick is placed on the surface of a flat horizontal disc as shown in the diagram below. The
disc is rotating at constant speed about a vertical axis through its centre. The brick does not
move relative to the disc.
Which of the diagrams below correctly represents the horizontal force or forces acting on the
brick?
(1)
6. Two satellites of equal mass, S1 and S2, orbit the Earth. S1 is orbiting at a distance r from the
Earth’s centre at speed v. S2 orbits at a distance 2r from the Earth’s centre at speed 2
v. The
ratio of the centripetal force on S1 to the centripetal force on S2 is
A. 8
1.
B. 4
1.
C. 4.
D. 8. (1)
7. A satellite of mass m and speed v orbits the Earth at a distance r from the centre of the Earth.
The gravitational field strength due to the Earth at the satellite is equal to
A. r
v.
B. r
v2
.
4
C. r
mv.
D. r
mv2
.
(1)
8. Which of the following expressions correctly relates the radius R of the circular orbit of a planet
round a star to the period T of the orbit?
A. R3 ∝ T
2
B. 3
1
R ∝ T
2
C. R2 ∝ T
3
D. 2
1
R ∝ T
3
(1)
9. Which one of the following graphs best represents the variation of the kinetic energy, KE, and
of the gravitational potential energy, GPE, of an orbiting satellite with its distance r from the
centre of the Earth?
0
0
0
0
0
0
0
0
Energy
Energy
Energy
Energy
KE
KE
KE
KE
GPE
GPE
GPE
GPE
r
r
r
r
A.
C.
B.
D.
(1)
5
10. This question is about circular motion.
A stone is attached to an inextensible string. The stone is made to rotate at constant speed v in a
horizontal circle. Diagram 1 below shows the stone in two positions A and B.
Diagram 1 Diagram 2
v
v
B
A
A
Diagram 2 above shows the velocity vector of the stone at point A.
(a) On diagram 2, draw vectors to show the change in velocity ∆v of the stone from point A
to point B. (3)
(b) Use your completed diagram 2 to explain why a force, directed towards the centre of the
circle, is necessary to cause circular motion.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
................................................................................................................................... (2)
(Total 5 marks)
11. This question is about atomic models. The diagram below (not to scale) shows a simple model
of the hydrogen atom in which the electron orbits the proton in a circular path of radius R.
electroncharge –e
protoncharge +e
R
6
(a) On the diagram, draw an arrow to show the direction of
(i) the acceleration of the electron (label this A); (1)
(ii) the velocity of the electron (label this V). (1)
(b) State an expression for the magnitude of the electrostatic force F acting on the electron.
.....................................................................................................................................
..................................................................................................................................... (1)
(c) The orbital speed of the electron is 2.2 × 106 m s
–1.
Deduce that the radius R of the orbit is 5.2 × 10–11
m.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
..................................................................................................................................... (3)
(d) A more complex model of the atom suggests that the orbital radius can only take certain
discrete values. This leads to the idea of discrete energy levels within the atom. Outline
the evidence that supports the existence of discrete energy levels.
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..................................................................................................................................... (3)
(Total 9 marks)
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12. This question is about circular motion.
A linear spring of negligible mass requires a force of 18.0 N to cause its length to increase by
1.0 cm.
A sphere of mass 75.0 g is attached to one end of the spring. The distance between the centre of
the sphere M and the other end P of the unstretched spring is 25.0 cm, as shown below.
25.0 cm
MP
The sphere is rotated at constant speed in a horizontal circle with centre P. The distance PM
increases to 26.5 cm.
(a) Explain why the spring increases in length when the sphere is moving in a circle.
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..................................................................................................................................... (2)
(b) Determine the speed of the sphere.
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..................................................................................................................................... (4)
(Total 6 marks)
13. This question is about gravitation.
A space probe is launched from the equator in the direction of the north pole of the Earth.
During the launch, the energy E given to the space probe of mass m is
E =e
3
4R
GMm
where G is the Gravitational constant and M and Re are, respectively, the mass and radius of the
Earth. Work done in overcoming frictional forces is not to be considered.
(a) (i) Explain what is meant by escape speed.
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(ii) Deduce that the space probe will not be able to travel into deep space.
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........................................................................................................................... (3)
The space probe is launched into a circular polar orbit of radius R.
(b) Derive expressions, in terms of G, M, Re, m and R, for
(i) the change in gravitational potential energy of the space probe as a result of
travelling from the Earth’s surface to its orbit.
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........................................................................................................................... (1)
(ii) the kinetic energy of the space probe when in its orbit.
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........................................................................................................................... (2)
(c) Using your answers in (b) and the total energy supplied to the space probe as given in (a),
determine the height of the orbit above the Earth’s surface.
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A space probe in a low orbit round the Earth will experience friction due to the Earth’s
atmosphere.
(d) (i) Describe how friction with the air reduces the energy of the space probe.
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(ii) Suggest why the rate of loss of energy of the space probe depends on the density of
the air and also the speed of the space probe.
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........................................................................................................................... (2)
(iii) State what will happen to the height of the space probe above the Earth’s surface
and to its speed as air resistance gradually reduces the total energy of the space
probe.
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(Total 18 marks)
14. This question is about satellite motion.
A satellite of mass m orbits a planet of mass M and radius R as shown below. (The diagram is
not to scale.)
planet mass M
R
x
satellite mass m
The radius of the circular orbit of the satellite is x. The planet may be assumed to behave as a
point mass with its mass concentrated at its centre.
(a) Deduce that the linear speed v of the satellite in its orbit is given by the expression
v = x
GM,
where G is the gravitational constant.
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..................................................................................................................................... (2)
(b) (i) Derive expressions, in terms of m, G, M and x, for the kinetic energy of the satellite
and for the gravitational potential energy of the satellite.
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Kinetic energy:
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Gravitational potential energy:
........................................................................................................................... (2)
(ii) Deduce an expression for the total energy of the satellite.
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The satellite is moved into an orbit closer to the planet where there is friction with the planet’s
atmosphere.
(c) (i) State the effect of these frictional forces on the total energy of the satellite.
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(ii) Apply your equation in (b)(ii) to deduce that, as a result of this friction, the radius
of the orbit will change continuously.
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........................................................................................................................... (2)
(iii) Describe the effect of this change in orbital radius on the speed of the satellite.
........................................................................................................................... (1)
(iv) The frictional forces will change as the orbit of the satellite changes. Suggest and
explain the effect on the motion of the satellite of these changing frictional forces.
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........................................................................................................................... (3)
(Total 13 marks)