Course Instructor: Momshad sir Worksheet on Energy, Work and Power

Mechanics
Student's Name:........................................................ AS & A LEVEL

9709/04/M/J/06
1 A car of mass 1200 kg travels on a horizontal straight road with constant acceleration a m s−2 .
(i) Given that the car’s speed increases from 10 m s−1 to 25 m s−1 while travelling a distance of 525 m,
find the value of a. [2]
The car’s engine exerts a constant driving force of 900 N. The resistance to motion of the car is constant
and equal to R N.
(ii) Find R. [2]

9709/04/M/J/06
2 A block of mass 50 kg is pulled up a straight hill and passes through points A and B with speeds 7 m s−1
and 3 m s−1 respectively. The distance AB is 200 m and B is 15 m higher than A. For the motion of
the block from A to B, find
(i) the loss in kinetic energy of the block, [2]
(ii) the gain in potential energy of the block. [2]
The resistance to motion of the block has magnitude 7.5 N.
(iii) Find the work done by the pulling force acting on the block. [2]
◦
The pulling force acting on the block has constant magnitude 45 N and acts at an angle α upwards
from the hill.

(iv) Find the value of α . [3]

9709/04/O/N/06
3

A box of mass 8 kg is pulled, at constant speed, up a straight path which is inclined at an angle of 15◦
to the horizontal. The pulling force is constant, of magnitude 30 N, and acts upwards at an angle of
10◦ from the path (see diagram). The box passes through the points A and B, where AB = 20 m and
B is above the level of A. For the motion from A to B, find
(i) the work done by the pulling force, [2]
(ii) the gain in potential energy of the box, [2]
(iii) the work done against the resistance to motion of the box. [1]

9709/04/O/N/06
4 A cyclist travels along a straight road working at a constant rate of 420 W. The total mass of the cyclist
and her cycle is 75 kg. Ignoring any resistance to motion, find the acceleration of the cyclist at an
instant when she is travelling at 5 m s−1 ,
(i) given that the road is horizontal,
(ii) given instead that the road is inclined at 1.5◦ to the horizontal and the cyclist is travelling up the
slope.
[5]
 9709/04/O/N/06
5

The diagram shows the vertical cross-section LMN of a fixed smooth surface. M is the lowest point
of the cross-section. L is 2.45 m above the level of M , and N is 1.2 m above the level of M . A particle
of mass 0.5 kg is released from rest at L and moves on the surface until it leaves it at N . Find
(i) the greatest speed of the particle, [3]
(ii) the kinetic energy of the particle at N . [2]
The particle is now projected from N , with speed v m s−1, along the surface towards M.
(iii) Find the least value of v for which the particle will reach L. [2]

9709/04/M/J/07
6 A car travels along a horizontal straight road with increasing speed until it reaches its maximum speed
of 30 m s−1 . The resistance to motion is constant and equal to R N, and the power provided by the
car’s engine is 18 kW.
(i) Find the value of R. [3]
(ii) Given that the car has mass 1200 kg, find its acceleration at the instant when its speed is 20 m s−1 .
[3]

9709/04/M/J/07
7

A lorry of mass 12 500 kg travels along a road that has a straight horizontal section AB and a straight
inclined section BC. The length of BC is 500 m. The speeds of the lorry at A, B and C are 17 m s−1 ,
25 m s−1 and 17 m s−1 respectively (see diagram).
(i) The work done against the resistance to motion of the lorry, as it travels from A to B, is 5000 kJ.
Find the work done by the driving force as the lorry travels from A to B. [4]

(ii) As the lorry travels from B to C, the resistance to motion is 4800 N and the work done by the
driving force is 3300 kJ. Find the height of C above the level of AB. [4]

9709/04/O/N/07
8 A car of mass 900 kg travels along a horizontal straight road with its engine working at a constant
rate of P kW. The resistance to motion of the car is 550 N. Given that the acceleration of the car is
0.2 m s−2 at an instant when its speed is 30 m s−1, find the value of P. [4]

9709/04/M/J/08
9 A block is being pulled along a horizontal floor by a rope inclined at 20◦ to the horizontal. The tension
in the rope is 851 N and the block moves at a constant speed of 2.5 m s−1.
(i) Show that the work done on the block in 12 s is approximately 24 kJ. [3]

(ii) Hence find the power being applied to the block, giving your answer to the nearest kW. [1]
9709/04/O/N/07
10

The diagram shows the vertical cross-section of a surface. A and B are two points on the cross-section,
and A is 5 m higher than B. A particle of mass 0.35 kg passes through A with speed 7 m s−1 , moving
on the surface towards B.

(i) Assuming that there is no resistance to motion, find the speed with which the particle reaches B.
[3]
(ii) Assuming instead that there is a resistance to motion, and that the particle reaches B with speed
11 m s−1 , find the work done against this resistance as the particle moves from A to B. [3]

9709/04/M/J/08 O
11
2.4 m

50° A C

B
OABC is a vertical cross-section of a smooth surface. The straight part OA has length 2.4 m and
makes an angle of 50◦ with the horizontal. A and C are at the same horizontal level and B is the lowest
point of the cross-section (see diagram). A particle P of mass 0.8 kg is released from rest at O and
moves on the surface. P remains in contact with the surface until it leaves the surface at C. Find
(i) the kinetic energy of P at A, [2]
(ii) the speed of P at C. [2]
The greatest speed of P is 8 m s−1 .

(iii) Find the depth of B below the horizontal through A and C. [3]

9709/04/O/N/08
12 A car of mass 1200 kg is travelling on a horizontal straight road and passes through a point A with
speed 25 m s−1 . The power of the car’s engine is 18 kW and the resistance to the car’s motion is 900 N.
(i) Find the deceleration of the car at A. [4]
(ii) Show that the speed of the car does not fall below 20 m s−1 while the car continues to move with
the engine exerting a constant power of 18 kW. [2]

9709/04/O/N/08
13 A load of mass 160 kg is lifted vertically by a crane, with constant acceleration. The load starts from
rest at the point O. After 7 s, it passes through the point A with speed 0.5 m s−1 . By considering
energy, find the work done by the crane in moving the load from O to A. [6]
9709/04/M/J/09
14
FN
Q

15°
C 100 m

P
A crate C is pulled at constant speed up a straight inclined path by a constant force of magnitude F N,
acting upwards at an angle of 15◦ to the path. C passes through points P and Q which are 100 m apart
(see diagram). As C travels from P to Q the work done against the resistance to C’s motion is 900 J,
and the gain in C ’s potential energy is 2100 J. Write down the work done by the pulling force as C
travels from P to Q, and hence find the value of F . [3]

9709/04/M/J/09
9 m s–1 5 m s–1
15 A 250 m
2.6°
B dm C D
A cyclist and his machine have a total mass of 80 kg. The cyclist starts from rest at the top A of a
straight path AB, and freewheels (moves without pedalling or braking) down the path to B. The path
AB is inclined at 2.6◦ to the horizontal and is of length 250 m (see diagram).

(i) Given that the cyclist passes through B with speed 9 m s−1 , find the gain in kinetic energy and
the loss in potential energy of the cyclist and his machine. Hence find the work done against the
resistance to motion of the cyclist and his machine. [3]
The cyclist continues to freewheel along a horizontal straight path BD until he reaches the point C,
where the distance BC is d m. His speed at C is 5 m s−1 . The resistance to motion is constant, and is
the same on BD as on AB.
(ii) Find the value of d. [3]
The cyclist starts to pedal at C , generating 425 W of power.
(iii) Find the acceleration of the cyclist immediately after passing through C. [3]

9709/42/O/N/09
16 A lorry of mass 15 000 kg moves with constant speed 14 m s−1 from the top to the bottom of a straight
hill of length 900 m. The top of the hill is 18 m above the level of the bottom of the hill. The total
work done by the resistive forces acting on the lorry, including the braking force, is 4.8 × 106 J. Find
(i) the loss in gravitational potential energy of the lorry, [1]
(ii) the work done by the driving force. [1]
On reaching the bottom of the hill the lorry continues along a straight horizontal road against a constant
resistance of 1600 N. There is no braking force acting. The speed of the lorry increases from 14 m s−1
at the bottom of the hill to 16 m s−1 at the point X , where X is 2500 m from the bottom of the hill.

(iii) By considering energy, find the work done by the driving force of the lorry while it travels from
the bottom of the hill to X. [3]

9709/42/O/N/09
17 A car of mass 1250 kg travels along a horizontal straight road with increasing speed. The power
provided by the car’s engine is constant and equal to 24 kW. The resistance to the car’s motion is
constant and equal to 600 N.
(i) Show that the speed of the car cannot exceed 40 m s−1 . [3]

(ii) Find the acceleration of the car at an instant when its speed is 15 m s−1 . [3]
9709/42/M/J/10
18 A car of mass 1150 kg travels up a straight hill inclined at 1.2◦ to the horizontal. The resistance to
motion of the car is 975 N. Find the acceleration of the car at an instant when it is moving with speed
16 m s−1 and the engine is working at a power of 35 kW. [4]

9709/42/M/J/10
19 P and Q are fixed points on a line of greatest slope of an inclined plane. The point Q is at a height of
0.45 m above the level of P. A particle of mass 0.3 kg moves upwards along the line PQ.
(i) Given that the plane is smooth and that the particle just reaches Q, find the speed with which it
passes through P. [3]

(ii) It is given instead that the plane is rough. The particle passes through P with the same speed
as that found in part (i), and just reaches a point R which is between P and Q. The work done
against the frictional force in moving from P to R is 0.39 J. Find the potential energy gained by
the particle in moving from P to R and hence find the height of R above the level of P. [4]

9709/42/O/N/10
20 A cyclist, working at a constant rate of 400 W, travels along a straight road which is inclined at 2◦ to
the horizontal. The total mass of the cyclist and his cycle is 80 kg. Ignoring any resistance to motion,
find, correct to 1 decimal place, the acceleration of the cyclist when he is travelling
(i) uphill at 4 m s−1 ,
(ii) downhill at 4 m s−1 .
[5]

9709/42/O/N/10
21 A block of mass 20 kg is pulled from the bottom to the top of a slope. The slope has length 10 m and
is inclined at 4.5◦ to the horizontal. The speed of the block is 2.5 m s−1 at the bottom of the slope and
1.5 m s−1 at the top of the slope.
(i) Find the loss of kinetic energy and the gain in potential energy of the block. [3]
(ii) Given that the work done against the resistance to motion is 50 J, find the work done by the
pulling force acting on the block. [2]

(iii) Given also that the pulling force is constant and acts at an angle of 15◦ upwards from the slope,
find its magnitude. [2]

9709/42/M/J/11
22 A load is pulled along horizontal ground for a distance of 76 m, using a rope. The rope is inclined at
5◦ above the horizontal and the tension in the rope is 65 N.
(i) Find the work done by the tension. [2]
At an instant during the motion the velocity of the load is 1.5 m s−1 .
(ii) Find the rate of working of the tension at this instant. [2]

9709/42/M/J/11
23 An object of mass 8 kg slides down a line of greatest slope of an inclined plane. Its initial speed at the
top of the plane is 3 m s−1 and its speed at the bottom of the plane is 8 m s−1 . The work done against
the resistance to motion of the object is 120 J. Find the height of the top of the plane above the level
of the bottom. [4]

9709/42/O/N/11
24 A racing cyclist, whose mass with his cycle is 75 kg, works at a rate of 720 W while moving on a
straight horizontal road. The resistance to the cyclist’s motion is constant and equal to R N.

(i) Given that the cyclist is accelerating at 0.16 m s−2 at an instant when his speed is 12 m s−1 , find
the value of R. [3]
(ii) Given that the cyclist’s acceleration is positive, show that his speed is less than 15 m s .
−1
[2]
9709/42/O/N/11
25 A lorry of mass 16 000 kg climbs a straight hill ABCD which makes an angle θ with the horizontal,
where sin θ = 201 . For the motion from A to B, the work done by the driving force of the lorry is

1200 kJ and the resistance to motion is constant and equal to 1240 N. The speed of the lorry is 15 m s−1
at A and 12 m s−1 at B.
(i) Find the distance AB. [5]
For the motion from B to D the gain in potential energy of the lorry is 2400 kJ.
(ii) Find the distance BD. [1]
For the motion from B to D the driving force of the lorry is constant and equal to 7200 N. From B to
C the resistance to motion is constant and equal to 1240 N and from C to D the resistance to motion
is constant and equal to 1860 N.
(iii) Given that the speed of the lorry at D is 7 m s−1 , find the distance BC. [4]

9709/42/M/J/12
26 A block is pulled in a straight line along horizontal ground by a force of constant magnitude acting at
an angle of 60◦ above the horizontal. The work done by the force in moving the block a distance of
5 m is 75 J. Find the magnitude of the force. [3]

9709/42/M/J/12
27 A car of mass 1250 kg travels from the bottom to the top of a straight hill which has length 400 m and
is inclined to the horizontal at an angle of α , where sin α = 0.125. The resistance to the car’s motion
is 800 N. Find the work done by the car’s engine in each of the following cases.
(i) The car’s speed is constant. [4]
(ii) The car’s initial speed is 6 m s , the car’s driving force is 3 times greater at the top of the hill
−1

than it is at the bottom, and the car’s power output is 5 times greater at the top of the hill than it
is at the bottom. [5]

9709/42/O/N/12
28 45 N
14°
A block is pushed along a horizontal floor by a force of magnitude 45 N acting at an angle of 14◦ to
the horizontal (see diagram). Find the work done by the force in moving the block a distance of 25 m.
[3]

9709/42/O/N/12
29 A car of mass 1250 kg moves from the bottom to the top of a straight hill of length 500 m. The top
of the hill is 30 m above the level of the bottom. The power of the car’s engine is constant and equal
to 30 000 W. The car’s acceleration is 4 m s−2 at the bottom of the hill and is 0.2 m s−2 at the top. The
resistance to the car’s motion is 1000 N. Find
(i) the car’s gain in kinetic energy, [5]
(ii) the work done by the car’s engine. [3]

9709/42/M/J/13
30 A and B are two points 50 metres apart on a straight path inclined at an angle  to the horizontal,
where sin  = 0.05, with A above the level of B. A block of mass 16 kg is pulled down the path from
A to B. The block starts from rest at A and reaches B with a speed of 10 m s−1 . The work done by the
pulling force acting on the block is 1150 J.
(i) Find the work done against the resistance to motion. [3]
The block is now pulled up the path from B to A. The work done by the pulling force and the work
done against the resistance to motion are the same as in the case of the downward motion.

(ii) Show that the speed of the block when it reaches A is the same as its speed when it started at B.
[2]

9709/42/M/J/13
31 A car of mass 1000 kg is travelling on a straight horizontal road. The power of its engine is constant
and equal to P kW. The resistance to motion of the car is 600 N. At an instant when the car’s speed is
25 m s−1 , its acceleration is 0.2 m s−2 . Find
(i) the value of P, [4]
(ii) the steady speed at which the car can travel. [3]
 9709/42/O/N/13
32 A box of mass 25 kg is pulled in a straight line along a horizontal floor. The box starts from rest at a
point A and has a speed of 3 m s−1 when it reaches a point B. The distance AB is 15 m. The pulling
force has magnitude 220 N and acts at an angle of !Å above the horizontal. The work done against
the resistance to motion acting on the box, as the box moves from A to B, is 3000 J. Find the value
of !. [5]

9709/42/O/N/13
33 The resistance to motion acting on a runner of mass 70 kg is kv N, where v m s−1 is the runner’s
speed and k is a constant. The greatest power the runner can exert is 100 W. The runner’s greatest
steady speed on horizontal ground is 4 m s−1.
(i) Show that k = 6.25. [2]
(ii) Find the greatest steady speed of the runner while running uphill on a straight path inclined at
an angle ! to the horizontal, where sin ! = 0.05. [4]

9709/42/M/J/14
34 A car of mass 600 kg travels along a straight horizontal road. The resistance to the car’s motion is
constant and equal to R N.
(i) Find the value of R, given that the car’s acceleration is 1.4 m s−2 at an instant when the car’s
speed is 18 m s−1 and its engine is working at a rate of 22.5 kW. [4]

(ii) Find the rate of working of the car’s engine when the car is moving with a constant speed of
15 m s−1 . [1]

9709/42/M/J/14
35
B

A
30

A light inextensible rope has a block A of mass 5 kg attached at one end, and a block B of mass 16 kg
attached at the other end. The rope passes over a smooth pulley which is fixed at the top of a rough
plane inclined at an angle of 30
to the horizontal. Block A is held at rest at the bottom of the plane
and block B hangs below the pulley (see diagram). The coefficient of friction between A and the
1
plane is . Block A is released from rest and the system starts to move. When each of the blocks
3
has moved a distance of x m each has speed v m s−1 .
(i) Write down the gain in kinetic energy of the system in terms of v. [1]

(ii) Find, in terms of x,

(a) the loss of gravitational potential energy of the system, [2]
(b) the work done against the frictional force. [3]

(iii) Show that 21v2 = 220x. [2]

9709/42/O/N/14
36 A train of mass 200 000 kg moves on a horizontal straight track. It passes through a point A with
speed 28 m s−1 and later it passes through a point B. The power of the train’s engine at B is 1.2 times
the power of the train’s engine at A. The driving force of the train’s engine at B is 0.96 times the
driving force of the train’s engine at A.
(i) Show that the speed of the train at B is 35 m s−1 . [2]
(ii) For the motion from A to B, find the work done by the train’s engine given that the work done
against the resistance to the train’s motion is 2.3 × 106 J. [3]
9709/42/M/J/15
37 One end of a light inextensible string is attached to a block. The string makes an angle of 60Å above
the horizontal and is used to pull the block in a straight line on a horizontal floor with acceleration
0.5 m s−2 . The tension in the string is 8 N. The block starts to move with speed 0.3 m s−1 . For the first
5 s of the block’s motion, find
(i) the distance travelled, [2]
(ii) the work done by the tension in the string. [2]

9709/42/M/J/15  
38 A plane is inclined at an angle of sin−1 18 to the horizontal. A and B are two points on the same line
of greatest slope with A higher than B. The distance AB is 12 m. A small object P of mass 8 kg is
released from rest at A and slides down the plane, passing through B with speed 4.5 m s−1 . For the
motion of P from A to B, find
(i) the increase in kinetic energy of P and the decrease in potential energy of P, [3]
(ii) the magnitude of the constant resisting force that opposes the motion of P. [2]

9709/42/O/N/15 A
39

2.5 m

4m

45Å
C
The diagram shows a vertical cross-section ABC of a surface. The part of the surface containing AB
is smooth and A is 2.5 m above the level of B. The part of the surface containing BC is rough and
is at 45Å to the horizontal. The distance BC is 4 m (see diagram). A particle P of mass 0.2 kg is
released from rest at A and moves in contact with the curve AB and then with the straight line BC.
The coefficient of friction between P and the part of the surface containing BC is 0.4. Find the speed
with which P reaches C. [6]

9709/42/O/N/15
 A car of mass 1600 kg moves with constant power 14 kW as it travels along a straight horizontal road.
The car takes 25 s to travel between two points A and B on the road.
(i) Find the work done by the car’s engine while the car travels from A to B. [2]
The resistance to the car’s motion is constant and equal to 235 N. The car has accelerations at A and
B of 0.5 m s−2 and 0.25 m s−2 respectively. Find
(ii) the gain in kinetic energy by the car in moving from A to B, [5]
(iii) the distance AB. [3]

9709/42/M/J/16
41 A particle of mass 8 kg is projected with a speed of 5 m s−1 up a line of greatest slope of a rough plane
5 . The motion of the particle is resisted by a
inclined at an angle ! to the horizontal, where sin ! = 13
constant frictional force of magnitude 15 N. The particle comes to instantaneous rest after travelling
a distance x m up the plane.
(i) Express the change in gravitational potential energy of the particle in terms of x. [2]

(ii) Use an energy method to find x. [4]

9709/42/M/J/16
42 A car of mass 1100 kg is moving on a road against a constant force of 1550 N resisting the motion.
(i) The car moves along a straight horizontal road at a constant speed of 40 m s−1 .
(a) Calculate, in kW, the power developed by the engine of the car. [2]
(b) Given that this power is suddenly decreased by 22 kW, find the instantaneous deceleration
of the car. [3]
(ii) The car now travels at constant speed up a straight road inclined at 8Å to the horizontal, with the
engine working at 80 kW. Assuming the resistance force remains the same, find this constant
speed. [3]

9709/42/O/N/16
43 A girl on a sledge starts, with a speed of 5 m s−1 , at the top of a slope of length 100 m which is at an
angle of 20Å to the horizontal. The sledge slides directly down the slope.
(i) Given that there is no resistance to the sledge’s motion, find the speed of the sledge at the bottom
of the slope. [3]

(ii) It is given instead that the sledge experiences a resistance to motion such that the total work done
against the resistance is 8500 J, and the speed of the sledge at the bottom of the slope is 21 m s−1 .
Find the total mass of the girl and the sledge. [3]

9709/42/O/N/16
44 A van of mass 3000 kg is pulling a trailer of mass 500 kg along a straight horizontal road at a constant
speed of 25 m s−1 . The system of the van and the trailer is modelled as two particles connected by a
light inextensible cable. There is a constant resistance to motion of 300 N on the van and 100 N on
the trailer.
(i) Find the power of the van’s engine. [2]

(ii) Write down the tension in the cable. [1]

−1
The van reaches the bottom of a hill inclined at 4Å to the horizontal with speed 25 m s . The power
of the van’s engine is increased to 25 000 W.

(iii) Assuming that the resistance forces remain the same, find the new tension in the cable at the
instant when the speed of the van up the hill is 20 m s−1 . [5]

9709/42/M/J/17
45 A carend
One starts
of afrom
lightrest and movesstring
inextensible in a straight linetofrom
is attached A with
pointThe
a block. constant
string makesacceleration s−2 the
an angle of 13Åmwith for
10 s. The car
horizontal. Thethen travels
tension in at
theconstant
string isspeed
20 N.for
The30string
s before
pullsdecelerating uniformly,
the block along coming
a horizontal to restatata
surface
point B. The
constant speed s−1isfor
of 1.5 mAB
distance 1.512
km.
s. The work done by the tension in the string is 50 J. Find 1. [3]

9709/42/M/J/17
46 A

5m
O
B D
30Å
6m
C

The diagram shows a wire ABCD consisting of a straight part AB of length 5 m and a part BCD in
the shape of a semicircle of radius 6 m and centre O. The diameter BD of the semicircle is horizontal
and AB is vertical. A small ring is threaded onto the wire and slides along the wire. The ring starts
from rest at A. The part AB of the wire is rough, and the ring accelerates at a constant rate of 2.5 m s−2
between A and B.
(i) Show that the speed of the ring as it reaches B is 5 m s−1 . [1]
The part BCD of the wire is smooth. The mass of the ring is 0.2 kg.

(ii) (a) Find the speed of the ring at C, where angle BOC = 30Å. [4]
(b) Find the greatest speed of the ring. [2]
 9709/42/M/J/17
47 A car of mass 1200 kg is moving on a straight road against a constant force of 850 N resisting the
motion.
(i) On a part of the road that is horizontal, the car moves with a constant speed of 42 m s−1 .
(a) Calculate, in kW, the power developed by the engine of the car. [2]
(b) Given that this power is suddenly increased by 6 kW, find the instantaneous acceleration of
the car. [3]
(ii) On a part of the road that is inclined at 1Å to the horizontal, the car moves up the hill at a constant
speed of 24 m s−1 , with the engine working at 80 kW. Find 1. [4]

9709/42/O/N/17
48 A cyclist is riding up a straight hill inclined at an angle ! to the horizontal, where sin ! = 0.04. The
total mass of the bicycle and rider is 80 kg. The cyclist is riding at a constant speed of 4 m s−1 . There is
a force resisting the motion. The work done by the cyclist against this resistance force over a distance
of 25 m is 600 J.
(i) Find the power output of the cyclist. [4]
−1
The cyclist reaches the top of the hill, where the road becomes horizontal, with speed 4 m s . The
cyclist continues to work at the same rate on the horizontal part of the road.

(ii) Find the speed of the cyclist 10 seconds after reaching the top of the hill, given that the work
done by the cyclist during this period against the resistance force is 1200 J. [4]

9709/42/M/J/18
49 A man has mass 80 kg. He runs along a horizontal road against a constant resistance force of
magnitude P N. The total work done by the man in increasing his speed from 4 m s−1 to 5.5 m s−1
while running a distance of 60 metres is 1200 J. Find the value of P. [4]

9709/42/O/N/18
50 A particle of mass 0.3 kg is released from rest above a tank containing water. The particle falls
vertically, taking 0.8 s to reach the water surface. There is no instantaneous change of speed when
the particle enters the water. The depth of water in the tank is 1.25 m. The water exerts a force on
the particle resisting its motion. The work done against this resistance force from the instant that the
particle enters the water until it reaches the bottom of the tank is 1.2 J.

(i) Use an energy method to find the speed of the particle when it reaches the bottom of the tank.
When the particle reaches the bottom of the tank, it bounces back vertically upwards with initial speed
[4]
−1
7 m s . As the particle rises through the water, it experiences a constant resistance force of 1.8 N.
The particle comes to instantaneous rest t seconds after it bounces on the bottom of the tank.
(ii) Find the value of t. [7]

9709/42/M/J/19
51 5.
A particle of mass 13 kg is on a rough plane inclined at an angle of 1 to the horizontal, where tan 1 = 12
The coefficient of friction between the particle and the plane is 0.3. A force of magnitude T N, acting
parallel to a line of greatest slope, moves the particle a distance of 2.5 m up the plane at a constant
speed. Find the work done by this force. [5]

9709/42/M/J/19
52 A car has mass 1000 kg. When the car is travelling at a steady speed of v m s−1 , where v > 2, the
resistance to motion of the car is Av + B
N, where A and B are constants. The car can travel along a
horizontal road at a steady speed of 18 m s−1 when its engine is working at 36 kW. The car can travel
up a hill inclined at an angle of 1 to the horizontal, where sin 1 = 0.05, at a steady speed of 12 m s−1
when its engine is working at 21 kW. Find A and B. [7]
 9709/42/O/N/19
53 A lorry of mass 25 000 kg travels along a straight horizontal road. There is a constant force of 3000 N
resisting the motion.
(i) Find the power required to maintain a constant speed of 30 m s−1 . [2]
The lorry comes to a straight hill inclined at 2Å to the horizontal. The driver switches off the engine
of the lorry at the point A which is at the foot of the hill. Point B is further up the hill. The speeds of
the lorry at A and B are 30 m s−1 and 25 m s−1 respectively. The resistance force is still 3000 N.
(ii) Use an energy method to find the height of B above the level of A. [5]

9709/42/M/J/20
54 A car of mass 1250 kg is moving on a straight road.
(a) On a horizontal section of the road, the car has a constant speed of 32 m s−1 and there is a constant
force of 750 N resisting the motion.
(i) Calculate, in kW, the power developed by the engine of the car. [2]
(ii) Given that this power is suddenly decreased by 8 kW, find the instantaneous deceleration of
the car. [3]
−1
(b) On a section of the road inclined at sin 0.096 to the horizontal, the resistance to the motion of
the car is 1000 + 8v
N when the speed of the car is v m s−1 . The car travels up this section of
the road at constant speed with the engine working at 60 kW.

Find this constant speed. [5]

9709/42/O/N/20
55 A car of mass 1800 kg is travelling along a straight horizontal road. The power of the car’s engine is
constant. There is a constant resistance to motion of 650 N.

(a) Find the power of the car’s engine, given that the car’s acceleration is 0.5 m s−2 when its speed
is 20 m s−1 . [3]
(b) Find the steady speed which the car can maintain with the engine working at this power. [2]

9709/42/M/J/21
56 A particle of mass 0.6 kg is projected with a speed of 4 m s−1 down a line of greatest slope of a smooth
plane inclined at 10Å to the horizontal.
Use an energy method to find the speed of the particle after it has moved 15 m down the plane. [3]

9709/42/M/J/21
57 A car of mass 1250 kg is pulling a caravan of mass 800 kg along a straight road. The resistances to the
motion of the car and caravan are 440 N and 280 N respectively. The car and caravan are connected
by a light rigid tow-bar.
(a) The car and caravan move along a horizontal part of the road at a constant speed of 30 m s−1 .
(i) Calculate, in kW, the power developed by the engine of the car. [2]
(ii) Given that this power is suddenly decreased by 8 kW, find the instantaneous deceleration of
the car and caravan and the tension in the tow-bar. [4]
(b) The car and caravan now travel along a part of the road inclined at sin−1 0.06 to the horizontal.
The car and caravan travel up the incline at constant speed with the engine of the car working at
28 kW.
(i) Find this constant speed. [3]
(ii) Find the increase in the potential energy of the caravan in one minute. [2]
9709/42/O/N/21 A C
58

1.8 m

The diagram shows a semi-circular track ABC of radius 1.8 m which is fixed in a vertical plane. The
points A and C are at the same horizontal level and the point B is at the bottom of the track. The
section AB is smooth and the section BC is rough. A small block is released from rest at A.
(a) Show that the speed of the block at B is 6 m s−1 . [2]
The block comes to instantaneous rest for the first time at a height of 1.2 m above the level of B. The
work done against the resistance force during the motion of the block from B to this point is 4.5 J.
(b) Find the mass of the block. [3]

9709/42/O/N/21
59 A railway engine of mass 75 000 kg is moving up a straight hill inclined at an angle ! to the horizontal,
where sin ! = 0.01. The engine is travelling at a constant speed of 30 m s−1 . The engine is working
at 960 kW. There is a constant force resisting the motion of the engine.
(a) Find the resistance force. [3]
The engine comes to a section of track which is horizontal. At the start of the section the engine is
travelling at 30 m s−1 and the power of the engine is now reduced to 900 kW. The resistance to motion
is no longer constant, but in the next 60 s the work done against the resistance force is 46 500 kJ.
(b) Find the speed of the engine at the end of the 60 s. [4]

9709/42/M/J/22
60 A car of mass 900 kg is moving up a hill inclined at sin−1 0.12 to the horizontal. The initial speed
of the car is 11 m s−1 . After 12 s, the car has travelled 150 m up the hill and has speed 16 m s−1 . The
engine of the car is working at a constant rate of 24 kW.
(a) Find the work done against the resistive forces during the 12 s. [5]
The car then travels along a straight horizontal road. There is a resistance to the motion of the car of
1520 + 4v
N when the speed of the car is v m s−1 . The car travels at a constant speed with the engine
working at a constant rate of 32 kW.
(b) Find this speed. [3]

9709/42/O/N/22
61 A car of mass 1200 kg is travelling along a straight horizontal road AB. There is a constant resistance
force of magnitude 500 N. When the car passes point A, it has a speed of 15 m s−1 and an acceleration
of 0.8 m s−2 .
(a) Find the power of the car’s engine at the point A. [3]
The car continues to work with this power as it travels from A to B. The car takes 53 seconds to travel
from A to B and the speed of the car at B is 32 m s−1 .
(b) Show that the distance AB is 1362.6 m. [3]

9709/42/M/J/23
62 A particle of mass 1.6 kg is dropped from a height of 9 m above horizontal ground. The speed of the
particle at the instant before hitting the ground is 12 m s−1 .
Find the work done against air resistance. [3]
 9709/42/M/J/23
63

PN 0.6 kg

35Å
A particle of mass 0.6 kg is placed on a rough plane which is inclined at an angle of 35Å to the
horizontal. The particle is kept in equilibrium by a horizontal force of magnitude P N acting in a
vertical plane containing a line of greatest slope (see diagram). The coefficient of friction between the
particle and plane is 0.4.
Find the least possible value of P. [6]

9709/42/O/N/23
64 A railway engine of mass 120 000 kg is towing a coach of mass 60 000 kg up a straight track inclined
at an angle of ! to the horizontal where sin ! = 0.02. There is a light rigid coupling, parallel to the
track, connecting the engine and coach. The driving force produced by the engine is 125 000 N and
there are constant resistances to motion of 22 000 N on the engine and 13 000 N on the coach.
(a) Find the acceleration of the engine and find the tension in the coupling. [5]
−1
At an instant when the engine is travelling at 30 m s , it comes to a section of track inclined upwards
at an angle " to the horizontal. The power produced by the engine is now 4 500 000 W and, as a result,
the engine maintains a constant speed.
(b) Assuming that the resistance forces remain unchanged, find the value of ". [4]

9709/42/O/N/23
65 A block of mass 15 kg slides down a line of greatest slope of an inclined plane. The top of the plane is
at a vertical height of 1.6 m above the level of the bottom of the plane. The speed of the block at the
top of the plane is 2 m s−1 and the speed of the block at the bottom of the plane is 4 m s−1 .
Find the work done against the resistance to motion of the block. [4]

9709/42/M/J/24
66 A car has mass 1400 kg. When the speed of the car is v m s -1 the magnitude of the resistance to motion
is kv 2 N where k is a constant.
(a) The car moves at a constant speed of 24 m s -1 up a hill inclined at an angle of a to the horizontal
where sin a = 0.12 . At this speed the magnitude of the resistance to motion is 480 N.

(i) Find the value of k. [1]

(ii) Find the power of the car’s engine. [3]
(b) The car now moves at a constant speed on a straight level road.

Given that its engine is working at 54 kW, find this speed. [3]