1 A car of mass 1200 kg travels along a horizontal straight road. The power of the car’s engine is 20 kW.

The resistance to the car’s motion is 400 N.

(i) Find the speed of the car at an instant when its acceleration is 0.5 m s−2 . [4]

(ii) Show that the maximum possible speed of the car is 50 m s−1 . [2]

The work done by the car’s engine as the car travels from a point A to a point B is 1500 kJ.

(iii) Given that the car is travelling at its maximum possible speed between A and B, find the time
taken to travel from A to B. [2]
9709/04/M/J/04

2 A car of mass 1250 kg travels down a straight hill with the engine working at a power of 22 kW. The
hill is inclined at 3◦ to the horizontal and the resistance to motion of the car is 1130 N. Find the speed
of the car at an instant when its acceleration is 0.2 m s−2 . [5]
9709/04/O/N/04

3 A small block is pulled along a rough horizontal floor at a constant speed of 1.5 m s−1 by a constant
force of magnitude 30 N acting at an angle of θ ◦ upwards from the horizontal. Given that the work
done by the force in 20 s is 720 J, calculate the value of θ . [3]
9709/04/M/J/05

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]
© UCLES 2006 9709/04/O/N/06

5 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]
© UCLES 2007 9709/04/M/J/07
6 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]
© UCLES 2007 9709/04/O/N/07

7 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]

© UCLES 2008 9709/04/M/J/08

8 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]
© UCLES 2008 9709/04/O/N/08

9 A car of mass 1000 kg moves along a horizontal straight road, passing through points A and B. The
power of its engine is constant and equal to 15 000 W. The driving force exerted by the engine is 750 N
at A and 500 N at B. Find the speed of the car at A and at B, and hence find the increase in the car’s
kinetic energy as it moves from A to B. [4]
9709/41/O/N/09

10 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/O/N/09

11 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]
9709/41/M/J/10
12 A car of mass 600 kg travels along a horizontal straight road, with its engine working at a rate of
40 kW. The resistance to motion of the car is constant and equal to 800 N. The car passes through the
point A on the road with speed 25 m s−1 . The car’s acceleration at the point B on the road is half its
acceleration at A. Find the speed of the car at B. [5]

9709/41/O/N/10

13 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

14 A car of mass 700 kg is travelling along a straight horizontal road. The resistance to motion is constant
and equal to 600 N.

(i) Find the driving force of the car’s engine at an instant when the acceleration is 2 m s−2 . [2]

(ii) Given that the car’s speed at this instant is 15 m s−1 , find the rate at which the car’s engine is
working. [2]
9709/41/M/J/11

15 A load of mass 1250 kg is raised by a crane from rest on horizontal ground, to rest at a height of
1.54 m above the ground. The work done against the resistance to motion is 5750 J.

(i) Find the work done by the crane. [3]

(ii) Assuming the power output of the crane is constant and equal to 1.25 kW, find the time taken to
raise the load. [2]
9709/41/M/J/11

16 A car of mass 1250 kg is travelling along a straight horizontal road with its engine working at a
constant rate of P W. The resistance to the car’s motion is constant and equal to R N. When the
speed of the car is 19 m s−1 its acceleration is 0.6 m s−2 , and when the speed of the car is 30 m s−1 its
acceleration is 0.16 m s−2 . Find the values of P and R. [6]
9709/43/M/J/11

17 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
18 A car of mass 600 kg travels along a straight horizontal road starting from a point A. The resistance
to motion of the car is 750 N.

(i) The car travels from A to B at constant speed in 100 s. The power supplied by the car’s engine
is constant and equal to 30 kW. Find the distance AB. [3]

(ii) The car’s engine is switched off at B and the car’s speed decreases until the car reaches C with a
speed of 20 m s−1 . Find the distance BC. [3]

(iii) The car’s engine is switched on at C and the power it supplies is constant and equal to 30 kW.
The car takes 14 s to travel from C to D and reaches D with a speed of 30 m s−1 . Find the distance
CD. [4]
9709/43/O/N/11

19 A car of mass 880 kg travels along a straight horizontal road with its engine working at a constant
rate of P W. The resistance to motion is 700 N. At an instant when the car’s speed is 16 m s−1 its
acceleration is 0.625 m s−2 . Find the value of P. [4]
9709/41/M/J/12

20 A car of mass 1200 kg moves in a straight line along horizontal ground. The resistance to motion of
the car is constant and has magnitude 960 N. The car’s engine works at a rate of 17 280 W.

(i) Calculate the acceleration of the car at an instant when its speed is 12 m s−1 . [3]

The car passes through the points A and B. While the car is moving between A and B it has constant
speed V m s−1 .

(ii) Show that V = 18. [2]

At the instant that the car reaches B the engine is switched off and subsequently provides no energy.
The car continues along the straight line until it comes to rest at the point C. The time taken for the
car to travel from A to C is 52.5 s.

(iii) Find the distance AC . [5]

9709/41/O/N/12

21 A train of mass 400 000 kg is moving on a straight horizontal track. The power of the engine is
constant and equal to 1500 kW and the resistance to the train’s motion is 30 000 N. Find
(i) the acceleration of the train when its speed is 37.5 m s−1 , [4]
(ii) the steady speed at which the train can move. [2]
9709/41/M/J/13
22 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]
© UCLES 2013 9709/42/M/J/13

23 A car has mass 800 kg. The engine of the car generates constant power P kW as the car moves along
a straight horizontal road. The resistance to motion is constant and equal to R N. When the car’s
speed is 14 m s−1 its acceleration is 1.4 m s−2 , and when the car’s speed is 25 m s−1 its acceleration is
0.33 m s−2 . Find the values of P and R. [6]
9709/43/M/J/13

24 A cyclist exerts a constant driving force of magnitude F N while moving up a straight hill inclined
36
at an angle ! to the horizontal, where sin ! = 325 . A constant resistance to motion of 32 N acts on
the cyclist. The total weight of the cyclist and his bicycle is 780 N. The cyclist’s acceleration is
−0.2 m s−2 .

(i) Find the value of F . [4]

The cyclist’s speed is 7 m s−1 at the bottom of the hill.

(ii) Find how far up the hill the cyclist travels before coming to rest. [2]
9709/41/O/N/13

25 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/O/N/13

26 A train is moving at constant speed V m s−1 along a horizontal straight track. Given that the power
of the train’s engine is 1330 kW and the total resistance to the train’s motion is 28 kN, find the value
of V . [3]

© UCLES 2014 9709/41/M/J/14

 27 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]

© UCLES 2014 9709/42/M/J/14

28 A car of mass 1250 kg travels up a straight hill inclined at an angle ! to the horizontal, where
sin ! = 0.02. The power provided by the car’s engine is 23 kW. The resistance to motion is constant
and equal to 600 N. Find the speed of the car at an instant when its acceleration is 0.5 m s−2 . [5]
9709/43/M/J/14

29 A car of mass 800 kg is moving on a straight horizontal road with its engine working at a rate of
22.5 kW. Find the resistance to the car’s motion at an instant when the car’s speed is 18 m s−1 and its
acceleration is 1.2 m s−2 . [4]
9709/41/O/N/14

30 A car of mass 1400 kg moves on a horizontal straight road. The resistance to the car’s motion is
constant and equal to 800 N and the power of the car’s engine is constant and equal to P W. At an
instant when the car’s speed is 18 m s−1 its acceleration is 0.5 m s−2 .

(i) Find the value of P. [3]

The car continues and passes through another point with speed 25 m s−1 .

(ii) Find the car’s acceleration at this point. [2]

© UCLES 2014 9709/43/O/N/14

31 A cyclist and her bicycle have a total mass of 84 kg. She works at a constant rate of P W while
moving on a straight road which is inclined to the horizontal at an angle 1, where sin 1 = 0.1. When
moving uphill, the cyclist’s acceleration is 1.25 m s−2 at an instant when her speed is 3 m s−1 . When
moving downhill, the cyclist’s acceleration is 1.25 m s−2 at an instant when her speed is 10 m s−1 . The
resistance to the cyclist’s motion, whether the cyclist is moving uphill or downhill, is R N. Find the
values of P and R. [8]
9709/41/M/J/15
32 The total mass of a cyclist and his cycle is 80 kg. The resistance to motion is zero.

(i) The cyclist moves along a horizontal straight road working at a constant rate of P W. Find the
value of P given that the cyclist’s speed is 5 m s−1 when his acceleration is 1.2 m s−2 . [2]

(ii) The cyclist moves up a straight hill inclined at an angle !, where sin ! = 0.035. Find the
acceleration of the cyclist at an instant when he is working at a rate of 450 W and has
speed 3.6 m s−1 . [3]
9709/42/M/J/15

33 A block is pulled along a horizontal floor by a horizontal rope. The tension in the rope is 500 N and
the block moves at a constant speed of 2.75 m s−1 . Find the work done by the tension in 40 s and find
the power applied by the tension. [4]
9709/43/M/J/15

34 A car of mass 860 kg travels along a straight horizontal road. The power provided by the car’s engine
is P W and the resistance to the car’s motion is R N. The car passes through one point with speed
4.5 m s−1 and acceleration 4 m s−2 . The car passes through another point with speed 22.5 m s−1 and
acceleration 0.3 m s−2 . Find the values of P and R. [6]
9709/43/M/J/15

35 A lorry of mass 24 000 kg is travelling up a hill which is inclined at 3Å to the horizontal. The power
developed by the lorry’s engine is constant, and there is a constant resistance to motion of 3200 N.

(i) When the speed of the lorry is 25 m s−1 , its acceleration is 0.2 m s−2 . Find the power developed
by the lorry’s engine. [4]

(ii) Find the steady speed at which the lorry moves up the hill if the power is 500 kW and the
resistance remains 3200 N. [2]
9709/41/O/N/15

36 A cyclist and his bicycle have a total mass of 90 kg. The cyclist starts to move with speed 3 m s−1
from the top of a straight hill, of length 500 m, which is inclined at an angle of sin−1 0.05 to the
horizontal. The cyclist moves with constant acceleration until he reaches the bottom of the hill with
speed 5 m s−1 . The cyclist generates 420 W of power while moving down the hill. The resistance to
the motion of the cyclist and his bicycle, R N, and the cyclist’s speed, v m s−1 , both vary.

420
(i) Show that R = + 43.56. [5]
v
(ii) Find the cyclist’s speed at the mid-point of the hill. Hence find the decrease in the value of R
when the cyclist moves from the top of the hill to the mid-point of the hill, and when the cyclist
moves from the mid-point of the hill to the bottom of the hill. [3]

9709/43/O/N/15
37 A constant resistance of magnitude 1350 N acts on a car of mass 1200 kg.

(i) The car is moving along a straight level road at a constant speed of 32 m s−1 . Find, in kW, the
rate at which the engine of the car is working. [2]

(ii) The car travels at a constant speed up a hill inclined at an angle of 1 to the horizontal, where
sin 1 = 0.1, with the engine working at 76.5 kW. Find this speed. [3]

9709/42/F/M/16

38 A car of mass 1200 kg is pulling a trailer of mass 800 kg up a hill inclined at an angle ! to the horizontal,
where sin ! = 0.1. The system of the car and the trailer is modelled as two particles connected by a
light inextensible cable. The driving force of the car’s engine is 2500 N and the resistances to the car
and trailer are 100 N and 150 N respectively.

(i) Find the acceleration of the system and the tension in the cable. [4]

(ii) When the car and trailer are travelling at a speed of 30 m s−1 , the driving force becomes zero.
The cable remains taut. Find the time, in seconds, before the system comes to rest. [3]

9709/42/F/M/16

39 A car of mass 1000 kg is moving along a straight horizontal road against resistances of total magnitude
300 N.

(i) Find, in kW, the rate at which the engine of the car is working when the car has a constant speed
of 40 m s−1 . [3]

(ii) Find the acceleration of the car when its speed is 25 m s−1 and the engine is working at 90% of
the power found in part (i). [3]

9709/41/M/J/16

40 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/M/J/16
41 The motion of a car of mass 1400 kg is resisted by a constant force of magnitude 650 N.

(i) Find the constant speed of the car on a horizontal road, assuming that the engine works at a rate
of 20 kW. [2]

(ii) The car is travelling at a constant speed of 10 m s−1 up a hill inclined at an angle of 1 to the
horizontal, where sin 1 = 17 . Find the power of the car’s engine. [3]

(iii) The car descends the same hill with the engine working at 80% of the power found in part (ii).
Find the acceleration of the car at an instant when the speed is 20 m s−1 . [3]
9709/43/M/J/16

42 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]

The van reaches the bottom of a hill inclined at 4Å to the horizontal with speed 25 m s−1 . 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/O/N/16

43 A crane is used to raise a block of mass 50 kg vertically upwards at constant speed through a height
of 3.5 m. There is a constant resistance to motion of 25 N.

(i) Find the work done by the crane. [3]

(ii) Given that the time taken to raise the block is 2 s, find the power of the crane. [2]
9709/43/O/N/16

44 A cyclist is cycling with constant power of 160 W along a horizontal straight road. There is a constant
resistance to motion of 20 N. At an instant when the cyclist’s speed is 5 m s−1 , his acceleration is
0.15 m s−2 .

(i) Show that the total mass of the cyclist and bicycle is 80 kg. [3]

The cyclist comes to a hill inclined at 2Å to the horizontal. When the cyclist starts climbing the hill,
he increases his power to a constant 300 W. The resistance to motion remains 20 N.

(ii) Show that the steady speed up the hill which the cyclist can maintain when working at this power
is 6.26 m s−1 , correct to 3 significant figures. [2]

(iii) Find the acceleration at an instant when the cyclist is travelling at 90% of the speed in part (ii).
[4]
9709/43/O/N/16
45 A car of mass 900 kg is moving on a straight horizontal road ABCD. There is a constant resistance
of magnitude 800 N in the sections AB and BC, and a constant resistance of magnitude R N in the
section CD. The power of the car’s engine is a constant 36 kW.

(i) The car moves from A to B at a constant speed in 120 s. Find the speed of the car and the distance
AB. [3]

The car’s engine is switched off at B.

(ii) The distance BC is 450 m. Find the speed of the car at C. [3]

(iii) The car comes to rest at D. The distance AD is 6637.5 m. Find the deceleration of the car and
the value of R. [4]
9709/42/F/M/17

46 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/M/J/17

47 A car of mass 1200 kg is travelling along a horizontal road.

(i) It is given that there is a constant resistance to motion.

(a) The engine of the car is working at 16 kW while the car is travelling at a constant speed of
40 m s−1 . Find the resistance to motion. [2]

(b) The power is now increased to 22.5 kW. Find the acceleration of the car at the instant it is
travelling at a speed of 45 m s−1 . [3]

(ii) It is given instead that the resistance to motion of the car is 590 + 2v
N when the speed of the
car is v m s−1 . The car travels at a constant speed with the engine working at 16 kW. Find this
speed. [3]
9709/43/M/J/17

48 A tractor of mass 3700 kg is travelling along a straight horizontal road at a constant speed of 12 m s−1 .
The total resistance to motion is 1150 N.

(i) Find the power output of the tractor’s engine. [1]

The tractor comes to a hill inclined at 4Å above the horizontal. The power output is increased to 25 kW
and the resistance to motion is unchanged.

(ii) Find the deceleration of the tractor at the instant it begins to climb the hill. [3]
 (iii) Find the constant speed that the tractor could maintain on the hill when working at this power.
[2]
9709/41/O/N/17

49 A lorry of mass 7850 kg travels on a straight hill which is inclined at an angle of 3Å to the horizontal.
There is a constant resistance to motion of 1480 N.

(i) Find the power of the lorry’s engine when the lorry is going up the hill at a constant speed of
10 m s−1 . [3]

(ii) Find the power of the lorry’s engine at an instant when the lorry is going down the hill at a speed
of 15 m s−1 with an acceleration of 0.8 m s−2 . [3]
9709/43/O/N/17

50 A car has mass 1250 kg.

(i) The car is moving along a straight level road at a constant speed of 36 m s−1 and is subject to a
constant resistance of magnitude 850 N. Find, in kW, the rate at which the engine of the car is
working. [2]

(ii) The car travels at a constant speed up a hill and is subject to the same resistance as in part (i). The
hill is inclined at an angle of 1Å to the horizontal, where sin 1Å = 0.1, and the engine is working
at 63 kW. Find the speed of the car. [3]

(iii) The car descends the same hill with the engine of the car working at a constant rate of 20 kW.
The resistance is not constant. The initial speed of the car is 20 m s−1 . Eight seconds later the
car has speed 24 m s−1 and has moved 176 m down the hill. Use an energy method to find the
total work done against the resistance during the eight seconds. [5]
9709/41/M/J/18

51 A train of mass 240 000 kg travels up a slope inclined at an angle of 4Å to the horizontal. There is a
constant resistance of magnitude 18 000 N acting on the train. At an instant when the speed of the
train is 15 m s−1 its deceleration is 0.2 m s−2 . Find the power of the engine of the train. [4]
9709/42/M/J/18

52 A car of mass 1400 kg travelling at a speed of v m s−1 experiences a resistive force of magnitude 40v N.
The greatest possible constant speed of the car along a straight level road is 56 m s−1 .

(i) Find, in kW, the greatest possible power of the car’s engine. [2]

(ii) Find the greatest possible acceleration of the car at an instant when its speed on a straight level
road is 32 m s−1 . [3]

(iii) The car travels down a hill inclined at an angle of 1Å to the horizontal at a constant speed of
50 m s−1 . The power of the car’s engine is 60 kW. Find the value of 1. [4]
9709/43/M/J/18
53 A high-speed train of mass 490 000 kg is moving along a straight horizontal track at a constant speed
of 85 m s−1 . The engines are supplying 4080 kW of power.

(i) Show that the resistance force is 48 000 N. [1]

1 . Given
(ii) The train comes to a hill inclined at an angle 1Å above the horizontal, where sin 1Å = 200
that the resistance force is unchanged, find the power required for the train to keep moving at the
same constant speed of 85 m s−1 . [3]
9709/41/O/N/18

54 A car of mass 1200 kg is driving along a straight horizontal road at a constant speed of 15 m s−1 . There
is a constant resistance to motion of 350 N.

(i) Find the power of the car’s engine. [1]

The car comes to a hill inclined at 1Å to the horizontal, still travelling at 15 m s−1 .

(ii) The car starts to descend the hill with reduced power and with an acceleration of 0.12 m s−2 .
Given that there is no change in the resistance force, find the new power of the car’s engine at the
instant when it starts to descend the hill. [3]

(iii) When the car is travelling at 20 m s−1 down the hill, the power is cut off and the car gradually
slows down. Assuming that the resistance force remains 350 N, find the distance travelled from
the moment when the power is cut off until the speed of the car is reduced to 18 m s−1 . [4]
9709/42/O/N/18

55 A van of mass 3200 kg travels along a horizontal road. The power of the van’s engine is constant and
equal to 36 kW, and there is a constant resistance to motion acting on the van.

(i) When the speed of the van is 20 m s−1 , its acceleration is 0.2 m s−2 . Find the resistance force.
[3]

When the van is travelling at 30 m s−1 , it begins to ascend a hill inclined at 1.5Å to the horizontal. The
power is increased and the resistance force is still equal to the value found in part (i).

(ii) Find the power required to maintain this speed of 30 m s−1 . [3]

(iii) The engine is now stopped, with the van still travelling at 30 m s−1 , and the van decelerates to
rest. Find the distance the van moves up the hill from the point at which the engine is stopped
until it comes to rest. [4]
9709/43/O/N/18

56 A car of mass 1500 kg is pulling a trailer of mass 300 kg along a straight horizontal road at a constant
speed of 20 m s−1 . The system of the car and trailer is modelled as two particles, connected by a light
rigid horizontal rod. The power of the car’s engine is 6000 W. There are constant resistances to motion
of R N on the car and 80 N on the trailer.

(i) Find the value of R. [2]

 The power of the car’s engine is increased to 12 500 W. The resistance forces do not change.

(ii) Find the acceleration of the car and trailer and the tension in the rod at an instant when the speed
of the car is 25 m s−1 . [5]
9709/42/F/M/19

57 A lorry has mass 12 000 kg.

(i) The lorry moves at a constant speed of 5 m s−1 up a hill inclined at an angle of 1 to the horizontal,
where sin 1 = 0.08. At this speed, the magnitude of the resistance to motion on the lorry is
1500 N. Show that the power of the lorry’s engine is 55.5 kW. [3]

When the speed of the lorry is v m s−1 the magnitude of the resistance to motion is kv2 N, where k is a
constant.

(ii) Show that k = 60. [1]

(iii) The lorry now moves at a constant speed on a straight level road. Given that its engine is still
working at 55.5 kW, find the lorry’s speed. [3]
9709/41/M/J/19

58 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/M/J/19

59 A car of mass 1400 kg is travelling up a hill inclined at an angle of 4Å to the horizontal. There is a
constant resistance to motion of magnitude 1550 N acting on the car.

(i) Given that the engine of the car is working at 30 kW, find the speed of the car at an instant when
its acceleration is 0.4 m s−2 . [4]

(ii) The greatest possible constant speed at which the car can travel up the hill is 40 m s−1 . Find the
maximum possible power of the engine. [3]
9709/43/M/J/19

60 A crane is lifting a load of 1250 kg vertically at a constant speed V m s−1 . Given that the power of the
crane is a constant 20 kW, find the value of V . [2]
9709/41/O/N/19