Power and driving force

7 A train, of mass 22 tonnes, moves along a straight horizontal track. A constant resistance force of 5000 N acts on the train. The power output of the engine of the train is 240 kW . Find the acceleration of the train when its speed is \(20 \mathrm {~ms} ^ { - 1 }\).

A car of mass 800 kg moves in a straight line along a horizontal road. There is a constant resistance to the motion of the car of magnitude 600 N. When the car is travelling at a speed of \(15 \text{ m s}^{-1}\) the power developed by the car is 27 kW. Determine the acceleration of the car when it is travelling at \(15 \text{ m s}^{-1}\). [4]

A car of mass \(M\) moves along a straight horizontal road. The total resistance to motion of the car is modelled as having constant magnitude \(R\). The engine of the car works at a constant rate \(RU\). Find the time taken for the car to accelerate from a speed of \(\frac{1}{4}U\) to a speed of \(\frac{1}{2}U\). [9]

5 A car, of mass 1000 kg , is travelling on a straight horizontal road. When the car travels at a speed of \(v \mathrm {~ms} ^ { - 1 }\), it experiences a resistance force of magnitude \(25 v\) newtons. The car has a maximum speed of \(72 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on the straight road.
Find the maximum power output of the car.
Fully justify your answer.

6 A car of mass 1300 kg is moving on a straight road.

1. On a horizontal section of the road, the car has a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and there is a constant force of 650 N resisting the motion.
   1. Calculate, in kW , the power developed by the engine of the car.
   2. Given that this power is suddenly increased by 9 kW , find the instantaneous acceleration of the car.
2. On a section of the road inclined at \(\sin ^ { - 1 } 0.08\) to the horizontal, the resistance to the motion of the car is \(( 1000 + 20 v ) \mathrm { N }\) when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car travels downwards along this section of the road at constant speed with the engine working at 11.5 kW . Find this constant speed.

1. A racing car of mass 750 kg is moving along a straight horizontal road at a constant speed of \(U \mathbf { k m ~ h } ^ { - \mathbf { 1 } }\). The engine of the racing car is working at a constant rate of 60 kW .

The resistance to the motion of the racing car is modelled as a force of magnitude \(37.5 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the racing car. Using the model,

1. find the value of \(U\) Later on, the racing car is accelerating up a straight road which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 5 } { 49 }\). The engine of the racing car is working at a constant rate of 60 kW . The total resistance to the motion of the racing car from non-gravitational forces is modelled as a force of magnitude \(37.5 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the racing car. At the instant when the acceleration of the racing car is \(2 \mathrm {~ms} ^ { - 2 }\), the speed of the racing car is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Using the model,
2. find the value of \(V\)

2 A car of mass 1200 kg is travelling at a constant speed of \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a straight horizontal road. The car experiences a total resistive force of 240 newtons.
Calculate the power of the car's engine.
Circle your answer.
[0pt] [1 mark]
900 W
4320 W
16000 W
21600 W

2 A car of mass 1200 kg is travelling at a constant speed of \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a straight horizontal road. The car experiences a total resistive force of 240 newtons.
Calculate the power of the car's engine.
Circle your answer.
[0pt] [1 mark]
900 W
4320 W
16000 W
21600 W

A car of mass 1250 kg is moving on a straight road.

1. On a horizontal section of the road, the car has a constant speed of \(32 \text{ ms}^{-1}\) and there is a constant force of 750 N resisting the motion.
   1. Calculate, in kW, the power developed by the engine of the car. [2]
   2. Given that this power is suddenly decreased by 8 kW, find the instantaneous deceleration of the car. [3]
2. On a section of the road inclined at \(\sin^{-1} 0.096\) to the horizontal, the resistance to the motion of the car is \((1000 + 8v)\) N when the speed of the car is \(v \text{ ms}^{-1}\). The car travels up this section of the road at constant speed with the engine working at 60 kW. Find this constant speed. [5]

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]

2 A car of mass 1400 kg is travelling at constant speed up a straight hill inclined at \(\alpha\) to the horizontal, where \(\sin \alpha = 0.1\). There is a constant resistance force of magnitude 600 N . The power of the car's engine is 22500 W .

1. Show that the speed of the car is \(11.25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   The car, moving with speed \(11.25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), comes to a section of the hill which is inclined at \(2 ^ { \circ }\) to the horizontal.
2. Given that the power and resistance force do not change, find the initial acceleration of the car up this section of the hill.

The maximum power produced by the engine of a small aeroplane of mass 2 tonnes is 128 kW. Air resistance opposes the motion directly and the lift force is perpendicular to the direction of motion. The magnitude of the air resistance is proportional to the square of the speed and the maximum steady speed in level flight is \(80 \text{ ms}^{-1}\).

1. Calculate the magnitude of the air resistance when the speed is \(60 \text{ ms}^{-1}\). [5]

The aeroplane is climbing at a constant angle of \(2°\) to the horizontal.

1. Find the maximum acceleration at an instant when the speed of the aeroplane is \(60 \text{ ms}^{-1}\). [4]

4 A cyclist is riding a bicycle along a straight road which is inclined at an angle of \(4 ^ { \circ }\) to the horizontal. The cyclist is working at a constant rate of 250 W . The combined mass of the cyclist and bicycle is 80 kg and the resistance to their motion is a constant 70 N . Determine the maximum constant speed at which the cyclist can ride the bicycle

• up the hill, and
• down the hill.

5. A straight road is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 20 }\). A lorry of mass 4800 kg moves up the road at a constant speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The non-gravitational resistance to the motion of the lorry is constant and has magnitude 2000 N .

1. Find, in kW to 3 significant figures, the rate of working of the lorry's engine.
   (5) The road becomes horizontal. The lorry's engine continues to work at the same rate and the resistance to motion remains the same. Find
2. the acceleration of the lorry immediately after the road becomes horizontal,
   (3)
3. the maximum speed, in \(\mathrm { m } \mathrm { s } ^ { - 1 }\) to 3 significant figures, at which the lorry will go along the horizontal road.
   (3)

4 A car of mass 1200 kg has a maximum speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when travelling on a horizontal road. The car experiences a resistance of \(k v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car and \(k\) is a constant. The maximum power of the car's engine is 45000 W .

1. Show that \(k = 50\).
2. Find the maximum possible acceleration of the car when it is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a horizontal road.
3. The car climbs a hill, which is inclined at an angle of \(10 ^ { \circ }\) to the horizontal, at a constant speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Calculate the power of the car's engine.

2 A train is travelling at maximum speed with its engine using its maximum power of 1800 kW When travelling at this speed the train experiences a total resistive force of 40000 N Find the maximum speed of the train. Circle your answer.
[0pt] [1 mark] \(22 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) \(45 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) \(54 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) \(90 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

2 A train is travelling at maximum speed with its engine using its maximum power of 1800 kW When travelling at this speed the train experiences a total resistive force of 40000 N Find the maximum speed of the train. Circle your answer.
[0pt] [1 mark] \(22 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) \(45 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) \(54 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) \(90 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

3 In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors and \(c\) is a positive real number.
The resultant of two forces \(c \mathbf { i N }\) and \(- \mathbf { i } + 2 \sqrt { c } \mathbf { j N }\) is denoted by \(R \mathrm {~N}\).

1. Show that the magnitude of \(R\) is \(c + 1\). A car of mass 900 kg travels along a straight horizontal road with constant resistance to motion of magnitude \(( c + 1 ) \mathrm { N }\). The car passes through point A on the road with speed \(6 \mathrm {~ms} ^ { - 1 }\), and 8 seconds later passes through a point B on the same road. The power developed by the car while travelling from A to B is zero. Furthermore, while travelling between A and B, the car's direction of motion is unchanged.
2. Determine the range of possible values of \(c\). The car later passes through a point C on the road. While travelling between B and C the power developed by the car is modelled as constant and equal to 18 kW . The car passes through C with speed \(5 \mathrm {~ms} ^ { - 1 }\) and acceleration \(3.5 \mathrm {~ms} ^ { - 2 }\).
3. Determine the value of \(c\).
4. Suggest how one of the modelling assumptions made in this question could be improved.

Kang is riding a motorbike along a straight, horizontal road. The motorbike has a maximum power of 75 000 W The maximum speed of the motorbike is \(50 \text{ m s}^{-1}\) When the speed of the motorbike is \(v \text{ m s}^{-1}\), the resistance force is \(kv\) newtons. Find the value of \(k\) Fully justify your answer. [4 marks]

2 A car of mass 1250 kg travels along a straight road inclined at \(2 ^ { \circ }\) to the horizontal. The resistance to the motion of the car is \(k v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car and \(k\) is a constant. The car travels at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up the slope and the engine of the car works at a constant rate of 21 kW .

1. Calculate the value of \(k\).
2. Calculate the constant speed of the car on a horizontal road.

3 A lorry has mass 12000 kg .

1. The lorry moves at a constant speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = 0.08\). 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 .
   When the speed of the lorry is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the magnitude of the resistance to motion is \(k v ^ { 2 } \mathrm {~N}\), where \(k\) is a constant.
2. Show that \(k = 60\).
3. 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.

1. A cyclist and his bicycle have a combined mass of 90 kg . He rides on a straight road up a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 21 }\). He works at a constant rate of 444 W and cycles up the hill at a constant speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

Find the magnitude of the resistance to motion from non-gravitational forces as he cycles up the hill.

4 The total mass of a cyclist and her bicycle is 70 kg . The cyclist is riding with constant power of 180 W up a straight hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.05\). At an instant when the cyclist's speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), her acceleration is \(- 0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). There is a constant resistance to motion of magnitude \(F \mathrm {~N}\).

1. Find the value of \(F\).
2. Find the steady speed that the cyclist could maintain up the hill when working at this power. [2]

2 A cyclist is travelling along a straight horizontal road. She is working at a constant rate of 150 W . At an instant when her speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), her acceleration is \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The resistance to motion is 20 N .

1. Find the total mass of the cyclist and her bicycle.
   The cyclist comes to a straight hill inclined at an angle \(\theta\) above the horizontal. She ascends the hill at constant speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). She continues to work at the same rate as before and the resistance force is unchanged.
2. Find the value of \(\theta\).

A toy railway locomotive of mass 0.8 kg is towing a truck of mass 0.4 kg on a straight horizontal track at a constant speed of \(2\,\text{m}\,\text{s}^{-1}\). There is a constant resistance force of magnitude 0.2 N on the locomotive, but no resistance force on the truck. There is a light rigid horizontal coupling connecting the locomotive and the truck.

1. State the tension in the coupling. [1]
2. Find the power produced by the locomotive's engine. [1] The power produced by the locomotive's engine is now changed to 1.2 W.
3. Find the magnitude of the tension in the coupling at the instant that the locomotive begins to accelerate. [5]

6 A car of mass 1600 kg is pulling a caravan of mass 800 kg . The car and the caravan are connected by a light rigid tow-bar. The resistances to the motion of the car and caravan are 400 N and 250 N respectively.

1. The car and caravan are travelling along a straight horizontal road.
   1. Given that the car and caravan have a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the power of the car's engine.
   2. The engine's power is now suddenly increased to 39 kW . Find the instantaneous acceleration of the car and caravan and find the tension in the tow-bar.
2. The car and caravan now travel up a straight hill, inclined at an angle of \(\sin ^ { - 1 } 0.05\) to the horizontal, at a constant speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car's engine is working at 32.5 kW . Find \(v\).

4 \includegraphics[max width=\textwidth, alt={}, center]{a1a43547-0a68-4346-884a-0c6d9302cf24-3_216_1219_255_415} \(A\) and \(B\) are two long straight parallel horizontal sections of railway track. An engine on track \(A\) is attached to a carriage of mass 6000 kg on track \(B\) by a light inextensible chain which remains horizontal and taut in the ensuing motion. The chain is 13 m in length and the points of attachment on the engine and carriage are a perpendicular distance of 5 m apart. The engine and carriage start at rest and then the engine accelerates uniformly to a speed of \(5.6 \mathrm {~ms} ^ { - 1 }\) while travelling 250 m . It is assumed that any resistance to motion can be ignored.

1. Find the work done on the carriage by the tension in the chain.
2. Find the magnitude of the tension in the chain. The mass of the engine is 10000 kg .
3. At a point further along the track the engine and the carriage are moving at a speed of \(8.4 \mathrm {~ms} ^ { - 1 }\) and the power of the engine is 68 kW . Find the acceleration of the engine at this instant.

2. A car is travelling along a straight horizontal road against resistances to motion which are constant and total 2000 N . When the engine of the car is working at a rate of \(H\) kilowatts, the maximum speed of the car is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the value of \(H\). The car driver wishes to overtake another vehicle so she increases the rate of working of the engine by \(20 \%\) and this results in an initial acceleration of \(0.32 \mathrm {~ms} ^ { - 2 }\). Assuming that the resistances to motion remain constant,
2. find the mass of the car.
   (4 marks)

6 A car of mass 1100 kg is moving on a road against a constant force of 1550 N resisting the motion.

1. The car moves along a straight horizontal road at a constant speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   1. Calculate, in kW , the power developed by the engine of the car.
   2. Given that this power is suddenly decreased by 22 kW , find the instantaneous deceleration of the car.
   3. The car now travels at constant speed up a straight road inclined at \(8 ^ { \circ }\) to the horizontal, with the engine working at 80 kW . Assuming the resistance force remains the same, find this constant speed.

1. A car of mass 900 kg is travelling up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 25 }\). The car is travelling at a constant speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the resistance to motion from non-gravitational forces has a constant magnitude of 800 N . The car takes 10 seconds to travel from \(A\) to \(B\), where \(A\) and \(B\) are two points on the road.
   1. Find the work done by the engine of the car as the car travels from \(A\) to \(B\).
   When the car is at \(B\) and travelling at a speed of \(14 \mathrm {~ms} ^ { - 1 }\) the rate of working of the engine of the car is suddenly increased to \(P \mathrm {~kW}\), resulting in an initial acceleration of the car of \(0.7 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The resistance to motion from non-gravitational forces still has a constant magnitude of 800 N .
2. Find the value of \(P\).
   

2 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 \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car's acceleration at the point \(B\) on the road is half its acceleration at \(A\). Find the speed of the car at \(B\).

6 A van of mass 3000 kg is pulling a trailer of mass 500 kg along a straight horizontal road at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). 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.

1. Find the power of the van's engine.
2. Write down the tension in the cable. The van reaches the bottom of a hill inclined at \(4 ^ { \circ }\) to the horizontal with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The power of the van's engine is increased to 25000 W .
3. 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 \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

A car of mass 750 kg is moving on a slope inclined at an angle \(\theta\) to the horizontal, where \(\sin\theta = 0.1\). When the car's engine is working at a constant power \(PW\), the car can travel at maximum speeds of \(14\text{ ms}^{-1}\) up the slope and \(28\text{ ms}^{-1}\) down the slope. In each case, the resistance to motion experienced by the car is proportional to the square of its speed. Find the value of \(P\) and determine the resistance to the motion of the car when its speed is \(10.5\text{ ms}^{-1}\). [10]

1 A particle, \(P\), of mass 2 kg moves in two dimensions. Its initial velocity is \(\binom { - 19.5 } { - 60 } \mathrm {~ms} ^ { - 1 }\).

1. Calculate the initial kinetic energy of \(P\). For \(t \geqslant 0 , P\) is acted upon only by a variable force \(\mathbf { F } = \binom { 4 t } { - 2 } \mathrm {~N}\), where \(t\) is the time in seconds.
2. Find

An engine is travelling along a straight horizontal track against negligible resistances. In travelling a distance of 750 m its speed increases from 5 m s\(^{-1}\) to 15 m s\(^{-1}\). Find the time taken if the engine was

1. exerting a constant tractive force, [2]
2. working at constant power. [9]

7 Find the power required to pump \(3 \mathrm {~m} ^ { 3 }\) of water per minute from a depth of 25 m and deliver it through a circular pipe of diameter 10 cm . Assume that friction may be neglected and that the density of water is \(1000 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\).

2. A car of mass 500 kg is moving at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\). The resistance to motion from non-gravitational forces is modelled as a constant force of magnitude 150 N .

1. Find the rate of working of the engine of the car. When the car is travelling up the road at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the engine is switched off. The car then comes to instantaneous rest, without braking, having moved a distance \(d\) metres up the road from the point where the engine was switched off. The resistance to motion from non-gravitational forces is again modelled as a constant force of magnitude 150 N .
2. Use the work-energy principle to find the value of \(d\).

5 A car of mass 1600 kg travels at constant speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a straight road inclined at an angle of \(\sin ^ { - 1 } 0.12\) to the horizontal.

1. Find the change in potential energy of the car in 30 s .
2. Given that the total work done by the engine of the car in this time is 1960 kJ , find the constant force resisting the motion.
3. Calculate, in kW , the power developed by the engine of the car.
4. Given that this power is suddenly decreased by \(15 \%\), find the instantaneous deceleration of the car.