6.02l Power and velocity: P = Fv

2 A minibus of mass 4000 kg is travelling along a straight horizontal road. The resistance to motion is 900 N .

1. Find the driving force when the acceleration of the minibus is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the power required for the minibus to maintain a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

5 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.

1. The car and caravan move along a horizontal part of the road at a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   1. Calculate, in kW , the power developed by the engine of the car.
   2. Given that this power is suddenly decreased by 8 kW , find the instantaneous deceleration of the car and caravan and the tension in the tow-bar.
2. 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 .
   1. Find this constant speed.
   2. Find the increase in the potential energy of the caravan in one minute.

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 \(\theta\).

5 A cyclist is riding along a straight horizontal road. The total mass of the cyclist and her bicycle is 70 kg . At an instant when the cyclist's speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), her acceleration is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). There is a constant resistance to motion of magnitude 30 N .

1. Find the power developed by the cyclist.
   The cyclist comes to the top of a hill inclined at \(5 ^ { \circ }\) to the horizontal. The cyclist stops pedalling and freewheels down the hill (so that the cyclist is no longer supplying any power). The magnitude of the resistance force remains at 30 N . Over a distance of \(d \mathrm {~m}\), the speed of the cyclist increases from \(6 \mathrm {~ms} ^ { - 1 }\) to \(12 \mathrm {~ms} ^ { - 1 }\).
2. Find the change in kinetic energy.
3. Use an energy method to find \(d\). \includegraphics[max width=\textwidth, alt={}, center]{4e555003-16f1-4453-ab25-c50929d4b5b3-10_725_785_260_680} Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley at \(B\) which is attached to two inclined planes. \(P\) lies on a smooth plane \(A B\) which is inclined at \(60 ^ { \circ }\) to the horizontal. \(Q\) lies on a plane \(B C\) which is inclined at \(30 ^ { \circ }\) to the horizontal. The string is taut and the particles can move on lines of greatest slope of the two planes (see diagram).

4 An athlete of mass 84 kg is running along a straight road.

1. Initially the road is horizontal and he runs at a constant speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The athlete produces a constant power of 60 W . Find the resistive force which acts on the athlete.
2. The athlete then runs up a 150 m section of the road which is inclined at \(0.8 ^ { \circ }\) to the horizontal. The speed of the athlete at the start of this section of road is \(3 \mathrm {~ms} ^ { - 1 }\) and he now produces a constant driving force of 24 N . The total resistive force which acts on the athlete along this section of road has constant magnitude 13 N . Use an energy method to find the speed of the athlete at the end of the 150 m section of road.

4 A lorry of mass 15000 kg moves on a straight horizontal road in the direction from \(A\) to \(B\). It passes \(A\) and \(B\) with speeds \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The power of the lorry's engine is constant and there is a constant resistance to motion of magnitude 6000 N . The acceleration of the lorry at \(B\) is 0.5 times the acceleration of the lorry at \(A\).

1. Show that the power of the lorry's engine is 200 kW , and hence find the acceleration of the lorry when it is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   The lorry begins to ascend a straight hill inclined at \(1 ^ { \circ }\) to the horizontal. It is given that the power of the lorry's engine and the resistance force do not change.
2. Find the steady speed up the hill that the lorry could maintain.

4 A car of mass 1700 kg is pulling a trailer of mass 300 kg along a straight horizontal road. The car and trailer are connected by a light inextensible cable which is parallel to the road. There are constant resistances to motion of 400 N on the car and 150 N on the trailer. The power of the car's engine is 14000 W . Find the acceleration of the car and the tension in the cable when the speed is \(20 \mathrm {~ms} ^ { - 1 }\).

1 A lorry of mass 16000 kg is travelling along a straight horizontal road. The engine of the lorry is working at constant power. The work done by the driving force in 10 s is 750000 J .

1. Find the power of the lorry's engine.
2. There is a constant resistance force acting on the lorry of magnitude 2400 N . Find the acceleration of the lorry at an instant when its speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

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.

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 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 .

1. Find the power of the car's engine, given that the car's acceleration is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) when its speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the steady speed which the car can maintain with the engine working at this power.

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 \(v\).

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.

5 A railway engine of mass 75000 kg is moving up a straight hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.01\). The engine is travelling at a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The engine is working at 960 kW . There is a constant force resisting the motion of the engine.

1. Find the resistance force.
   The engine comes to a section of track which is horizontal. At the start of the section the engine is travelling at \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) 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 46500 kJ .
2. Find the speed of the engine at the end of the 60 s .

4 A car of mass 1400 kg is moving on a straight road against a constant force of 1250 N resisting the motion.

1. The car moves along a horizontal section of the road at a constant speed of \(36 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   1. Calculate the work done against the resisting force during the first 8 seconds.
   2. Calculate, in kW , the power developed by the engine of the car.
   3. Given that this power is suddenly increased by 12 kW , find the instantaneous acceleration of the car.
2. The car now travels at a constant speed of \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a section of the road inclined at \(\theta ^ { \circ }\) to the horizontal, with the engine working at 64 kW . Find the value of \(\theta\).

6 A car of mass 1750 kg is pulling a caravan of mass 500 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 650 N and 150 N respectively.

1. The car and caravan are moving along a straight horizontal road at a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   1. Find the power of the car's engine.
   2. The engine's power is now suddenly increased to 40 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 \(\sin ^ { - 1 } 0.14\) to the horizontal, at a constant speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car's engine is working at 31 kW . The resistances to the motion of the car and caravan are unchanged. Find \(v\).

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.

4 A car has mass 1600 kg .

1. The car is moving along a straight horizontal road at a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is subject to a constant resistance of magnitude 480 N . Find, in kW , the rate at which the engine of the car is working.
   The car now moves down a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = 0.09\). The engine of the car is working at a constant rate of 12 kW . The speed of the car is \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the hill. Ten seconds later the car has travelled 280 m down the hill and has speed \(32 \mathrm {~ms} ^ { - 1 }\).
2. Given that the resistance is not constant, use an energy method to find the total work done against the resistance during the ten seconds.

3 A cyclist is riding along a straight horizontal road. The total mass of the cyclist and his bicycle is 90 kg . The power exerted by the cyclist is 250 W . At an instant when the cyclist's speed is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), his acceleration is \(0.1 \mathrm {~ms} ^ { - 2 }\).

1. Find the value of the constant resistance to motion acting on the cyclist.
   The cyclist comes to the bottom of a hill inclined at \(2 ^ { \circ }\) to the horizontal.
2. Given that the power and resistance to motion are unchanged, find the steady speed which the cyclist could maintain when riding up the hill.

2 A constant resistance of magnitude 1350 N acts on a car of mass 1200 kg .

1. The car is moving along a straight level road at a constant speed of \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find, in kW , the rate at which the engine of the car is working.
2. The car travels at a constant speed down a hill inclined at an angle of \(\theta ^ { \circ }\) to the horizontal, where \(\sin \theta ^ { \circ } = \frac { 1 } { 20 }\), with the engine working at 31.5 kW . Find the speed of the car.

1 A crate of mass 800 kg is lifted vertically, at constant speed, by the cable of a crane. Find

1. the tension in the cable,
2. the power applied to the crate in increasing the height by 20 m in 50 s .

6 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 .

1. Find the speed of the car at an instant when its acceleration is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Show that the maximum possible speed of the car is \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The work done by the car's engine as the car travels from a point \(A\) to a point \(B\) is 1500 kJ .
3. Given that the car is travelling at its maximum possible speed between \(A\) and \(B\), find the time taken to travel from \(A\) to \(B\).

3 A car travels along a horizontal straight road with increasing speed until it reaches its maximum speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to motion is constant and equal to \(R \mathrm {~N}\), and the power provided by the car's engine is 18 kW .

1. Find the value of \(R\).
2. Given that the car has mass 1200 kg , find its acceleration at the instant when its speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

2 A block is being pulled along a horizontal floor by a rope inclined at \(20 ^ { \circ }\) to the horizontal. The tension in the rope is 851 N and the block moves at a constant speed of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that the work done on the block in 12 s is approximately 24 kJ .
2. Hence find the power being applied to the block, giving your answer to the nearest kW .

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

1. Find the driving force of the car's engine at an instant when the acceleration is \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Given that the car's speed at this instant is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the rate at which the car's engine is working.