6.02l Power and velocity: P = Fv

2 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 .

1. Find the work done by the crane.
2. Assuming the power output of the crane is constant and equal to 1.25 kW , find the time taken to raise the load.

1 A load is pulled along horizontal ground for a distance of 76 m , using a rope. The rope is inclined at \(5 ^ { \circ }\) above the horizontal and the tension in the rope is 65 N .

1. Find the work done by the tension. At an instant during the motion the velocity of the load is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the rate of working of the tension at this instant.

2 A car of mass 1250 kg is travelling along a straight horizontal road with its engine working at a constant rate of \(P \mathrm {~W}\). The resistance to the car's motion is constant and equal to \(R \mathrm {~N}\). When the speed of the car is \(19 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), and when the speed of the car is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(0.16 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the values of \(P\) and \(R\).

1 A car of mass 880 kg travels along a straight horizontal road with its engine working at a constant rate of \(P \mathrm {~W}\). The resistance to motion is 700 N . At an instant when the car's speed is \(16 \mathrm {~ms} ^ { - 1 }\) its acceleration is \(0.625 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the value of \(P\).

4 A car of mass 1230 kg increases its speed from \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 24.5 s . The table below shows corresponding values of time \(t \mathrm {~s}\) and speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Using the values in the table, find the average acceleration of the car for \(0 < t < 0.5\) and for \(16.3 < t < 24.5\). While the car is increasing its speed the power output of its engine is constant and equal to \(P \mathrm {~W}\), and the resistance to the car's motion is constant and equal to \(R \mathrm {~N}\).
2. Assuming that the values obtained in part (i) are approximately equal to the accelerations at \(v = 5\) and at \(v = 20\), find approximations for \(P\) and \(R\).

4 A train of mass 400000 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 30000 N . Find

1. the acceleration of the train when its speed is \(37.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
2. the steady speed at which the train can move.

3 A car has mass 800 kg . The engine of the car generates constant power \(P \mathrm {~kW}\) as the car moves along a straight horizontal road. The resistance to motion is constant and equal to \(R \mathrm {~N}\). When the car's speed is \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), and when the car's speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(0.33 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the values of \(P\) and \(R\).

2 A car of mass 1250 kg travels up a straight hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.02\). 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 \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

5 A lorry of mass 16000 kg travels at constant speed from the bottom, \(O\), to the top, \(A\), of a straight hill. The distance \(O A\) is 1200 m and \(A\) is 18 m above the level of \(O\). The driving force of the lorry is constant and equal to 4500 N .

1. Find the work done against the resistance to the motion of the lorry. On reaching \(A\) the lorry continues along a straight horizontal road against a constant resistance of 2000 N . The driving force of the lorry is not now constant, and the speed of the lorry increases from \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(A\) to \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the point \(B\) on the road. The distance \(A B\) is 2400 m .
2. Use an energy method to find \(F\), where \(F \mathrm {~N}\) is the average value of the driving force of the lorry while moving from \(A\) to \(B\).
3. Given that the driving force at \(A\) is 1280 N greater than \(F \mathrm {~N}\) and that the driving force at \(B\) is 1280 N less than \(F \mathrm {~N}\), show that the power developed by the lorry's engine is the same at \(B\) as it is at \(A\).

5 A cyclist and her bicycle have a total mass of 84 kg . She works at a constant rate of \(P \mathrm {~W}\) while moving on a straight road which is inclined to the horizontal at an angle \(\theta\), where \(\sin \theta = 0.1\). When moving uphill, the cyclist's acceleration is \(1.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at an instant when her speed is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When moving downhill, the cyclist's acceleration is \(1.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at an instant when her speed is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to the cyclist's motion, whether the cyclist is moving uphill or downhill, is \(R \mathrm {~N}\). Find the values of \(P\) and \(R\).

2 The total mass of a cyclist and his cycle is 80 kg . The resistance to motion is zero.

1. The cyclist moves along a horizontal straight road working at a constant rate of \(P \mathrm {~W}\). Find the value of \(P\) given that the cyclist's speed is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when his acceleration is \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. The cyclist moves up a straight hill inclined at an angle \(\alpha\), where \(\sin \alpha = 0.035\). Find the acceleration of the cyclist at an instant when he is working at a rate of 450 W and has speed \(3.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

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

1. Find, in kW , the rate at which the engine of the car is working when the car has a constant speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the acceleration of the car when its speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the engine is working at \(90 \%\) of the power found in part (i).

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.

5 The motion of a car of mass 1400 kg is resisted by a constant force of magnitude 650 N .

1. Find the constant speed of the car on a horizontal road, assuming that the engine works at a rate of 20 kW .
2. The car is travelling at a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 7 }\). Find the power of the car's engine.
3. 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 \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

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

1. It is given that there is a constant resistance to motion.
   1. The engine of the car is working at 16 kW while the car is travelling at a constant speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the resistance to motion.
   2. 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 \(45 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   3. It is given instead that the resistance to motion of the car is \(( 590 + 2 v ) \mathrm { N }\) when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car travels at a constant speed with the engine working at 16 kW . Find this speed.

2 A train of mass 240000 kg travels up a slope inclined at an angle of \(4 ^ { \circ }\) to the horizontal. There is a constant resistance of magnitude 18000 N acting on the train. At an instant when the speed of the train is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its deceleration is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the power of the engine of the train.

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.

3 A car of mass 1400 kg is travelling up a hill inclined at an angle of \(4 ^ { \circ }\) to the horizontal. There is a constant resistance to motion of magnitude 1550 N acting on the car.

1. 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 \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. The greatest possible constant speed at which the car can travel up the hill is \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the maximum possible power of the engine. \includegraphics[max width=\textwidth, alt={}, center]{539be201-7bfc-4ba0-8378-c7aec4473ac7-06_643_419_255_863} Two particles \(A\) and \(B\), of masses 1.3 kg and 0.7 kg respectively, are connected by a light inextensible string which passes over a smooth fixed pulley. Particle \(A\) is 1.75 m above the floor and particle \(B\) is 1 m above the floor (see diagram). The system is released from rest with the string taut, and the particles move vertically. When the particles are at the same height the string breaks.

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 up a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = 0.1\), with the engine working at 76.5 kW . Find this speed.

1 A car of mass 1000 kg travels along a horizontal straight road with its engine working at a constant rate of 20 kW . The resistance to motion of the car is 600 N . Find the acceleration of the car at an instant when its speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1 A motorcycle of mass 100 kg is travelling on a horizontal straight road. Its engine is working at a rate of 8 kW . At an instant when the speed of the motorcycle is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find, at this instant,

1. the force produced by the engine,
2. the resistance to motion of the motorcycle.

3 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 \(3 ^ { \circ }\) 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 \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

3 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 \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),

1. given that the road is horizontal,
2. given instead that the road is inclined at \(1.5 ^ { \circ }\) to the horizontal and the cyclist is travelling up the slope.

3 A car of mass 1200 kg is travelling on a horizontal straight road and passes through a point \(A\) with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The power of the car's engine is 18 kW and the resistance to the car's motion is 900 N .

1. Find the deceleration of the car at \(A\).
2. Show that the speed of the car does not fall below \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) while the car continues to move with the engine exerting a constant power of 18 kW .

1 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 15000 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 \(B\).