6.02l Power and velocity: P = Fv

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]

A crate of mass 600 kg is being pulled up a line of greatest slope of a rough plane at a constant speed of \(2\) m s\(^{-1}\) by a rope attached to a winch. The plane is inclined at an angle of \(30°\) to the horizontal and the rope is parallel to the plane. The winch is working at a constant rate of 8 kW. Find the coefficient of friction between the crate and the plane. [5]

A car of mass 1400 kg is moving along a straight horizontal road against a resistance of magnitude 350 N.

1. Find, in kW, the rate at which the engine of the car is working when it is travelling at a constant speed of \(20 \text{ m s}^{-1}\). [2]
2. Find the acceleration of the car when its speed is \(20 \text{ m s}^{-1}\) and the engine is working at 15 kW. [3]

A constant resistance of magnitude 1400 N acts on a car of mass 1250 kg.

1. The car is moving along a straight level road at a constant speed of 28 m s\(^{-1}\). Find, in kW, the rate at which the engine of the car is working. [2]
2. The car now travels at a constant speed up a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = 0.12\), with the engine working at 43.5 kW. Find this speed. [3]
3. On another occasion, the car pulls a trailer of mass 600 kg up the same hill. The system of the car and the trailer is modelled as particles connected by a light inextensible cable. The car's engine produces a driving force of 5000 N and the resistance to the motion of the trailer is 300 N. The resistance to the motion of the car remains 1400 N. Find the acceleration of the system and the tension in the cable. [4]

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 \(15 \text{ m s}^{-1}\) and an acceleration of \(0.8 \text{ m s}^{-2}\).

1. 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 \(32 \text{ m s}^{-1}\).

1. Show that the distance \(AB\) is 1362.6 m. [3]

A railway engine of mass 120000 kg is towing a coach of mass 60000 kg up a straight track inclined at an angle of \(\alpha\) to the horizontal where \(\sin \alpha = 0.02\). There is a light rigid coupling, parallel to the track, connecting the engine and coach. The driving force produced by the engine is 125000 N and there are constant resistances to motion of 22000 N on the engine and 13000 N on the coach.

1. Find the acceleration of the engine and find the tension in the coupling. [5]

At an instant when the engine is travelling at 30 m s\(^{-1}\), it comes to a section of track inclined upwards at an angle \(\beta\) to the horizontal. The power produced by the engine is now 4500000 W and, as a result, the engine maintains a constant speed.

1. Assuming that the resistance forces remain unchanged, find the value of \(\beta\). [4]

A car has mass \(1200\) kg. When the car is travelling at a speed of \(v \text{ ms}^{-1}\), there is a resistive force of magnitude \(kv\) N. The maximum power of the car's engine is \(92.16\) kW.

1. The car travels along a straight level road.
   1. The car has a greatest possible constant speed of \(48 \text{ ms}^{-1}\). Show that \(k = 40\). [1]
   2. At an instant when its speed is \(45 \text{ ms}^{-1}\), find the greatest possible acceleration of the car. [3]
2. The car now travels at a constant speed up a hill inclined at an angle of \(\sin^{-1} 0.15\) to the horizontal. Find the greatest possible speed of the car going up the hill. [4]

A car of mass 1600 kg travels up a slope inclined at an angle of \(\sin^{-1}\) 0.08 to the horizontal. There is a constant resistance of magnitude 240 N acting on the car.

1. It is given that the car travels at a constant speed of 32 ms\(^{-1}\). Find the power of the engine of the car. [3]
2. Find the acceleration of the car when its speed is 24 ms\(^{-1}\) and the engine is working at 95\% of the power found in (a). [3]

A car of mass \(1200 \text{ kg}\) travels along a horizontal straight road. The power provided by the car's engine is constant and equal to \(20 \text{ kW}\). The resistance to the car's motion is constant and equal to \(500 \text{ N}\). The car passes through the points \(A\) and \(B\) with speeds \(10 \text{ m s}^{-1}\) and \(25 \text{ m s}^{-1}\) respectively. The car takes \(30.5 \text{ s}\) to travel from \(A\) to \(B\).

1. Find the acceleration of the car at \(A\). [4]
2. By considering work and energy, find the distance \(AB\). [8]

\includegraphics{figure_5} 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 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).

1. 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 \(AB\).

1. Find the value of \(d\). [3]

The cyclist starts to pedal at \(C\), generating 425 W of power.

1. Find the acceleration of the cyclist immediately after passing through \(C\). [3]

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]

A car of mass \(1150 \text{ kg}\) travels up a straight hill inclined at \(1.2°\) to the horizontal. The resistance to motion of the car is \(975 \text{ N}\). Find the acceleration of the car at an instant when it is moving with speed \(16 \text{ m s}^{-1}\) and the engine is working at a power of \(35 \text{ kW}\). [4]

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 \(4.5 \text{ m s}^{-1}\) and acceleration \(4 \text{ m s}^{-2}\). The car passes through another point with speed \(22.5 \text{ m s}^{-1}\) and acceleration \(0.3 \text{ m s}^{-2}\). Find the values of \(P\) and \(R\). [6]

A car of mass \(1200\) kg is moving on a straight road against a constant force of \(850\) N resisting the motion.

1. On a part of the road that is horizontal, the car moves with a constant speed of \(42\) m s\(^{-1}\).
   1. Calculate, in kW, the power developed by the engine of the car. [2]
   2. Given that this power is suddenly increased by \(6\) kW, find the instantaneous acceleration of the car. [3]
2. On a part of the road that is inclined at \(\theta°\) 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 \(\theta\). [4]

A car has mass \(1250 \text{ kg}\).

1. The car is moving along a straight level road at a constant speed of \(36 \text{ m s}^{-1}\) and is subject to a constant resistance of magnitude \(850 \text{ N}\). Find, in kW, the rate at which the engine of the car is working. [2]
2. 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 \(\theta°\) to the horizontal, where \(\sin \theta° = 0.1\), and the engine is working at \(63 \text{ kW}\). Find the speed of the car. [3]
3. The car descends the same hill with the engine of the car working at a constant rate of \(20 \text{ kW}\). The resistance is not constant. The initial speed of the car is \(20 \text{ m s}^{-1}\). Eight seconds later the car has speed \(24 \text{ m s}^{-1}\) and has moved \(176 \text{ m}\) down the hill. Use an energy method to find the total work done against the resistance during the eight seconds. [5]

A car of mass \(1400\text{ kg}\) travelling at a speed of \(v\text{ m s}^{-1}\) experiences a resistive force of magnitude \(40v\text{ N}\). The greatest possible constant speed of the car along a straight level road is \(56\text{ m s}^{-1}\).

1. Find, in kW, the greatest possible power of the car's engine. [2]
2. Find the greatest possible acceleration of the car at an instant when its speed on a straight level road is \(32\text{ m s}^{-1}\). [3]
3. The car travels down a hill inclined at an angle of \(\theta°\) to the horizontal at a constant speed of \(50\text{ m s}^{-1}\). The power of the car's engine is \(60\text{ kW}\). Find the value of \(\theta\). [4]

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 \(\theta\) to the horizontal, where \(\sin \theta = 0.05\), at a steady speed of 12 m s\(^{-1}\) when its engine is working at 21 kW. Find \(A\) and \(B\). [7]

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.

1. 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]
2. The distance \(BC\) is \(450\) m. Find the speed of the car at \(C\). [3]
3. 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]

The car's engine is switched off at \(B\).

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.

1. Find the value of \(R\). [2]
2. The power of the car's engine is increased to \(12\,500\) W. The resistance forces do not change. 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]

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 \(0.2 \text{ m s}^{-2}\) at an instant when its speed is \(30 \text{ m s}^{-1}\), find the value of \(P\). [4]

A cyclist is riding up a straight hill inclined at an angle \(α\) to the horizontal, where \(\sin α = 0.04\). 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.

1. Find the power output of the cyclist. [4]

The cyclist reaches the top of the hill, where the road becomes horizontal, with speed 4 m s\(^{-1}\). The cyclist continues to work at the same rate on the horizontal part of the road.

1. 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]

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.

1. Find the power of the car's engine. [1]

The car comes to a hill inclined at \(1°\) to the horizontal, still travelling at \(15\) m s\(^{-1}\).

1. 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]
2. 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]

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]

A lorry of mass 25 000 kg travels along a straight horizontal road. There is a constant force of 3000 N resisting the motion.

1. Find the power required to maintain a constant speed of 30 m s\(^{-1}\). [2]

The lorry comes to a straight hill inclined at 2° 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.

1. Use an energy method to find the height of \(B\) above the level of \(A\). [5]

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

1. 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]
2. 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]