6.02k Power: rate of doing work

6 A train, of mass 60 tonnes, travels on a straight horizontal track. It has a maximum speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when its engine is working at 800 kW .

1. Find the magnitude of the resistance force acting on the train when the train is travelling at its maximum speed.
2. When the train is travelling at \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the power is turned off. Assume that the resistance force is constant and is equal to that found in part (a). Also assume that this resistance force is the only horizontal force acting on the train. Use an energy method to find how far the train travels when slowing from \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(36 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   (4 marks)

1 A particle \(Q\) of mass \(m \mathrm {~kg}\) is acted on by a single force so that it moves with constant acceleration \(\mathbf { a } = \binom { 1 } { 2 } \mathrm {~ms} ^ { - 2 }\). Initially \(Q\) is at the point \(O\) and is moving with velocity \(\mathbf { u } = \binom { 2 } { - 5 } \mathrm {~ms} ^ { - 1 }\). After \(Q\) has been moving for 5 seconds it reaches the point \(A\).

1. Use the equation \(\mathbf { v } . \mathbf { v } = \mathbf { u } . \mathbf { u } + 2 \mathbf { a } . \mathbf { x }\) to show that at \(A\) the kinetic energy of \(Q\) is 37 m J .
2. 1. Show that the power initially generated by the force is - 8 mW W.
   2. The power in part (b)(i) is negative. Explain what this means about the initial motion of \(Q\).
3. 1. Find the time at which the power generated by the force is instantaneously zero.
   2. Find the minimum kinetic energy of \(Q\) in terms of \(m\).

1 A car of mass 800 kg is driven with its engine generating a power of 15 kW .

1. The car is first driven along a straight horizontal road and accelerates from rest. Assuming that there is no resistance to motion, find the speed of the car after 6 seconds.
2. The car is next driven at constant speed up a straight road inclined at an angle \(\theta\) to the horizontal. The resistance to motion is now modelled as being constant with magnitude 150 N . Given that \(\sin \theta = \frac { 1 } { 20 }\), find the speed of the car.
3. The car is now driven at a constant speed of \(30 \mathrm {~ms} ^ { - 1 }\) along the horizontal road pulling a trailer of mass 150 kg which is attached by means of a light rigid horizontal towbar. Assuming that the resistance to motion of the car is three times the resistance to motion of the trailer, find

3. \begin{figure}[h] \end{figure} A van of mass 900 kg is moving up a straight road inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 25 }\). The van is towing a trailer of mass 200 kg . The trailer is attached to the van by a rigid towbar which is parallel to the direction of motion of the van and the trailer, as shown in Figure 1. The resistance to the motion of the van from non-gravitational forces is modelled as a constant force of magnitude 400 N .
The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 240 N . The towbar is modelled as a light rod.
The engine of the van is working at a constant rate of 15 kW .

1. Find the acceleration of the van at the instant when the speed of the van is \(12 \mathrm {~ms} ^ { - 1 }\) At the instant when the speed of the van is \(14 \mathrm {~ms} ^ { - 1 }\), the trailer is passing the point \(A\) on the slope and the towbar breaks. The trailer continues to move up the slope until it comes to rest at the point \(B\).
   The resistance to the motion of the trailer from non-gravitational forces is still modelled as a constant force of magnitude 240 N .
2. Use the work-energy principle to find the distance \(A B\).

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 \(1000 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\).

8 A car of mass 1 tonne reaches the foot of an incline travelling at \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It reaches the top of the incline 50 seconds later travelling at \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The length of the incline is 1200 m and the angle made with the horizontal is \(\sin ^ { - 1 } \left( \frac { 1 } { 8 } \right)\). The constant resistance to motion is 400 N . Find the average power developed by the engine of the car.

A circular wheel is modelled as a uniform disc of mass \(6\) kg and radius \(0.25\) m. It is rotating with angular speed \(2\) rad s\(^{-1}\) about a fixed smooth axis perpendicular to its plane and passing through its centre. A braking force of constant magnitude is applied tangentially to the rim of the wheel. The wheel comes to rest \(5\) s after the braking force is applied. Find the magnitude of the braking force and the angle turned through by the wheel while the braking force acts. [7]

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 \(2.75 \text{ m s}^{-1}\). Find the work done by the tension in 40 s and find the power applied by the tension. [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 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 weightlifter performs an exercise in which he raises a mass of 200 kg from rest vertically through a distance of 0.7 m and holds it at that height.

1. Find the work done by the weightlifter. [2]
2. Given that the time taken to raise the mass is 1.2 s, find the average power developed by the weightlifter. [2]

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]

A car of mass 750 kg is moving up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{1}{15}\). The resistance to motion of the car from non-gravitational forces has constant magnitude \(R\) newtons. The power developed by the car's engine is 15 kW and the car is moving at a constant speed of 20 m s\(^{-1}\).

1. Show that \(R = 260\). (4)

The power developed by the car's engine is now increased to 18 kW. The magnitude of the resistance to motion from non-gravitational forces remains at 260 N. At the instant when the car is moving up the road at 20 m s\(^{-1}\) the car's acceleration is \(a\) m s\(^{-2}\).

1. Find the value of \(a\). (4)

A car of mass 1000 kg moves with constant speed \(V\) m s\(^{-1}\) up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{1}{30}\). The engine of the car is working at a rate of 12 kW. The resistance to motion from non-gravitational forces has magnitude 500 N. Find the value of \(V\). [5]

A caravan of mass 600 kg is towed by a car of mass 900 kg along a straight horizontal road. The towbar joining the car to the caravan is modelled as a light rod parallel to the road. The total resistance to motion of the car is modelled as having magnitude 300 N. The total resistance to motion of the caravan is modelled as having magnitude 150 N. At a given instant the car and the caravan are moving with speed 20 m s\(^{-1}\) and acceleration 0.2 m s\(^{-2}\).

1. Find the power being developed by the car's engine at this instant. [5]
2. Find the tension in the towbar at this instant. [2]

A pump raises water from a reservoir at a depth of 25 m below ground level. The water is delivered at ground level with speed 12 ms\(^{-1}\) through a pipe of radius 4 cm. Find

1. the potential and kinetic energy given to the water each second, [5 marks]
2. the rate, in kW, at which the pump is working. [2 marks]

[1 m\(^3\) of water has a mass of 1000 kg.]

A van of mass 1600 kg is moving with constant speed down a straight road inclined at 7° to the horizontal. The non-gravitational resistance to the van's motion has a constant magnitude of 2000 N and the engine of the van is working at a rate of 1.5 kW. Find

1. the constant speed of the van, [5 marks]
2. the acceleration of the van if the resistance is suddenly reduced to 1900 N. [2 marks]

A motor-cycle and its rider have a total mass of 460 kg. The maximum rate at which the cycle's engine can work is 25 920 W and the maximum speed of the cycle on a horizontal road is 36 ms\(^{-1}\). A variable resisting force acts on the cycle and has magnitude \(kv^2\), where \(v\) is the speed of the cycle in ms\(^{-1}\).

1. Show that \(k = \frac{5}{8}\). [4 marks]
2. Find the acceleration of the cycle when it is moving at 25 ms\(^{-1}\) on the horizontal road, with its engine working at full power. [5 marks]

A car of mass 700 kg is travelling up a hill which is inclined at a constant angle of \(5°\) to the horizontal. At a certain point \(P\) on the hill the car's speed is 20 m s\(^{-1}\). The point \(Q\) is 400 m further up the hill from \(P\), and at \(Q\) the car's speed is 15 m s\(^{-1}\).

1. Calculate the work done by the car's engine as the car moves from \(P\) to \(Q\), assuming that any resistances to the car's motion may be neglected. [4]

Assume instead that the resistance to the car's motion between \(P\) and \(Q\) is a constant force of magnitude 200 N.

1. Given that the acceleration of the car at \(Q\) is zero, show that the power of the engine as the car passes through \(Q\) is 12.0 kW, correct to 3 significant figures. [3]
2. Given that the power of the car's engine at \(P\) is the same as at \(Q\), calculate the car's retardation at \(P\). [3]

A block is being pushed in a straight line along horizontal ground by a force of 18 N inclined at 15° below the horizontal. The block moves a distance of 6 m in 5 s with constant speed. Find

1. the work done by the force, [3]
2. the power with which the force is working. [2]

A car of mass 1500 kg travels along a straight horizontal road. The resistance to the motion of the car is \(kv^{\frac{3}{2}}\) N, where \(v\) ms\(^{-1}\) is the speed of the car and \(k\) is a constant. At the instant when the engine produces a power of 15000 W, the car has speed 15 ms\(^{-1}\) and is accelerating at 0.4 ms\(^{-2}\).

1. Find the value of \(k\). [4]

It is given that the greatest steady speed of the car on this road is 30 ms\(^{-1}\).

1. Find the greatest power that the engine can produce. [3]

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]

A car of mass 1400 kg is travelling on a straight horizontal road against a constant resistance to motion of 600 N. At a certain instant the car is accelerating at \(0.3 \text{ m s}^{-2}\) and the engine of the car is working at a rate of 23 kW.

1. Find the speed of the car at this instant. [3]

Subsequently the car moves up a hill inclined at \(10°\) to the horizontal at a steady speed of \(12 \text{ m s}^{-1}\). The resistance to motion is still a constant 600 N.

1. Calculate the power of the car's engine as it moves up the hill. [3]