Find power at constant speed

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 \(B\).

2 A high-speed train of mass 490000 kg is moving along a straight horizontal track at a constant speed of \(85 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The engines are supplying 4080 kW of power.

1. Show that the resistance force is 48000 N .
2. The train comes to a hill inclined at an angle \(\theta ^ { \circ }\) above the horizontal, where \(\sin \theta ^ { \circ } = \frac { 1 } { 200 }\). Given that the resistance force is unchanged, find the power required for the train to keep moving at the same constant speed of \(85 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

2. A van of mass 1200 kg is travelling along a straight horizontal road. The resistance to the motion of the van has a constant magnitude of 650 N and the van's engine is working at a rate of 30 kW .

1. Find the acceleration of the van when its speed is \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The van now travels up a straight road which is inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 12 }\). The resistance to the motion of the van from non-gravitational forces has a constant magnitude of 650 N . The van moves up the road at a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
2. Find, in kW , the rate at which the van's engine is now working.
   "

1. A car of mass 400 kg is moving up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 14 }\). The resistance to motion of the car from non-gravitational forces is modelled as a constant force of magnitude \(R\) newtons. When the car is moving at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the power developed by the car's engine is 10 kW .

Find the value of \(R\).

2. A lorry of mass 1800 kg travels along a straight horizontal road. The lorry's engine is working at a constant rate of 30 kW . When the lorry's speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its acceleration is \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The magnitude of the resistance to the motion of the lorry is \(R\) newtons.

1. Find the value of \(R\). The lorry now travels up a straight road which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 12 }\). The magnitude of the non-gravitational resistance to motion is \(R\) newtons. The lorry travels at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the new rate of working of the lorry's engine.

1. A lorry of mass 2000 kg is moving down a straight road inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 25 }\). The resistance to motion is modelled as a constant force of magnitude 1600 N . The lorry is moving at a constant speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

Find, in kW , the rate at which the lorry's engine is working.

5 A train consisting of an engine and eight carriages moves on a straight horizontal track. A constant resistive force of 2400 N acts on the engine.
A constant resistive force of 300 N acts on each of the eight carriages.
The maximum speed of the train on the track is \(120 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) Find the maximum power output of the engine.
Fully justify your answer.

3 A van, of mass 1500 kg , travels at a constant speed of \(22 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a slope inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 25 }\). The van experiences a resistance force of 8000 N .
Find the power output of the van's engine, giving your answer in kilowatts.

5 A train consists of an engine and five carriages. A constant resistance force of 3000 N acts on the engine, and a constant resistance force of 400 N acts on each of the five carriages. The maximum speed of the train on a horizontal track is \(90 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).

1. Show that this speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Hence find the maximum power output of the engine. Give your answer in kilowatts.
   (3 marks)

5 A train, of mass 600 tonnes, travels at constant speed up a slope inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 40 }\). The speed of the train is \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it experiences total resistance forces of 200000 N . Find the power produced by the train, giving your answer in kilowatts.

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 crate of mass 200 kg is being pulled at constant speed along horizontal ground by a horizontal rope attached to a winch. The winch is working at a constant rate of 4.5 kW and there is a constant resistance to the motion of the crate of magnitude 600 N.

1. Find the time that it takes for the crate to move a distance of 15 m. [2] The rope breaks after the crate has moved 15 m.
2. Find the time taken, after the rope breaks, for the crate to come to rest. [3]

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 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]