6.02k Power: rate of doing work

4. 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 \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(\mathrm { R } = 260\). 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 \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the car's acceleration is a \(\mathrm { m } \mathrm { s } ^ { - 2 }\).
2. Find the value of a.

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 \(R\).

5. A car of mass 1000 kg is towing a trailer of mass 1500 kg along a straight horizontal road. The tow-bar joining the car to the trailer is modelled as a light rod parallel to the road. The total resistance to motion of the car is modelled as having constant magnitude 750 N . The total resistance to motion of the trailer is modelled as of magnitude \(R\) newtons, where \(R\) is a constant. When the engine of the car is working at a rate of 50 kW , the car and the trailer travel at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(R = 1250\). When travelling at \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the driver of the car disengages the engine and applies the brakes. The brakes provide a constant braking force of magnitude 1500 N to the car. The resisting forces of magnitude 750 N and 1250 N are assumed to remain unchanged. Calculate
2. the deceleration of the car while braking,
3. the thrust in the tow-bar while braking,
4. the work done, in kJ , by the braking force in bringing the car and the trailer to rest.
5. Suggest how the modelling assumption that the resistances to motion are constant could be refined to be more realistic.

4. The resistance to the motion of a cyclist is modelled as \(k v ^ { 2 } \mathrm {~N}\), where \(k\) is a constant and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the cyclist. The total mass of the cyclist and his bicycle is 100 kg . The cyclist freewheels down a slope inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 20 }\), at a constant speed of \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(k = 4\). The cyclist ascends a slope inclined at an angle \(\beta\) to the horizontal, where \(\sin \beta = \frac { 1 } { 40 }\), at a constant speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the rate at which the cyclist is working.
   (6 marks)

5. A straight road is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 20 }\). A lorry of mass 4800 kg moves up the road at a constant speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The non-gravitational resistance to the motion of the lorry is constant and has magnitude 2000 N .

1. Find, in kW to 3 significant figures, the rate of working of the lorry's engine.
   (5) The road becomes horizontal. The lorry's engine continues to work at the same rate and the resistance to motion remains the same. Find
2. the acceleration of the lorry immediately after the road becomes horizontal,
   (3)
3. the maximum speed, in \(\mathrm { m } \mathrm { s } ^ { - 1 }\) to 3 significant figures, at which the lorry will go along the horizontal road.
   (3)

2 \includegraphics[max width=\textwidth, alt={}, center]{34024618-0ff9-44a1-ac57-d4d7e8a3655e-2_431_421_881_861} A uniform disc of radius 0.4 m is free to rotate without friction in a vertical plane about a horizontal axis through its centre. The moment of inertia of the disc about the axis is \(0.2 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). One end of a light inextensible string is attached to a point on the rim of the disc and the string is wound round the rim. The other end of the string is attached to a particle of mass 1.5 kg which hangs freely (see diagram). The system is released from rest. Find

1. the angular acceleration of the disc,
2. the speed of the particle when the disc has turned through an angle of \(\frac { 1 } { 6 } \pi\).

2 \includegraphics[max width=\textwidth, alt={}, center]{d3e9a568-a9ea-483e-8e65-90fdc4a69781-2_431_421_881_861} A uniform disc of radius 0.4 m is free to rotate without friction in a vertical plane about a horizontal axis through its centre. The moment of inertia of the disc about the axis is \(0.2 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). One end of a light inextensible string is attached to a point on the rim of the disc and the string is wound round the rim. The other end of the string is attached to a particle of mass 1.5 kg which hangs freely (see diagram). The system is released from rest. Find

1. the angular acceleration of the disc,
2. the speed of the particle when the disc has turned through an angle of \(\frac { 1 } { 6 } \pi\).

3 \includegraphics[max width=\textwidth, alt={}, center]{b486decd-75b8-44bd-889f-2472f1163871-2_570_419_1539_863} A uniform disc, of mass 2 kg and radius 0.2 m , is free to rotate in a vertical plane about a smooth horizontal axis through its centre. One end of a light inextensible string is attached to a point on the rim of the disc and the string is wound round the rim. The other end of the string is attached to a small block of mass 4 kg , which hangs freely (see diagram). The system is released from rest. During the subsequent motion, the block experiences a constant resistance to its motion, of magnitude \(R \mathrm {~N}\). Given that the angular speed of the disc after it has turned through 2 radians is \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\), find \(R\) and the tension in the string.
[0pt] [9]

5 A model train has mass 100 kg . When the train is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the resistance to its motion is \(3 v ^ { 2 } \mathrm {~N}\) and the power output of the train is \(\frac { 3000 } { v } \mathrm {~W}\).

1. Show that the driving force acting on the train is 120 N at an instant when the train is moving with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the acceleration of the train at an instant when it is moving horizontally with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The train moves with constant speed up a straight hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 98 }\).
3. Calculate the speed of the train.

4 A car of mass 1200 kg has a maximum speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when travelling on a horizontal road. The car experiences a resistance of \(k v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car and \(k\) is a constant. The maximum power of the car's engine is 45000 W .

1. Show that \(k = 50\).
2. Find the maximum possible acceleration of the car when it is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a horizontal road.
3. The car climbs a hill, which is inclined at an angle of \(10 ^ { \circ }\) to the horizontal, at a constant speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Calculate the power of the car's engine.

4 A car of mass 800 kg experiences a resistance of magnitude \(k v ^ { 2 } \mathrm {~N}\), where \(k\) is a constant and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the car's speed. The car's engine is working at a constant rate of \(P \mathrm {~W}\). At an instant when the car is travelling on a horizontal road with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(0.75 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At an instant when the car is ascending a hill of constant slope \(12 ^ { \circ }\) to the horizontal with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Show that \(k = 0.900\), correct to 3 decimal places, and find \(P\). The power is increased to \(1.5 P \mathrm {~W}\).
2. Calculate the maximum steady speed of the car on a horizontal road.

1 Find the average power exerted by a climber of mass 75 kg when climbing a vertical distance of 40 m in 2 minutes.

4 A car of mass 700 kg is moving along a horizontal road against a constant resistance to motion of 400 N . At an instant when the car is travelling at \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the driving force of the car at this instant.
2. Find the power at this instant. The maximum steady speed of the car on a horizontal road is \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the maximum power of the car. The car now moves at maximum power against the same resistance up a slope of constant angle \(\theta ^ { \circ }\) to the horizontal. The maximum steady speed up the slope is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
4. Find \(\theta\).

6 A car, of mass 1200 kg , moves on a straight horizontal road where it has a maximum speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) When the car travels at a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) it experiences a resistance force which can be modelled as being of magnitude 30 v newtons. 6

1. Show that the power output of the car is 48000 W , when it is travelling at its maximum speed. 6
2. Find the maximum acceleration of the car when it is travelling at a speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) [0pt] [4 marks]

5 The engine of a car of mass 1200 kg produces a maximum power of 40 kW .
In an initial model of the motion of the car the total resistance to motion is assumed to be constant.

1. Given that the greatest steady speed of the car on a straight horizontal road is \(42 \mathrm {~ms} ^ { - 1 }\), find the magnitude of the resistance force. The car is attached to a trailer of mass 200 kg by a light rigid horizontal tow bar. The greatest steady speed of the car and trailer on the road is now \(30 \mathrm {~ms} ^ { - 1 }\). The resistance to motion of the trailer may also be assumed constant.
2. Find the magnitude of the resistance force on the trailer. The car and trailer again travel along the road. At one instant their speed is \(15 \mathrm {~ms} ^ { - 1 }\) and their acceleration is \(0.57 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
3. (a) Find the power of the engine of the car at this instant.
   (b) Find the magnitude of the tension in the tow bar at this instant. In a refined model of the motion of the car and trailer the resistance to the motion of each is assumed to be zero until they reach a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the speed is \(10 \mathrm {~ms} ^ { - 1 }\) or above the same constant resistance forces as in the first model are assumed to apply to each. The car and trailer start at rest on the road and accelerate, using maximum power.
4. Without carrying out any further calculations,
   (a) explain whether the time taken to attain a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) would be predicted to be lower, the same or higher using the refined model compared with the original model,
   (b) explain whether the greatest steady speed of the system would be predicted to be lower, the same or higher using the refined model compared with the original model.

3 A car of mass 1500 kg has an engine with maximum power 60 kW . When the car is travelling at \(10 \mathrm {~ms} ^ { - 1 }\) along a straight horizontal road using maximum power, its acceleration is \(3.3 \mathrm {~ms} ^ { - 2 }\). In an initial model of the motion of the car it is assumed that the resistance to motion is constant.

1. Using this initial model, find the greatest possible steady speed of the car along the road. In a refined model the resistance to motion is assumed to be proportional to the speed of the car.
2. Using this refined model, find the greatest possible steady speed of the car along the road. The greatest possible steady speed of the car on the road is measured and found to be \(21.6 \mathrm {~ms} ^ { - 1 }\).
3. Explain what this value means about the models used in parts (a) and (b).

4 A cyclist is riding a bicycle along a straight road which is inclined at an angle of \(4 ^ { \circ }\) to the horizontal. The cyclist is working at a constant rate of 250 W . The combined mass of the cyclist and bicycle is 80 kg and the resistance to their motion is a constant 70 N . Determine the maximum constant speed at which the cyclist can ride the bicycle

• up the hill, and
• down the hill.

8 As part of an industrial process a single pump causes the intake of a liquid chemical to the bottom end of a tube, draws it up the tube and then discharges it through a nozzle at the top end of the tube. The tube is straight and narrow, 35 m long and inclined at an angle of \(26 ^ { \circ }\) to the horizontal. The chemical arrives at the intake at the bottom end of the tube with a speed of \(6.2 \mathrm {~ms} ^ { - 1 }\). At the top end of the tube the chemical is discharged horizontally with a speed of \(14.3 \mathrm {~ms} ^ { - 1 }\) (see diagram). In total, the pump discharges 1500 kg of chemical through the nozzle each hour. \includegraphics[max width=\textwidth, alt={}, center]{98053e88-1aec-4b0d-ae5f-ece4ad340266-5_405_1175_685_242} In order to model the changes to the mechanical energy of the chemical during the entire process of intake, drawing and discharge, the following modelling assumptions are made.

• At any instant the total resistance to the motion of all the liquid in the tube is 40 N .
• All other resistances to motion are ignored.
• The liquid in the tube moves at a constant speed of \(6.2 \mathrm {~ms} ^ { - 1 }\).
  1. 1. Find the difference between the total amount of energy output by the pump each hour and the total amount of mechanical energy gained by the chemical each hour.
     2. Give one reason why the model underestimates the power of the engine.

3 A crate of mass 45 kg is sliding with a speed of \(0.8 \mathrm {~ms} ^ { - 1 }\) in a straight line across a smooth horizontal floor. One end of a light inextensible rope is attached to the crate. At a certain instant a builder takes the other end of the rope and starts to pull, applying a constant force of 80 N for 5 seconds. While the builder is pulling the crate, the rope makes a constant angle of \(40 ^ { \circ }\) above the horizontal. Both the rope and the velocity of the crate lie in the same vertical plane (see diagram).
It may be assumed that there is no resistance to the motion of the crate.

1. Determine the work done by the builder in pulling the crate.
2. 1. Find the kinetic energy of the crate at the instant when the builder stops pulling the crate.
   2. Explain why the answers to part (a) and part (b)(i) are not equal.
3. Find the average power developed by the builder in pulling the crate.
4. Calculate the total impulse exerted on the crate by the builder.

4 A rower is rowing a boat in a straight line across a lake. The combined mass of the rower, boat and oars is 240 kg . The maximum power that the rower can generate is 450 W . In a model of the motion of the boat it is assumed that the total resistance to the motion of the boat is 150 N at any instant when the boat is in motion.

1. Find the maximum possible acceleration of the boat, according to the model, at an instant when its speed is \(0.5 \mathrm {~ms} ^ { - 1 }\). At one stage in its motion the boat is travelling at a constant speed and the rower is generating power at an average rate of 210 W , which is assumed to be constant. The boat passes a pole and then, after travelling 350 m , a second pole.
2. Determine how long it takes, according to the model, for the boat to travel between the two poles.
3. State a reason why the assumption that the rower's generated power is constant may be unrealistic.

4 A particle \(B\) of mass 5 kg is at rest at the bottom of a slope which is angled at \(\sin ^ { - 1 } 0.2\) above the horizontal. A constant force \(D\) initially acts directly up the slope on \(B\). The total resistance to the motion of \(B\) is modelled as being a constant 12 N .
At the instant that \(D\) stops acting, the speed of \(B\) is \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) has moved 90 m up the slope.
Determine the average power of \(D\) over the time that \(D\) has been acting on \(B\).

6 A motorbike and its rider, together denoted by \(M\), have a combined mass of 360 kg . The resistive force experienced by \(M\) when it is in motion is modelled as being proportional to the speed it is moving at. All motion of \(M\) is on a straight horizontal road. It is found that with the engine of the motorbike working at a rate of 12 kW , the maximum constant speed that \(M\) can move at is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Determine the speed of \(M\) such that with the engine working at a rate of 12 kW the acceleration of \(M\) is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1 A car of mass 1200 kg is driven on a long straight horizontal road. There is a constant force of 250 N resisting the motion of the car. The engine of the car is working at a constant power of 10 kW .

1. The car can travel at constant speed \(v \mathrm {~ms} ^ { - 1 }\) along the road. Find \(v\).
2. Find the acceleration of the car at an instant when its speed is \(30 \mathrm {~ms} ^ { - 1 }\).

3 The mass of a truck is 6000 kg and the maximum power that its engine can generate is 90 kW . In a model of the motion of the truck it is assumed that while it is moving the total resistance to its motion is constant. At first the truck is driven along a straight horizontal road. The greatest constant speed that it can be driven at when it is using maximum power is \(25 \mathrm {~ms} ^ { - 1 }\).

1. Find the value of the resistance to motion. The truck is being driven along the horizontal road with the engine working at 60 kW .
2. Find the acceleration of the truck at the instant when its speed is \(10 \mathrm {~ms} ^ { - 1 }\). The truck is now driven down a straight road which is inclined at an angle \(\theta\) below the horizontal. The greatest constant speed that the truck can be driven at maximum power is \(40 \mathrm {~ms} ^ { - 1 }\).
3. Determine the value of \(\theta\).