6.02l Power and velocity: P = Fv

A car has mass 550 kg. When the car travels along a straight horizontal road there is a constant resistance to the motion of magnitude \(R\) newtons, the engine of the car is working at a rate of \(P\) watts and the car maintains a constant speed of 30 m s\(^{-1}\). When the car travels up a line of greatest slope of a hill which is inclined at \(\theta\) to the horizontal, where \(\sin \theta = \frac{1}{14}\), with the engine working at a rate of \(P\) watts, it maintains a constant speed of 25 m s\(^{-1}\). The non-gravitational resistance to motion when the car travels up the hill is a constant force of magnitude \(R\) newtons.

1. 1. Find the value of \(R\).
   2. Find the value of \(P\). [8]
2. Find the acceleration of the car when it travels along the straight horizontal road at 20 m s\(^{-1}\) with the engine working at 50 kW. [4]

A van of mass 1500 kg is driving up a straight road inclined at an angle \(α\) to the horizontal, where \(\sin α = \frac{1}{16}\). The resistance to motion due to non-gravitational forces is modelled as a constant force of magnitude 1000 N. Given that initially the speed of the van is 30 m s\(^{-1}\) and that the van's engine is operating at a rate of 60 kW,

1. calculate the magnitude of the initial deceleration of the van. [4]

When travelling up the same hill, the rate of working of the van's engine is increased to 80 kW. Using the same model for the resistance due to non-gravitational forces,

1. calculate in m s\(^{-1}\) the constant speed which can be sustained by the van at this rate of working. [4]

1. Give one reason why the use of this model for resistance may mean that your answer to part (b) is too high. [1]

A particle \(P\) moves in a straight line so that, at time \(t\) seconds, its acceleration \(a\) m s\(^{-2}\) is given by $$a = \begin{cases} 4t - t^2, & 0 \leq t \leq 3, \\ \frac{27}{t^2}, & t > 3. \end{cases}$$ At \(t = 0\), \(P\) is at rest. Find the speed of \(P\) when

1. \(t = 3\), [3]

1. \(t = 6\). [5]

Figure 1 shows the path taken by a cyclist in travelling on a section of a road. When the cyclist comes to the point \(A\) on the top of a hill, she is travelling at 8 m s\(^{-1}\). She descends a vertical distance of 20 m to the bottom of the hill. The road then rises to the point \(B\) through a vertical distance of 12 m. When she reaches the point \(B\), her speed is 5 m s\(^{-1}\). The total mass of the cyclist and the cycle is 80 kg and the total distance along the road from \(A\) to \(B\) is 500 m. By modelling the resistance to the motion of the cyclist as of constant magnitude 20 N,

1. find the work done by the cyclist in moving from \(A\) to \(B\). [5]

At \(B\) the road is horizontal. Given that at \(B\) the cyclist is accelerating at 0.5 m s\(^{-2}\),

1. find the power generated by the cyclist at \(B\). [4]

A car of mass 1000 kg is moving along a straight horizontal road with a constant acceleration of \(j\) m s\(^{-2}\). The resistance to motion is modelled as a constant force of magnitude 1200 N. When the car is travelling at 12 m s\(^{-1}\), the power generated by the engine of the car is 24 kW.

1. Calculate the value of \(j\). [4]

When the car is travelling at 14 m s\(^{-1}\), the engine is switched off and the car comes to rest, without braking, in a distance of \(d\) metres. Assuming the same model for resistance,

1. use the work-energy principle to calculate the value of \(d\). [3]

1. Give a reason why the model used for the resistance to motion may not be realistic. [1]

A uniform ladder \(AB\), of mass \(m\) and length \(2a\), has one end \(A\) on rough horizontal ground. The other end \(B\) rests against a smooth vertical wall. The ladder is in a vertical plane perpendicular to the wall. The ladder makes an angle \(α\) with the horizontal, where \(\tan α = \frac{4}{3}\). A child of mass \(2m\) stands on the ladder at \(C\) where \(AC = \frac{1}{4}a\), as shown in Fig. 1. The ladder and the child are in equilibrium. By modelling the ladder as a rod and the child as a particle, calculate the least possible value of the coefficient of friction between the ladder and the ground. [9]

A van of mass 1500 kg is driving up a straight road inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{1}{12}\). The resistance to motion due to non-gravitational forces is modelled as a constant force of magnitude 1000 N. Given that initially the speed of the van is 30 m s\(^{-1}\) and that the van's engine is working at a rate of 60 kW,

1. calculate the magnitude of the initial deceleration of the van. [4]

When travelling up the same hill, the rate of working of the van's engine is increased to 80 kW. Using the same model for the resistance due to non-gravitational forces,

1. calculate in m s\(^{-1}\) the constant speed which can be sustained by the van at this rate of working. [4]
2. Give one reason why the use of this model for resistance may mean that your answer to part (b) is too high. [1]

A car of mass 1000 kg is moving along a straight horizontal road with a constant acceleration of \(f\) m s\(^{-2}\). The resistance to motion is modelled as a constant force of magnitude 1200 N. When the car is travelling at 12 m s\(^{-1}\), the power generated by the engine of the car is 24 kW.

1. Calculate the value of \(f\). [4]

When the car is travelling at 14 m s\(^{-1}\), the engine is switched off and the car comes to rest, without braking, in a distance of \(d\) metres. Assuming the same model for resistance,

1. use the work-energy principle to calculate the value of \(d\). [3]
2. Give a reason why the model used for the resistance to motion may not be realistic. [1]

A car of mass 1000 kg is moving along a straight horizontal road. The resistance to motion is modelled as a constant force of magnitude \(R\) newtons. The engine of the car is working at a rate of 12 kW. When the car is moving with speed 15 m s\(^{-1}\), the acceleration of the car is 0.2 m s\(^{-2}\).

1. Show that \(R = 600\). [4]

The car now moves with constant speed \(U\) m s\(^{-1}\) downhill on a straight road inclined at \(\theta\) to the horizontal, where \(\sin \theta = \frac{1}{30}\). The engine of the car is now working at a rate of 7 kW. The resistance to motion from non-gravitational forces remains of magnitude \(R\) newtons.

1. Calculate the value of \(U\). [5]

A car of mass 800 kg is moving at a constant speed of 15 m s\(^{-1}\) down a straight road inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{3}{4}\). The resistance to motion from non-gravitational forces is modelled as a constant force of magnitude 900 N.

1. Find, in kW, the rate of working of the engine of the car. [4]

When the car is travelling down the road at 15 m s\(^{-1}\), the engine is switched off. The car comes to rest in time \(T\) seconds after the engine is switched off. The resistance to motion from non-gravitational forces is again modelled as a constant force of magnitude 900 N.

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

A car of mass 1000 kg is moving at a constant speed of 16 m s\(^{-1}\) up a straight road inclined at an angle \(\theta\) to the horizontal. The rate of working of the engine of the car is 20 kW and the resistance to motion from non-gravitational forces is modelled as a constant force of magnitude 550 N.

1. Show that \(\sin \theta = \frac{1}{14}\). [5]

When the car is travelling up the road at 16 m s\(^{-1}\), the engine is switched off. The car comes to rest, without braking, having moved a distance \(y\) metres from the point where the engine was switched off. The resistance to motion from non-gravitational forces is again modelled as a constant force of magnitude 550 N.

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

A cyclist and her bicycle have a total mass of \(70\) kg. She cycles along a straight horizontal road with constant speed \(3.5 \text{ ms}^{-1}\). She is working at a constant rate of \(490\) W.

1. Find the magnitude of the resistance to motion. [4]

The cyclist now cycles down a straight road which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{1}{14}\), at a constant speed \(U \text{ ms}^{-1}\). The magnitude of the non-gravitational resistance to motion is modelled as \(40U\) newtons. She is now working at a constant rate of \(24\) W.

1. Find the value of \(U\). [7]

A cyclist and her cycle have a combined mass of \(75\) kg. The cyclist is cycling up a straight road inclined at \(5°\) to the horizontal. The resistance to the motion of the cyclist from non-gravitational forces is modelled as a constant force of magnitude \(20\) N. At the instant when the cyclist has a speed of \(12\) m s\(^{-1}\), she is decelerating at \(0.2\) m s\(^{-2}\).

1. Find the rate at which the cyclist is working at this instant. [5]

When the cyclist passes the point \(A\) her speed is \(8\) m s\(^{-1}\). At \(A\) she stops working but does not apply the brakes. She comes to rest at the point \(B\). The resistance to motion from non-gravitational forces is again modelled as a constant force of magnitude \(20\) N.

1. Use the work-energy principle to find the distance \(AB\). [5]

\includegraphics{figure_1} Figure 1 shows the path taken by a cyclist in travelling on a section of a road. When the cyclist comes to the point \(A\) on the top of a hill, she is travelling at 8 m s\(^{-1}\). She descends a vertical distance of 20 m to the bottom of the hill. The road then rises to the point \(B\) through a vertical distance of 12 m. When she reaches \(B\), her speed is 5 m s\(^{-1}\). The total mass of the cyclist and the cycle is 80 kg and the total distance along the road from \(A\) to \(B\) is 500 m. By modelling the resistance to the motion of the cyclist as of constant magnitude 20 N,

1. find the work done by the cyclist in moving from \(A\) to \(B\). [5]

At \(B\) the road is horizontal. Given that at \(B\) the cyclist is accelerating at 0.5 m s\(^{-2}\),

1. find the power generated by the cyclist at \(B\). [4]

A girl and her bicycle have a combined mass of 64 kg. She cycles up a straight stretch of road which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{1}{14}\). She cycles at a constant speed of 5 m s\(^{-1}\). When she is cycling at this speed, the resistance to motion from non-gravitational forces has magnitude 20 N.

1. Find the rate at which the cyclist is working. [4]

She now turns round and comes down the same road. Her initial speed is 5 m s\(^{-1}\), and the resistance to motion is modelled as remaining constant with magnitude 20 N. She free-wheels down the road for a distance of 80 m. Using this model,

1. find the speed of the cyclist when she has travelled a distance of 80 m. [5]

The cyclist again moves down the same road, but this time she pedals down the road. The resistance is now modelled as having magnitude proportional to the speed of the cyclist. Her initial speed is again 5 m s\(^{-1}\) when the resistance to motion has magnitude 20 N.

1. Find the magnitude of the resistance to motion when the speed of the cyclist is 8 m s\(^{-1}\). [1]

The cyclist works at a constant rate of 200 W.

1. Find the magnitude of her acceleration when her speed is 8 m s\(^{-1}\). [4]

A car of mass 1200 kg moves along a straight horizontal road with a constant speed of 24 m s\(^{-1}\). The resistance to motion of the car has magnitude 600 N.

1. Find, in kW, the rate at which the engine of the car is working. [2]

The car now moves up a hill inclined at \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{1}{20}\). The resistance to motion of the car from non-gravitational forces remains of magnitude 600 N. The engine of the car now works at a rate of 30 kW.

1. Find the acceleration of the car when its speed is 20 m s\(^{-1}\). [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 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 car has mass 1200 kg. The maximum power of the car's engine is 32 kW. The resistance to motion due to non-gravitational forces is modelled as a force of constant magnitude 800 N. When the car is travelling on a horizontal road at constant speed \(V\) m s\(^{-1}\), the engine of the car is working at maximum power.

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

The car now travels downhill on a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{1}{40}\). The resistance to motion due to non-gravitational forces is still modelled as a force of constant magnitude 800 N. Given that the engine of the car is again working at maximum power,

1. find the acceleration of the car when its speed is 20 m s\(^{-1}\). [4]

A cyclist and her bicycle have a combined mass of \(100\) kg. She is working at a constant rate of \(80\) W and is moving in a straight line on a horizontal road. The resistance to motion is proportional to the square of her speed. Her initial speed is \(4\) m s\(^{-1}\) and her maximum possible speed under these conditions is \(20\) m s\(^{-1}\). When she is at a distance \(x\) m from a fixed point \(O\) on the road, she is moving with speed \(v\) m s\(^{-1}\) away from \(O\).

1. Show that $$v \frac{dv}{dx} = \frac{8000 - v^3}{10000v}.$$ [5]
2. Find the distance she travels as her speed increases from \(4\) m s\(^{-1}\) to \(8\) m s\(^{-1}\). [5]
3. Use the trapezium rule, with 2 intervals, to estimate how long it takes for her speed to increase from \(4\) m s\(^{-1}\) to \(8\) m s\(^{-1}\). [4]

A small car, of mass 850 kg, moves on a straight horizontal road. Its engine is working at its maximum rate of 25 kW, and a constant resisting force of magnitude 900 N opposes the car's motion.

1. Find the acceleration of the car when it is moving with speed 15 ms\(^{-1}\). [3 marks]
2. Find the maximum speed of the car on the horizontal road. [3 marks]

With the engine still working at 25 kW and the non-gravitational resistance remaining at 900 N, the car now climbs a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{1}{10}\).

1. Find the maximum speed of the car on this hill. [4 marks]

A lorry of mass 4200 kg can develop a maximum power of 84 kW. On any road the lorry experiences a non-gravitational resisting force which is directly proportional to its speed. When the lorry is travelling at 20 ms\(^{-1}\) the resisting force has magnitude 2400 N. Find the maximum speed of the lorry when it is

1. travelling on a horizontal road, [4 marks]
2. climbing a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{1}{7}\). [6 marks]

An engine of mass \(20\,000\) kg climbs a hill inclined at \(10°\) to the horizontal. The total non-gravitational resistance to its motion has magnitude \(35\,000\) N and the maximum speed of the engine on the hill is \(15\) ms\(^{-1}\).

1. Find, in kW, the maximum rate at which the engine can work. [4 marks]
2. Find the maximum speed of the engine when it is travelling on a horizontal track against the same non-gravitational resistance as before. [3 marks]

A car, of mass 1100 kg, pulls a trailer of mass 550 kg along a straight horizontal road by means of a rigid tow-bar. The car is accelerating at 1.2 ms\(^{-2}\) and the resistances to the motion of the car and trailer have magnitudes 500 N and 200 N respectively.

1. Show that the driving force produced by the engine of the car is 2680 N. [3 marks]
2. Find the tension in the tow-bar between the car and the trailer. [3 marks]
3. Find the rate, in kW, at which the car's engine is working when the car is moving with speed 18 ms\(^{-1}\). [2 marks]

When the car is moving at 18 ms\(^{-1}\) it starts to climb a straight hill which is inclined at \(6°\) to the horizontal. If the car's engine continues to work at the same rate and the resistances to motion remain the same as previously,

1. find the acceleration of the car at the instant when it starts to climb the hill. [3 marks]
2. Show that tension in the tow-bar remains unchanged. [3 marks]

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]