6.02l Power and velocity: P = Fv

1. \hspace{0pt} [In this question use \(\mathrm { g } = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) ]

A jogger of mass 60 kg runs along a straight horizontal road at a constant speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The total resistance to the motion of the jogger is modelled as a constant force of magnitude 30 N .

1. Find the rate at which the jogger is working. The jogger now comes to a hill which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 1 } { 15 }\). Because of the hill, the jogger reduces her speed to \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and maintains this constant speed as she runs up the hill. The total resistance to the motion of the jogger from non-gravitational forces continues to be modelled as a constant force of magnitude 30 N .
2. Find the rate at which she has to work in order to run up the hill at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. A car of mass 600 kg pulls a trailer of mass 150 kg along a straight horizontal road. The trailer is connected to the car by a light inextensible towbar, which is parallel to the direction of motion of the car. The resistance to the motion of the trailer is modelled as a constant force of magnitude 200 N . At the instant when the speed of the car is \(v \mathrm {~ms} ^ { - 1 }\), the resistance to the motion of the car is modelled as a force of magnitude \(( 200 + \lambda v ) \mathrm { N }\), where \(\lambda\) is a constant.

When the engine of the car is working at a constant rate of 15 kW , the car is moving at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

1. Show that \(\lambda = 8\) Later on, the car is pulling the trailer up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 15 }\) The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 200 N at all times. At the instant when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the car from non-gravitational forces is modelled as a force of magnitude \(( 200 + 8 v ) \mathrm { N }\). The engine of the car is again working at a constant rate of 15 kW .
   When \(v = 10\), the towbar breaks. The trailer comes to instantaneous rest after moving a distance \(d\) metres up the road from the point where the towbar broke.
2. Find the acceleration of the car immediately after the towbar breaks.
3. Use the work-energy principle to find the value of \(d\).
   

1. A truck of mass 1200 kg is moving along a straight horizontal road.

At the instant when the speed of the truck is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the truck is modelled as a force of magnitude \(( 900 + 9 v ) \mathrm { N }\). The engine of the truck is working at a constant rate of 25 kW .

1. Find the deceleration of the truck at the instant when \(v = 25\) Later on, the truck is moving up a straight road that is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\) At the instant when the speed of the truck is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the truck from non-gravitational forces is modelled as a force of magnitude ( \(900 + 9 v\) ) N. When the engine of the truck is working at a constant rate of 25 kW the truck is moving up the road at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the value of \(V\).

1. A van of mass 900 kg is moving along a straight horizontal road.

At the instant when the speed of the van is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the van is modelled as a force of magnitude \(( 500 + 7 v ) \mathrm { N }\). When the engine of the van is working at a constant rate of 18 kW , the van is moving along the road at a constant speed \(V \mathrm {~ms} ^ { - 1 }\)

1. Find the value of \(V\). Later on, the van is moving up a straight road that is inclined to the horizontal at an angle \(\theta\), where \(\sin \theta = \frac { 1 } { 21 }\) At the instant when the speed of the van is \(v \mathrm {~ms} ^ { - 1 }\), the resistance to the motion of the van from non-gravitational forces is modelled as a force of magnitude \(( 500 + 7 v ) \mathrm { N }\). The engine of the van is again working at a constant rate of 18 kW .
2. Find the acceleration of the van at the instant when \(v = 15\)

2. \begin{figure}[h] \end{figure} A van of mass 600 kg is moving up a straight road which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 15 }\). The van is towing a trailer of mass 150 kg . The van is attached to the trailer by a 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 200 N .
The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 100 N . The towbar is modelled as a light rod.
The engine of the van is working at a constant rate of 12 kW .
Find the tension in the towbar at the instant when the speed of the van is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

1. A car of mass 1000 kg moves in a straight line along a horizontal road at a constant speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to the motion of the car is a constant force of magnitude 400 N.

The engine of the car is working at a constant rate of 16 kW .

1. Find the value of \(U\). The car now pulls a trailer of mass 600 kg in a straight line along the road using a tow rope which is parallel to the direction of motion. The resistance to the motion of the car is again a constant force of magnitude 400 N . The resistance to the motion of the trailer is a constant force of magnitude 300 N . The engine of the car is working at a constant rate of 16 kW .
   The tow rope is modelled as being light and inextensible.
   Using the model,
2. find the tension in the tow rope at the instant when the speed of the car is \(\frac { 20 } { 3 } \mathrm {~ms} ^ { - 1 }\)

1. A car of mass 1000 kg moves in a straight line along a horizontal road at a constant speed of \(72 \mathrm {~km} \mathrm {~h} ^ { - 1 }\)

• The resistance to the motion of the car is modelled as a constant force of magnitude 900 N

The engine of the car is working at a constant rate of \(P \mathrm {~kW}\).
Using the model,

1. find the value of \(P\). The car now travels in a straight line up a road which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 2 } { 49 }\)
   At the instant when the engine of the car is working at a constant rate of 30 kW and the car is moving up the road at \(10 \mathrm {~ms} ^ { - 1 }\), the acceleration of the car is \(a \mathrm {~ms} ^ { - 2 }\) Using the refined model,
2. find the value of \(a\). Later on, when the engine of the car is again working at a constant rate of 30 kW , the car is moving up the road at a constant speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Using the refined model,
3. find the value of \(U\).

1. A car of mass 600 kg is moving along a straight horizontal road.

At the instant when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the car is modelled as a force of magnitude \(( 200 + 2 v ) \mathrm { N }\). The engine of the car is working at a constant rate of 12 kW .

1. Find the acceleration of the car at the instant when \(v = 20\) Later on the car is moving up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 14 }\) At the instant when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the car from non-gravitational forces is modelled as a force of magnitude ( \(200 + 2 v ) \mathrm { N }\). The engine is again working at a constant rate of 12 kW .
   At the instant when the car has speed \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the car is decelerating at \(0.05 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the value of \(w\).

1. A cyclist and her cycle have a combined mass of 60 kg . The cyclist is moving along a straight horizontal road and is working at a constant rate of 200 W .

When she has travelled a distance \(x\) metres, her speed is \(v \mathrm {~ms} ^ { - 1 }\) and the magnitude of the resistance to motion is \(3 v ^ { 2 } \mathrm {~N}\).

1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} x } = \frac { 200 - 3 v ^ { 3 } } { 60 v ^ { 2 } }\) The distance travelled by the cyclist as her speed increases from \(2 \mathrm {~ms} ^ { - 1 }\) to \(4 \mathrm {~ms} ^ { - 1 }\) is \(D\) metres.
2. Find the exact value of \(D\)

1. A car of mass 500 kg moves along a straight horizontal road.

The engine of the car produces a constant driving force of 1800 N .
The car accelerates from rest from the fixed point \(O\) at time \(t = 0\) and at time \(t\) seconds the car is \(x\) metres from \(O\), moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the car has magnitude \(2 v ^ { 2 } \mathrm {~N}\). At time \(T\) seconds, the car is at the point \(A\), moving with speed \(10 \mathrm {~ms} ^ { - 1 }\).

1. Show that \(T = \frac { 25 } { 6 } \ln 2\)
2. Show that the distance from \(O\) to \(A\) is \(125 \ln \frac { 9 } { 8 } \mathrm {~m}\).

1. Use an energy method to find the magnitude of the frictional force acting on the block. Calculate the coefficient of friction between the block and the plane.
2. Calculate the power of the tension in the string when the block has a speed of \(7 \mathrm {~ms} ^ { - 1 }\). Fig. 3.1 shows a thin planar uniform rigid rectangular sheet of metal, OPQR, of width 1.65 m and height 1.2 m . The mass of the sheet is \(M \mathrm {~kg}\). The sides OP and PQ have thin rigid uniform reinforcements attached with masses \(0.6 M \mathrm {~kg}\) and \(0.4 M \mathrm {~kg}\), respectively. Fig. 3.1 also shows coordinate axes with origin at O . The sheet with its reinforcements is to be used as an inn sign.

1. Calculate the coordinates of the centre of mass of the inn sign. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_421_492_210_1334} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
   \end{figure} The inn sign has a weight of 300 N . It hangs in equilibrium with QR horizontal when vertical forces \(Y _ { \mathrm { Q } } \mathrm { N }\) and \(Y _ { \mathrm { R } } \mathrm { N }\) act at Q and R respectively.
2. Calculate the value of \(Y _ { \mathrm { Q } }\) and show that \(Y _ { \mathrm { R } } = 120\). The inn sign is hung from a framework, ABCD , by means of two light vertical inextensible wires attached to the sign at Q and R and the framework at B and C , as shown in Fig. 3.2. QR and BC are horizontal. The framework is made from light rigid rods \(\mathrm { AB } , \mathrm { BC } , \mathrm { CA }\) and CD freely pin-jointed together at \(\mathrm { A } , \mathrm { B }\) and C and to a vertical wall at A and D . Fig. 3.3 shows the dimensions of the framework in metres as well as the external forces \(X _ { \mathrm { A } } \mathrm { N } , Y _ { \mathrm { A } } \mathrm { N }\) acting at A and \(X _ { \mathrm { D } } \mathrm { N } , Y _ { \mathrm { D } } \mathrm { N }\) acting at D . You are given that \(\sin \alpha = \frac { 5 } { 13 } , \cos \alpha = \frac { 12 } { 13 } , \sin \beta = \frac { 4 } { 5 }\) and \(\cos \beta = \frac { 3 } { 5 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_543_526_1420_253} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_629_793_1343_964} \captionsetup{labelformat=empty} \caption{Fig. 3.3}
   \end{figure}
3. Mark on the diagram in your Printed Answer Book all the forces acting on the pin-joints at \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , including those internal to the rods, when the inn sign is hanging from the framework.
4. Show that \(X _ { \mathrm { D } } = 261\).
5. Calculate the forces internal to the rods \(\mathrm { AB } , \mathrm { BC }\) and CD , stating whether each rod is in tension or thrust (compression). Calculate also the values of \(Y _ { \mathrm { D } }\) and \(Y _ { \mathrm { A } }\). [Your working in this part should correspond to your diagram in part (iii).]

4 \includegraphics[max width=\textwidth, alt={}, center]{a1a43547-0a68-4346-884a-0c6d9302cf24-3_216_1219_255_415} \(A\) and \(B\) are two long straight parallel horizontal sections of railway track. An engine on track \(A\) is attached to a carriage of mass 6000 kg on track \(B\) by a light inextensible chain which remains horizontal and taut in the ensuing motion. The chain is 13 m in length and the points of attachment on the engine and carriage are a perpendicular distance of 5 m apart. The engine and carriage start at rest and then the engine accelerates uniformly to a speed of \(5.6 \mathrm {~ms} ^ { - 1 }\) while travelling 250 m . It is assumed that any resistance to motion can be ignored.

1. Find the work done on the carriage by the tension in the chain.
2. Find the magnitude of the tension in the chain. The mass of the engine is 10000 kg .
3. At a point further along the track the engine and the carriage are moving at a speed of \(8.4 \mathrm {~ms} ^ { - 1 }\) and the power of the engine is 68 kW . Find the acceleration of the engine at this instant.

2 A pump is pumping still water from the base of a well at a constant rate of 300 kg per minute. The well is 4.5 m deep and water is released from the pump at ground level in a horizontal jet with a speed of \(6.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Ignoring any energy losses due to resistance, calculate the power generated by the pump.

4 A car has a mass of 850 kg and its engine can generate a maximum power of 35 kW . The total resistance to motion of the car is modelled as \(k v \mathrm {~N}\) where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car and \(k\) is a constant. When the car is moving in a straight line on a straight horizontal road, the greatest constant speed that it can attain is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(k = 56\).
2. Find the greatest possible acceleration of the car on the road at an instant when it is moving with a speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A trailer of mass 240 kg is attached to the car by means of a light inextensible tow bar which is parallel to the surface of the road. The resistance to motion of the trailer is modelled as a constant force of magnitude 350 N . The car and trailer move on the horizontal road. At a certain instant the car's engine is working at a rate of 30 kW and the acceleration of the car is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
3. (a) Find the speed of the car at this instant.
   (b) Find the magnitude of the tension in the tow bar at this instant. The car and trailer now move in a straight line on a straight road inclined at \(8 ^ { \circ }\) to the horizontal.
4. Find the difference between their greatest possible constant speed travelling up the slope and their greatest possible constant speed travelling down the slope.

1 A particle, \(P\), of mass 2 kg moves in two dimensions. Its initial velocity is \(\binom { - 19.5 } { - 60 } \mathrm {~ms} ^ { - 1 }\).

1. Calculate the initial kinetic energy of \(P\). For \(t \geqslant 0 , P\) is acted upon only by a variable force \(\mathbf { F } = \binom { 4 t } { - 2 } \mathrm {~N}\), where \(t\) is the time in seconds.
2. Find

2 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

1 A body, \(P\), of mass 2 kg moves under the action of a single force \(\mathbf { F } \mathrm { N }\). At time \(t \mathrm {~s}\), the velocity of the body is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where $$\mathbf { v } = \left( t ^ { 2 } - 3 \right) \mathbf { i } + \frac { 5 } { 2 t + 1 } \mathbf { j } \text { for } t \geq 2 .$$

1. Obtain \(\mathbf { F }\) in terms of \(t\).
2. Calculate the rate at which the force \(\mathbf { F }\) is working at \(t = 4\).
3. By considering the change in kinetic energy of \(P\), calculate the work done by the force \(\mathbf { F }\) during the time interval \(2 \leq t \leq 4\).

4 A car has a maximum speed of \(42 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is moving on a horizontal road. When the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it experiences a resistance force of magnitude \(30 v\) newtons.

1. Show that the maximum power of the car is 52920 W .
2. The car has mass 1200 kg . It travels, from rest, up a slope inclined at \(5 ^ { \circ }\) to the horizontal.
   1. Show that, when the car is travelling at its maximum speed \(\mathrm { V } \mathrm { m } \mathrm { s } ^ { - 1 }\) up the slope, $$V ^ { 2 } + 392 \sin 5 ^ { \circ } V - 1764 = 0$$
   2. Hence find \(V\).

7 A motorcycle has a maximum power of 72 kilowatts. The motorcycle and its rider are travelling along a straight horizontal road. When they are moving at a speed of \(\mathrm { V } \mathrm { m } \mathrm { s } ^ { - 1 }\), they experience a total resistance force of magnitude \(k V\) newtons, where \(k\) is a constant.

1. The maximum speed of the motorcycle and its rider is \(60 \mathrm {~ms} ^ { - 1 }\). Show that \(k = 20\).
2. When the motorcycle is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the rider allows the motorcycle to freewheel so that the only horizontal force acting is the resistance force. When the motorcycle has been freewheeling for \(t\) seconds, its speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the magnitude of the resistance force is \(20 v\) newtons. The mass of the motorcycle and its rider is 500 kg .
   1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = - \frac { v } { 25 }\).
   2. Hence find the time that it takes for the speed of the motorcycle to reduce from \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
      (6 marks)

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)

4 A van, of mass 1500 kg , has a maximum speed of \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a straight horizontal road. When the van travels at a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it experiences a resistance force of magnitude \(40 v\) newtons.

1. Show that the maximum power of the van is 100000 watts.
2. The van is travelling along a straight horizontal road. Find the maximum possible acceleration of the van when its speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. The van starts to climb a hill which is inclined at \(6 ^ { \circ }\) to the horizontal. Find the maximum possible constant speed of the van as it travels in a straight line up the hill.
   (6 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 train is travelling at maximum speed with its engine using its maximum power of 1800 kW When travelling at this speed the train experiences a total resistive force of 40000 N Find the maximum speed of the train. Circle your answer.
[0pt] [1 mark] \(22 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) \(45 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) \(54 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) \(90 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

5 A car, of mass 1000 kg , is travelling on a straight horizontal road. When the car travels at a speed of \(v \mathrm {~ms} ^ { - 1 }\), it experiences a resistance force of magnitude \(25 v\) newtons. The car has a maximum speed of \(72 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on the straight road.
Find the maximum power output of the car.
Fully justify your answer.

4 A car has mass 1000 kg and travels on a straight horizontal road. The maximum speed of the car on this road is \(48 \mathrm {~ms} ^ { - 1 }\) In a simple model, it is assumed that the car experiences a resistance force that is proportional to its speed. When the car travels at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the magnitude of the resistance force is 600 newtons. 4

1. Show that the maximum power of the car is 69120 W
   4
2. Find the maximum acceleration of the car when it is travelling at \(25 \mathrm {~ms} ^ { - 1 }\) 4
3. Find the maximum acceleration of the car when it is travelling at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) 4
4. Comment on the validity of the model in the context of your answers to parts (b) and (c).