6.02l Power and velocity: P = Fv

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

2 A car of mass 1500 kg has an engine with maximum power 60 kW . When the car is travelling at \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) 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). \includegraphics[max width=\textwidth, alt={}, center]{aa25b8a6-9a5a-4de2-9534-18db8a175c34-03_583_378_169_255} As shown in the diagram, \(A B\) is a long thin rod which is fixed vertically with \(A\) above \(B\). One end of a light inextensible string of length 1 m is attached to \(A\) and the other end is attached to a particle \(P\) of mass \(m _ { 1 } \mathrm {~kg}\). One end of another light inextensible string of length 1 m is also attached to \(P\). Its other end is attached to a small smooth ring \(R\), of mass \(m _ { 2 } \mathrm {~kg}\), which is free to move on \(A B\). Initially, \(P\) moves in a horizontal circle of radius 0.6 m with constant angular velocity \(\omega \mathrm { rads } ^ { - 1 }\). The magnitude of the tension in string \(A P\) is denoted by \(T _ { 1 } \mathrm {~N}\) while that in string \(P R\) is denoted by \(T _ { 2 } \mathrm {~N}\).
   1. By considering forces on \(R\), express \(T _ { 2 }\) in terms of \(m _ { 2 }\).
   2. Show that
      1. \(T _ { 1 } = \frac { 49 } { 4 } \left( m _ { 1 } + m _ { 2 } \right)\),
      2. \(\omega ^ { 2 } = \frac { 49 \left( m _ { 1 } + 2 m _ { 2 } \right) } { 4 m _ { 1 } }\).
      3. Deduce that, in the case where \(m _ { 1 }\) is much bigger than \(m _ { 2 } , \omega \approx 3.5\). In a different case, where \(m _ { 1 } = 2.5\) and \(m _ { 2 } = 2.8 , P\) slows down. Eventually the system comes to rest with \(P\) and \(R\) hanging in equilibrium.
   3. Find the total energy lost by \(P\) and \(R\) as the angular velocity of \(P\) changes from the initial value of \(\omega \mathrm { rads } ^ { - 1 }\) to zero.

2 A particle \(P\) of mass 2.4 kg is moving in a straight line \(O A\) on a horizontal plane. \(P\) is acted on by a force of magnitude 30 N in the direction of motion. The distance \(O A\) is 10 m . \begin{enumerate}[label=(\alph*)] \item Find the work done by this force as \(P\) moves from \(O\) to \(A\). The motion of \(P\) is resisted by a constant force of magnitude \(R \mathrm {~N}\). The velocity of \(P\) increases from \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(O\) to \(18 \mathrm {~ms} ^ { - 1 }\) at \(A\). \item Find the value of \(R\). \item Find the average power used in overcoming the resistance force on \(P\) as it moves from \(O\) to \(A\). When \(P\) reaches \(A\) it collides directly with a particle \(Q\) of mass 1.6 kg which was at rest at \(A\) before the collision. The impulse exerted on \(Q\) by \(P\) as a result of the collision is 17.28 Ns . \item

1. Find the speed of \(Q\) after the collision.
2. Hence show that the collision is inelastic. It is required to model the motion of a car of mass \(m \mathrm {~kg}\) travelling at a constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) around a circular portion of banked track. The track is banked at \(30 ^ { \circ }\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{b9741472-f230-4e2d-9c8b-47f7e168e938-03_355_565_269_274} In a model, the following modelling assumptions are made.
   For a particular portion of banked track, \(r = 24\).
   (b) Find the value of \(v\) as predicted by the model. A car is being driven on this portion of the track at the constant speed calculated in part (b). The driver finds that in fact he can drive a little slower or a little faster than this while still moving in the same horizontal circle.
   (c) Explain

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 \(A B\).

1 A lorry moves along a straight horizontal road. The engine of the lorry produces a constant power of 80 kW . The mass of the lorry is 10 tonnes and the resistance to motion is constant at 4000 N .

1. Express the driving force of the lorry in terms of its velocity and hence, using Newton's second law, write down a differential equation which connects the velocity of the lorry and the time for which it has been moving.
2. Hence find the time taken, in seconds, for the lorry to accelerate from \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

12 A train of mass 250 tonnes is ascending an incline of \(\sin ^ { - 1 } \left( \frac { 1 } { 500 } \right)\) and working at 400 kW against resistance to motion which may be regarded as a constant force of 20000 N .

1. Find the constant speed, \(V\), with which the train can ascend the incline working at this power.
2. The train begins to ascend the incline at \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the same power and against the same resistance. Find the distance covered in reaching a speed of \(\frac { 3 } { 4 } V\).

7 A cyclist and her machine have a combined mass of 90 kg and she is riding along a straight horizontal road. She is working at a constant power of 75 W . At time \(t\) seconds her speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the resistance to motion is \(k v \mathrm {~N}\), where \(k\) is a constant.

1. If the cyclist's maximum steady speed is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), show that \(k = \frac { 3 } { 4 }\).
2. Use Newton's second law to show that $$\frac { 25 } { v } - \frac { v } { 4 } = 30 \frac { \mathrm {~d} v } { \mathrm {~d} t } .$$
3. Find the time taken for the cyclist to accelerate from a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to a speed of \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

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

9 A car of mass 800 kg is descending a straight hill which is inclined at \(2 ^ { \circ }\) to the horizontal. The car passes through the points \(A\) and \(B\) with speeds \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The distance \(A B\) is 400 m .

1. Assuming that resistances to motion are negligible, calculate the work done by the car's engine over the distance from \(A\) to \(B\).
2. Assuming also that the driving force produced by the car's engine remains constant, calculate the power of the car's engine at the mid-point of \(A B\).

7 A cyclist and her machine have a combined mass of 90 kg and she is riding along a straight horizontal road. She is working at a constant power of 75 W . At time \(t\) seconds her speed is \(v \mathrm {~ms} ^ { - 1 }\) and the resistance to motion is \(k v \mathrm {~N}\), where \(k\) is a constant.

1. If the cyclist's maximum steady speed is \(10 \mathrm {~ms} ^ { - 1 }\), show that \(k = \frac { 3 } { 4 }\).
2. Use Newton's second law to show that $$\frac { 25 } { v } - \frac { v } { 4 } = 30 \frac { \mathrm {~d} v } { \mathrm {~d} t } .$$
3. Find the time taken for the cyclist to accelerate from a speed of \(3 \mathrm {~ms} ^ { - 1 }\) to a speed of \(7 \mathrm {~ms} ^ { - 1 }\).

10 A cyclist and her bicycle have a combined mass of 90 kg and she is riding along a straight horizontal road. She is working at a constant power of 75 W . At time \(t\) seconds her speed is \(v \mathrm {~ms} ^ { - 1 }\) and the resistance to motion is \(k v \mathrm {~N}\), where \(k\) is a constant.

1. Given that the steady speed at which the cyclist can move is \(10 \mathrm {~ms} ^ { - 1 }\), show that \(k = \frac { 3 } { 4 }\).
2. Show that $$\frac { 25 } { v } - \frac { v } { 4 } = 30 \frac { \mathrm {~d} v } { \mathrm {~d} t } .$$
3. Find the time taken for the cyclist to accelerate from a speed of \(3 \mathrm {~ms} ^ { - 1 }\) to a speed of \(7 \mathrm {~ms} ^ { - 1 }\).

10 The engine of a lorry of mass 4000 kg works at a constant rate of 75 kW . Resistance to motion is modelled by a constant resistive force. On a horizontal road the lorry travels at a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the work done by the engine in travelling for 1 minute on the horizontal road.
2. The lorry travels at a constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a slope of angle \(\sin ^ { - 1 } 0.05\) to the horizontal. Find the value of \(v\).

7 A car has mass 800 kg .

1. The car accelerates from \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in climbing a hill with a vertical height of 16 m . Ignoring resistive forces, find the work done by the engine.
2. The engine produces a constant power output of 189 kW . The car now travels along horizontal ground. Modelling the resistive force as \(7 v ^ { 2 } \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed, find the value of \(v\) for which the speed of the car is constant.

10 A cyclist and her bicycle have a combined mass of 90 kg and she is riding along a straight horizontal road. She is working at a constant power of 75 W . At time \(t\) seconds her speed is \(v \mathrm {~ms} ^ { - 1 }\) and the resistance to motion is \(k v \mathrm {~N}\), where \(k\) is a constant.

1. Given that the steady speed at which the cyclist can move is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), show that \(k = \frac { 3 } { 4 }\).
2. Show that $$\frac { 25 } { v } - \frac { v } { 4 } = 30 \frac { \mathrm {~d} v } { \mathrm {~d} t } .$$
3. Find the time taken for the cyclist to accelerate from a speed of \(3 \mathrm {~ms} ^ { - 1 }\) to a speed of \(7 \mathrm {~ms} ^ { - 1 }\).

10 A cyclist and her bicycle have a combined mass of 90 kg and she is riding along a straight horizontal road. She is working at a constant power of 75 W . At time \(t\) seconds her speed is \(v \mathrm {~ms} ^ { - 1 }\) and the resistance to motion is \(k v \mathrm {~N}\), where \(k\) is a constant.

1. Given that the steady speed at which the cyclist can move is \(10 \mathrm {~ms} ^ { - 1 }\), show that \(k = \frac { 3 } { 4 }\).
2. Show that $$\frac { 25 } { v } - \frac { v } { 4 } = 30 \frac { \mathrm {~d} v } { \mathrm {~d} t } .$$
3. Find the time taken for the cyclist to accelerate from a speed of \(3 \mathrm {~ms} ^ { - 1 }\) to a speed of \(7 \mathrm {~ms} ^ { - 1 }\).

5 When a car of mass 990 kg moves at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a horizontal straight road, the power of its engine is 8.8 kW .

1. Find the magnitude of the resistance to the motion of the car at this speed.
2. Assuming that the resistance has magnitude \(k v ^ { 2 }\) newtons when the speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the value of the constant \(k\). The power of the engine is now increased to 22 kW and remains constant at this value.
3. Using the model in part (ii), show that $$\frac { \mathrm { d } v } { \mathrm {~d} x } = \frac { 20000 - v ^ { 3 } } { 900 v ^ { 2 } } .$$
4. Hence show that the car moves about 300 m as its speed increases from \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

A car of mass \(1800\text{ kg}\) is towing a trailer of mass \(400\text{ kg}\) along a straight horizontal road. The car and trailer are connected by a light rigid tow-bar. The car is accelerating at \(1.5\text{ m s}^{-2}\). There are constant resistance forces of \(250\text{ N}\) on the car and \(100\text{ N}\) on the trailer.

1. Find the tension in the tow-bar. [2]
2. Find the power of the engine of the car at the instant when the speed is \(20\text{ m s}^{-1}\). [3]

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]

Two racing cars \(A\) and \(B\) are at rest alongside each other at a point \(O\) on a straight horizontal test track. The mass of \(A\) is 1200 kg. The engine of \(A\) produces a constant driving force of 4500 N. When \(A\) arrives at a point \(P\) its speed is 25 m s\(^{-1}\). The distance \(OP\) is \(d\) m. The work done against the resistance force experienced by \(A\) between \(O\) and \(P\) is 75 000 J.

1. Show that \(d = 100\). [3]

Car \(B\) starts off at the same instant as car \(A\). The two cars arrive at \(P\) simultaneously and with the same speed. The engine of \(B\) produces a driving force of 3200 N and the car experiences a constant resistance to motion of 1200 N.

1. Find the mass of \(B\). [3]
2. Find the steady speed which \(B\) can maintain when its engine is working at the same rate as it is at \(P\). [3]

A car of mass \(900\text{kg}\) is moving up a hill inclined at \(\sin^{-1} 0.12\) to the horizontal. The initial speed of the car is \(11\text{ms}^{-1}\). After \(12\text{s}\), the car has travelled \(150\text{m}\) up the hill and has speed \(16\text{ms}^{-1}\). The engine of the car is working at a constant rate of \(24\text{kW}\).

1. Find the work done against the resistive forces during the \(12\text{s}\). [5]
2. The car then travels along a straight horizontal road. There is a resistance to the motion of the car of \((1520 + 4v)\text{N}\) when the speed of the car is \(v\text{ms}^{-1}\). The car travels at a constant speed with the engine working at a constant rate of \(32\text{kW}\). Find this speed. [3]

A car of mass \(1200\) kg is travelling along a straight horizontal road. The power of the car's engine is constant and is equal to \(16\) kW. There is a constant resistance to motion of magnitude \(500\) N.

1. Find the acceleration of the car at an instant when its speed is \(20\) m s\(^{-1}\). [3]
2. Assuming that the power and the resistance forces remain unchanged, find the steady speed at which the car can travel. [2]

The car comes to the bottom of a straight hill of length \(316\) m, inclined at an angle to the horizontal of \(\sin^{-1}(\frac{4}{65})\). The power remains constant at \(16\) kW, but the magnitude of the resistance force is no longer constant and changes such that the work done against the resistance force in ascending the hill is \(128400\) J. The time taken to ascend the hill is \(15\) s.

1. Given that the car is travelling at a speed of \(20\) m s\(^{-1}\) at the bottom of the hill, find its speed at the top of the hill. [6]

A car has mass 1400 kg. When the speed of the car is \(v\text{ ms}^{-1}\) the magnitude of the resistance to motion is \(kv^2\) N where \(k\) is a constant.

1. The car moves at a constant speed of \(24\text{ ms}^{-1}\) up a hill inclined at an angle of \(\alpha\) to the horizontal where \(\sin \alpha = 0.12\). At this speed the magnitude of the resistance to motion is 480 N.
   1. Find the value of \(k\). [1]
   2. Find the power of the car's engine. [3]
2. The car now moves at a constant speed on a straight level road. Given that its engine is working at 54 kW, find this speed. [3]

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]