6.02l Power and velocity: P = Fv

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.

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 \(\theta\).

2 The resistance to the motion of a car is \(k v ^ { \frac { 3 } { 2 } } \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the car's speed and \(k\) is a constant. The power exerted by the car's engine is 15000 W , and the car has constant speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a horizontal road.

1. Show that \(k = 4.8\). With the engine operating at a much lower power, the car descends a hill of inclination \(\alpha\), where \(\sin \alpha = \frac { 1 } { 15 }\). At an instant when the speed of the car is \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its acceleration is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Given that the mass of the car is 700 kg , calculate the power of the engine. \includegraphics[max width=\textwidth, alt={}, center]{941c0c81-a74f-49c0-acb7-1c23266fc2c8-02_579_447_1658_849} A particle \(P\) of mass 0.4 kg is attached to one end of each of two light inextensible strings which are both taut. The other end of the longer string is attached to a fixed point \(A\), and the other end of the shorter string is attached to a fixed point \(B\), which is vertically below \(A\). The string \(A P\) makes an angle of \(30 ^ { \circ }\) with the vertical and is 0.5 m long. The string \(B P\) makes an angle of \(60 ^ { \circ }\) with the vertical. \(P\) moves with constant angular speed in a horizontal circle with centre vertically below \(B\) (see diagram). The tension in the string \(A P\) is twice the tension in the string \(B P\). Calculate

5 A car of mass 1500 kg travels up a line of greatest slope of a straight road inclined at \(5 ^ { \circ }\) to the horizontal. The power of the car's engine is constant and equal to 25 kW and the resistance to the motion of the car is constant and equal to 750 N . The car passes through point \(A\) with speed \(10 \mathrm {~ms} ^ { - 1 }\).

1. Find the acceleration of the car at \(A\). The car later passes through a point \(B\) with speed \(20 \mathrm {~ms} ^ { - 1 }\). The car takes 28s to travel from \(A\) to \(B\).
2. Find the distance \(A B\).

6 A car of mass 700 kg is travelling up a hill which is inclined at a constant angle of \(5 ^ { \circ }\) to the horizontal. At a certain point \(P\) on the hill the car's speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The point \(Q\) is 400 m further up the hill from \(P\), and at \(Q\) the car's speed is \(15 \mathrm {~ms} ^ { - 1 }\).

1. Calculate the work done by the car's engine as the car moves from \(P\) to \(Q\), assuming that any resistances to the car's motion may be neglected. Assume instead that the resistance to the car's motion between \(P\) and \(Q\) is a constant force of magnitude 200 N.
2. Given that the acceleration of the car at \(Q\) is zero, show that the power of the engine as the car passes through \(Q\) is 12.0 kW , correct to 3 significant figures.
3. Given that the power of the car's engine at \(P\) is the same as at \(Q\), calculate the car's retardation at \(P\).

1 A child of mass 35 kg runs up a flight of stairs in 10 seconds. The vertical distance climbed is 4 m . Assuming that the child's speed is constant, calculate the power output.

4 A car of mass 900 kg is travelling at a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a level road. The total resistance to motion is 450 N .

1. Calculate the power output of the car's engine. A roof box of mass 50 kg is mounted on the roof of the car. The total resistance to motion of the vehicle increases to 500 N .
2. The car's engine continues to work at the same rate. Calculate the maximum speed of the car on the level road. The power output of the car's engine increases to 15000 W . The resistance to motion of the car, with roof box, remains 500 N .
3. Calculate the instantaneous acceleration of the car on the level road when its speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
4. The car climbs a hill which is at an angle of \(5 ^ { \circ }\) to the horizontal. Calculate the instantaneous retardation of the car when its speed is \(26 \mathrm {~ms} ^ { - 1 }\).

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.

14 J
18J
42 J 3 The time taken for the moon to make one complete orbit around Earth is approximately 27.3 days. Model this orbit as circular, with a radius of \(3.84 \times 10 ^ { 8 }\) metres.
Find the approximate speed of the moon relative to Earth, in metres per second.
4 A particle \(P\), of mass \(m \mathrm {~kg}\), collides with a particle \(Q\), of mass 2 kg Immediately before the collision the velocity of \(P\) is \(\left[ \begin{array} { c } 4 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(Q\) is \(\left[ \begin{array} { c } - 3 \\ 5 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) As a result of the collision the particles coalesce into a single particle which moves with velocity \(\left[ \begin{array} { l } k \\ 0 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(k\) is a constant. Find the value of \(k\).
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.
6 The magnitude of the gravitational force \(F\) between two planets of masses \(m _ { 1 }\) and \(m _ { 2 }\) with centres at a distance \(d\) apart is given by $$F = \frac { G m _ { 1 } m _ { 2 } } { d ^ { 2 } }$$ where \(G\) is a constant.
6

1. Show that \(G\) must have dimensions \(L ^ { 3 } M ^ { - 1 } T ^ { - 2 }\), where \(L\) represents length, \(M\) represents mass and \(T\) represents time.
   6
2. The lifetime \(t\) of a planet is thought to depend on its mass \(m\), its radius \(r\), the constant \(G\) and a dimensionless constant \(k\) such that $$t = k m ^ { a } r ^ { b } G ^ { c }$$ where \(a , b\) and \(c\) are constants.
   Determine the values of \(a , b\) and \(c\).
   7 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) As part of a competition, Jo-Jo makes a small pop-up rocket.
   It is operated by pressing the rocket vertically downwards to compress a light spring, which is positioned underneath the rocket. The rocket is released from rest and moves vertically upwards.
   The mass of the rocket is 18 grams and the stiffness constant of the spring is \(60 \mathrm { Nm } ^ { - 1 }\) Initially the spring is compressed by 3 cm
   7
   1. Find the speed of the rocket when the spring first reaches its natural length.
      7
   2. By considering energy find the distance that the rocket rises. 7
   3. In order to win a prize in the competition, the rocket must reach a point which is 15 cm vertically above its starting position. With reference to the assumptions you have made, determine if Jo-Jo wins a prize or not. Fully justify your answer.
      8 Two smooth spheres \(A\) and \(B\) have the same radius and are free to move on a smooth horizontal surface. The masses of \(A\) and \(B\) are \(2 m\) and \(m\) respectively.
      Both \(A\) and \(B\) are initially at rest.
      The sphere \(A\) is set in motion directly towards \(B\) with speed \(3 u\) and at the same time \(B\) is set in motion directly towards \(A\) with speed \(2 u\). Subsequently \(A\) and \(B\) collide directly. \(A\) The coefficient of restitution between the spheres is \(e\).
      8
   4. Show that the speed of \(B\) after the collision is given by $$\frac { 2 u ( 2 + 5 e ) } { 3 }$$ \section*{Question 8 continues on the next page} 8
   5. Given that the direction of the velocity of \(A\) is reversed during the collision, find the range of possible values of \(e\). Fully justify your answer.
      [0pt] [4 marks]
      8
   6. Given that the magnitude of the impulse that \(A\) exerts on \(B\) is \(\frac { 19 m u } { 3 }\), find the value of \(e\).
      

      Question number	Additional page, if required. Write the question numbers in the left-hand margin.
      	\(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

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.

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 \(30 \mathrm {~ms} ^ { - 1 }\).

2 A car has a mass of 800 kg . The engine of the car is working at a constant power of 15 kW . In an initial model of the motion of the car it is assumed that the car is subject to a constant resistive force of magnitude \(R N\). The car is initially driven on a straight horizontal road. At the instant that its speed is \(20 \mathrm {~ms} ^ { - 1 }\) its acceleration is \(0.4 \mathrm {~ms} ^ { - 2 }\).

1. Show that \(R = 430\).
2. Hence find the maximum constant speed at which the car can be driven along this road, according to the initial model. In a revised model the resistance to the motion of the car at any instant is assumed to be 60 v where \(v\) is the speed of the car at that instant. The car is now driven up a straight road which is inclined at an angle \(\alpha\) above the horizontal where \(\sin \alpha = 0.2\).
3. Determine the speed of the car at the instant that its acceleration is \(0.15 \mathrm {~ms} ^ { - 2 }\) up the slope, according to the revised model.

3 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 \(\mathbf { v . v } = \mathbf { u . u } + 2 \mathbf { a x }\) 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 .
   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 has mass 1200 kg . The total resistance to the car's motion is constant and equal to 250 N .

1. The car is driven along a straight horizontal road with its engine working at 10 kW . Find the acceleration of the car at the instant that its speed is \(5 \mathrm {~ms} ^ { - 1 }\). The maximum power that the car's engine can generate is 20 kW .
2. Find the greatest constant speed at which the car can be driven along a straight horizontal road. The car is driven up a straight road which is inclined at an angle \(\theta\) above the horizontal where \(\sin \theta = 0.05\).
3. Find the greatest constant speed at which the car can be driven up this road.

2 The coordinates of two points, \(A\) and \(B\), are \(( - 1,6 )\) and \(( 5,12 )\) respectively, where the units of the coordinate axes are metres. A particle \(P\) moves from \(A\) to \(B\) under the action of several forces. The force \(\mathbf { F } = 7 \mathbf { i } - 2 \mathbf { j } \mathbf { N }\) is one of the forces acting on \(P\).

1. Calculate the work done by \(\mathbf { F }\) on \(P\) as \(P\) moves from \(A\) to \(B\). At the instant when \(P\) reaches \(B\) its velocity is \(- \mathbf { i } - 5 \mathbf { j } \mathrm {~ms} ^ { - 1 }\).
2. Find the power generated by \(\mathbf { F }\) at the instant that \(P\) reaches \(B\). One end of a light elastic string was attached to the origin of the coordinate system and the other to \(P\) when \(P\) was at \(A\), before it moved to \(B\). The natural length of the string is 8 m and its modulus of elasticity is 24 N .
3. At the instant that \(P\) reaches \(B\), find the following.

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 \(\theta\).

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 \(2 \leq t \leq 4\).