6.02i Conservation of energy: mechanical energy principle

4 A ball \(B\) of mass 1.7 kg is connected to one end of a light elastic spring of natural length 1.2 m . The other end of the spring is attached to a point \(O\) on the ceiling of a large room. The modulus of elasticity of the spring is 50 N . The ball is held 3.2 m vertically below \(O\) and projected upwards with an initial speed of \(0.5 \mathrm {~ms} ^ { - 1 }\). In order to model the motion of \(B\) (before any collision with the ceiling) the following assumptions are made.

• Air resistance is ignored.
• \(B\) is small.
• The fully compressed length of the spring is negligible. \begin{enumerate}[label=(\roman*)]
• Determine whether, according to the model, \(B\) reaches \(O\).
• Without doing any further calculations, explain whether the answer to part (i) could change in each of the following different cases.
  1. A new model is used in which air resistance is taken into account.
  2. The spring is replaced by an elastic string with the same natural length and modulus of elasticity.
  3. \(\quad B\) is initially projected downwards rather than upwards.

7 A smooth track \(A B\) is in the shape of an arc of a circle with centre \(O\) and radius 1.4 m . The track is fixed in a vertical plane with \(A\) above the level of \(B\) and a point \(C\) on the track vertically below \(O\). Angle \(A O C\) is \(60 ^ { \circ }\) and angle \(C O B\) is \(30 ^ { \circ }\). Point \(C\) is 2.5 m vertically above the point \(F\), which lies in a horizontal plane. A particle of mass 0.4 kg is placed at \(A\) and projected down the track with an initial velocity of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle first hits the plane at point \(H\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{a8c9d007-e67f-4637-9e74-630ba9a91442-5_767_1265_488_415}

1. Find the magnitude of the contact force between the particle and the track when the particle is at \(B\). [5]
2. Find the distance \(F H\).

6 A fairground game involves a player kicking a ball, \(B\), from rest so as to project it with a horizontal velocity of magnitude \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball is attached to one end of a light rod of length \(l \mathrm {~m}\). The other end of the rod is smoothly hinged at a fixed point \(O\) so that \(B\) can only move in the vertical plane which contains \(O\), a fixed barrier and a bell which is fixed \(l \mathrm {~m}\) vertically above \(O\). Initially \(B\) is vertically below \(O\). The barrier is positioned so that when \(B\) collides directly with the barrier, \(O B\) makes an angle \(\theta\) with the downwards vertical through \(O\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{bf86ac88-0fd1-4d49-a705-9b8d06fbac2a-4_643_659_584_724} The coefficient of restitution between \(B\) and the barrier is \(e . B\) rebounds from the barrier, passes through its original position and continues on a circular path towards the bell. The bell will only ring if the ball strikes it with a speed of at least \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The player wins the game if the player causes the bell to ring having kicked \(B\) so that it first collides with the barrier. You may assume that \(B\) and the bell are small and that the barrier has negligible thickness. Show that, whatever the position of the barrier, the player cannot win the game if \(u ^ { 2 } < 4 g l + \frac { V ^ { 2 } } { e ^ { 2 } }\). \section*{END OF QUESTION PAPER}

6 A particle \(P\), of mass \(m \mathrm {~kg}\), is placed at the point \(Q\) on the top of a smooth upturned hemisphere of radius 3 metres and centre \(O\). The plane face of the hemisphere is fixed to a horizontal table. The particle is set into motion with an initial horizontal velocity of \(2 \mathrm {~ms} ^ { - 1 }\). When the particle is on the surface of the hemisphere, the angle between \(O P\) and \(O Q\) is \(\theta\) and the particle has speed \(v \mathrm {~ms} ^ { - 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-005_419_1013_607_511}

1. Show that \(v ^ { 2 } = 4 + 6 g ( 1 - \cos \theta )\).
2. Find the value of \(\theta\) when the particle leaves the hemisphere.

7 A particle, of mass 10 kg , is attached to one end of a light elastic string of natural length 0.4 metres and modulus of elasticity 100 N . The other end of the string is fixed to the point \(O\).

1. Find the length of the elastic string when the particle hangs in equilibrium directly below \(O\).
2. The particle is pulled down and held at a point \(P\), which is 1 metre vertically below \(O\). Show that the elastic potential energy of the string when the particle is in this position is 45 J .
3. The particle is released from rest at the point \(P\). In the subsequent motion, the particle has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is \(x\) metres below \(\boldsymbol { O }\).
   1. Show that, while the string is taut, $$v ^ { 2 } = 39.6 x - 25 x ^ { 2 } - 14.6$$
   2. Find the value of \(x\) when the particle comes to rest for the first time after being released, given that the string is still taut.

8 Two small blocks, \(A\) and \(B\), of masses 0.8 kg and 1.2 kg respectively, are stuck together. A spring has natural length 0.5 metres and modulus of elasticity 49 N . One end of the spring is attached to the top of the block \(A\) and the other end of the spring is attached to a fixed point \(O\).

1. The system hangs in equilibrium with the blocks stuck together, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-017_385_239_669_881} Find the extension of the spring.
2. Show that the elastic potential energy of the spring when the system is in equilibrium is 1.96 J .
3. The system is hanging in this equilibrium position when block \(B\) falls off and block \(A\) begins to move vertically upwards. Block \(A\) next comes to rest when the spring is compressed by \(x\) metres.
   1. Show that \(x\) satisfies the equation $$x ^ { 2 } + 0.16 x - 0.008 = 0$$
   2. Find the value of \(x\).

1 A child, of mass 35 kg , slides down a slide in a water park. The child, starting from rest, slides from the point \(A\) to the point \(B\), which is 10 metres vertically below the level of \(A\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-2_259_595_685_705}

1. In a simple model, all resistance forces are ignored. Use an energy method to find the speed of the child at \(B\).
2. State one resistance force that has been ignored in answering part (a).
3. In fact, when the child slides down the slide, she reaches \(B\) with a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that the slide is 20 metres long and the sum of the resistance forces has a constant magnitude of \(F\) newtons, use an energy method to find the value of \(F\).
   (4 marks)

3 A light inextensible string has length \(2 a\). One end of the string is attached to a fixed point \(O\) and a particle of mass \(m\) is attached to the other end. Initially, the particle is held at the point \(A\) with the string taut and horizontal. The particle is then released from rest and moves in a circular path. Subsequently, it passes through the point \(B\), which is directly below \(O\). The points \(O , A\) and \(B\) are as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-3_426_437_575_772}

1. Show that the speed of the particle at \(B\) is \(2 \sqrt { a g }\).
2. Find the tension in the string as the particle passes through \(B\). Give your answer in terms of \(m\) and \(g\).

8 Two small blocks, \(A\) and \(B\), of masses 0.8 kg and 1.2 kg respectively, are stuck together. A spring has natural length 0.5 metres and modulus of elasticity 49 N . One end of the spring is attached to the top of the block \(A\) and the other end of the spring is attached to a fixed point \(O\).

1. The system hangs in equilibrium with the blocks stuck together, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-8_385_239_669_881} Find the extension of the spring.
2. Show that the elastic potential energy of the spring when the system is in equilibrium is 1.96 J .
3. The system is hanging in this equilibrium position when block \(B\) falls off and block \(A\) begins to move vertically upwards. Block \(A\) next comes to rest when the spring is compressed by \(x\) metres.
   1. Show that \(x\) satisfies the equation $$x ^ { 2 } + 0.16 x - 0.008 = 0$$
   2. Find the value of \(x\).

2 A stone, of mass 6 kg , is thrown vertically upwards with a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point at a height of 4 metres above ground level.

1. Calculate the initial kinetic energy of the stone.
2. 1. Show that the kinetic energy of the stone when it hits the ground is 667 J , correct to three significant figures.
   2. Hence find the speed of the stone when it hits the ground.
   3. State two modelling assumptions that you have made.

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)

7 A hollow cylinder, of internal radius 4 m , is fixed so that its axis is horizontal. The point \(O\) is on this axis. A particle, of mass 6 kg , is set in motion so that it moves on the smooth inner surface of the cylinder in a vertical circle about \(O\). Its speed at the point \(A\), which is vertically below \(O\), is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{851cb2a3-5bc8-4af9-b1fc-a143d37beebe-5_746_739_504_662} When the particle is at the point \(B\), at a height of 2 m above \(A\), find:

1. its speed;
2. the normal reaction between the cylinder and the particle.

9 A bungee jumper, of mass 80 kg , is attached to one end of a light elastic cord, of natural length 16 metres and modulus of elasticity 784 N . The other end of the cord is attached to a horizontal platform, which is at a height of 65 metres above the ground. The bungee jumper steps off the platform at the point where the cord is attached and falls vertically. The bungee jumper can be modelled as a particle. Hooke's law can be assumed to apply throughout the motion and air resistance can be assumed to be negligible.

1. Find the length of the cord when the acceleration of the bungee jumper is zero.
2. The cord extends by \(x\) metres beyond its natural length before the bungee jumper first comes to rest.
   1. Show that \(x ^ { 2 } - 32 x - 512 = 0\).
   2. Find the distance above the ground at which the bungee jumper first comes to rest.

7 A smooth hemisphere, of radius \(a\) and centre \(O\), is fixed with its plane face on a horizontal surface. A particle, of mass \(m\), can move freely on the surface of the hemisphere. The particle is placed at the point \(A\), the highest point of the hemisphere, and is set in motion along the surface with speed \(u\).

1. While the particle is in contact with the hemisphere at a point \(P , O P\) makes an angle \(\theta\) with the upward vertical. \includegraphics[max width=\textwidth, alt={}, center]{06b431ca-d3a8-46d6-b9f8-bac08d3fd51e-5_366_1246_715_395} Show that the speed of the particle at \(P\) is $$\left( u ^ { 2 } + 2 g a [ 1 - \cos \theta ] \right) ^ { \frac { 1 } { 2 } }$$
2. The particle leaves the surface of the hemisphere when \(\theta = \alpha\). Find \(\cos \alpha\) in terms of \(a , u\) and \(g\).

8 A bungee jumper, of mass 49 kg , is attached to one end of a light elastic cord of natural length 22 metres and modulus of elasticity 1078 newtons. The other end of the cord is attached to a horizontal platform, which is at a height of 60 metres above the ground. The bungee jumper steps off the platform at the point where the cord is attached, and falls vertically. The bungee jumper can be modelled as a particle. Assume that Hooke's Law applies whilst the cord is taut and that air resistance is negligible throughout the motion. When the bungee jumper has fallen \(x\) metres, his speed is \(v \mathrm {~ms} ^ { - 1 }\).

1. By considering energy, show that, when \(x\) is greater than 22, $$5 v ^ { 2 } = 318 x - 5 x ^ { 2 } - 2420$$
2. Explain why \(x\) must be greater than 22 for the equation in part (a) to be valid. ( 1 mark)
3. Find the maximum value of \(x\).
4. 1. Show that the speed of the bungee jumper is a maximum when \(x = 31.8\).
   2. Hence find the maximum speed of the bungee jumper.

7 A small bead, of mass \(m\), is suspended from a fixed point \(O\) by a light inextensible string, of length \(a\). The bead is then set into circular motion with the string taut at \(B\), where \(B\) is vertically below \(O\), with a horizontal speed \(u\). \includegraphics[max width=\textwidth, alt={}, center]{03994596-21ad-4201-8d64-ba2d7b7e0a77-5_451_458_461_760}

1. Given that the string does not become slack, show that the least value of \(u\) required for the bead to make complete revolutions about \(O\) is \(\sqrt { 5 a g }\).
2. In the case where \(u = \sqrt { 5 a g }\), find, in terms of \(g\) and \(m\), the tension in the string when the bead is at the point \(C\), which is at the same horizontal level as \(O\), as shown in the diagram.
3. State one modelling assumption that you have made in your solution.

8

1. Hooke's law states that the tension in a stretched string of natural length \(l\) and modulus of elasticity \(\lambda\) is \(\frac { \lambda x } { l }\) when its extension is \(x\). Using this formula, prove that the work done in stretching a string from an unstretched position to a position in which its extension is \(e\) is \(\frac { \lambda e ^ { 2 } } { 2 l }\).
   (3 marks)
2. A particle, of mass 5 kg , is attached to one end of a light elastic string of natural length 0.6 metres and modulus of elasticity 150 N . The other end of the string is fixed to a point \(O\).
   1. Find the extension of the elastic string when the particle hangs in equilibrium directly below \(O\).
   2. The particle is pulled down and held at the point \(P\), which is 0.9 metres vertically below \(O\). Show that the elastic potential energy of the string when the particle is in this position is 11.25 J .
   3. The particle is released from rest at the point \(P\). In the subsequent motion, the particle has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is \(x\) metres above \(\boldsymbol { P }\). Show that, while the string is taut, $$v ^ { 2 } = 10.4 x - 50 x ^ { 2 }$$
   4. Find the value of \(x\) when the particle comes to rest for the first time after being released, given that the string is still taut.

2 A slide at a water park may be modelled as a smooth plane of length 20 metres inclined at \(30 ^ { \circ }\) to the vertical. Anne, who has a mass of 55 kg , slides down the slide. At the top of the slide, she has an initial velocity of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down the slide.

1. Calculate Anne's initial kinetic energy.
2. By using conservation of energy, find the kinetic energy and the speed of Anne after she has travelled the 20 metres.
3. State one modelling assumption which you have made.

6 A block, of mass 5 kg , is attached to one end of a length of elastic string. The other end of the string is fixed to a vertical wall. The block is placed on a horizontal surface. The elastic string has natural length 1.2 m and modulus of elasticity 180 N . The block is pulled so that it is 2 m from the wall and is then released from rest. Whilst taut, the string remains horizontal. It may be assumed that, after the string becomes slack, it does not interfere with the movement of the block. \includegraphics[max width=\textwidth, alt={}, center]{9cfa110c-ee11-447a-b21a-3f436432e27d-5_396_960_660_534}

1. Calculate the elastic potential energy when the block is 2 m from the wall.
2. If the horizontal surface is smooth, find the speed of the block when it hits the wall.
3. The surface is in fact rough and the coefficient of friction between the block and the surface is \(\mu\). Find \(\mu\) if the block comes to rest just as it reaches the wall.

7 In crazy golf, a golf ball is hit so that it starts to move in a vertical circle on the inside of a smooth cylinder. Model the golf ball as a particle, \(P\), of mass \(m\). The circular path of the golf ball has radius \(a\) and centre \(O\). At time \(t\), the angle between \(O P\) and the horizontal is \(\theta\), as shown in the diagram. The golf ball has speed \(u\) at the lowest point of its circular path. \includegraphics[max width=\textwidth, alt={}, center]{9cfa110c-ee11-447a-b21a-3f436432e27d-6_739_742_719_641}

1. Show that, while the golf ball is in contact with the cylinder, the reaction of the cylinder on the golf ball is $$\frac { m u ^ { 2 } } { a } - 3 m g \sin \theta - 2 m g$$
2. Given that \(u = \sqrt { 3 a g }\), the golf ball will not complete a vertical circle inside the cylinder. Find the angle which \(O P\) makes with the horizontal when the golf ball leaves the surface of the cylinder.
   (4 marks)

6. A light elastic spring \(A B\) has natural length \(2 a\) and modulus of elasticity \(2 m n ^ { 2 } a\), where \(n\) is a constant. A particle \(P\) of mass \(m\) is attached to the end \(A\) of the spring. At time \(t = 0\), the spring, with \(P\) attached, lies at rest and unstretched on a smooth horizontal plane. The other end \(B\) of the spring is then pulled along the plane in the direction \(A B\) with constant acceleration \(f\). At time \(t\) the extension of the spring is \(x\).

1. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + n ^ { 2 } x = f .$$
2. Find \(x\) in terms of \(n , f\) and \(t\). Hence find
3. the maximum extension of the spring,
4. the speed of \(P\) when the spring first reaches its maximum extension.
   1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors due east and due north respectively]
   A man cycles at a constant speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on level ground and finds that when his velocity is \(u \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the velocity of the wind appears to be \(v ( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(v\) is a positive constant. When the man cycles with velocity \(\frac { 1 } { 5 } u ( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), the velocity of the wind appears to be \(w \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(w\) is a positive constant. Find, in terms of \(u\), the true velocity of the wind.
   

7. \begin{figure}[h] \end{figure} Figure 3 shows a framework \(A B C\), consisting of two uniform rods rigidly joined together at \(B\) so that \(\angle A B C = 90 ^ { \circ }\). The \(\operatorname { rod } A B\) has length \(2 a\) and mass \(4 m\), and the \(\operatorname { rod } B C\) has length \(a\) and mass \(2 m\). The framework is smoothly hinged at \(A\) to a fixed point, so that the framework can rotate in a fixed vertical plane. One end of a light elastic string, of natural length \(2 a\) and modulus of elasticity \(3 m g\), is attached to \(A\). The string passes through a small smooth ring \(R\) fixed at a distance \(2 a\) from \(A\), on the same horizontal level as \(A\) and in the same vertical plane as the framework. The other end of the string is attached to \(B\). The angle \(A R B\) is \(\theta\), where \(0 < \theta < \frac { \pi } { 2 }\).

1. Show that the potential energy \(V\) of the system is given by $$V = 8 a m g \sin 2 \theta + 5 a m g \cos 2 \theta + \text { constant }$$
2. Find the value of \(\theta\) for which the system is in equilibrium.
3. Determine the stability of this position of equilibrium. A smooth uniform sphere \(S\), of mass \(m\), is moving on a smooth horizontal plane when it collides obliquely with another smooth uniform sphere \(T\), of the same radius as \(S\) but of mass \(2 m\), which is at rest on the plane. Immediately before the collision the velocity of \(S\) makes an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\), with the line joining the centres of the spheres. Immediately after the collision the speed of \(T\) is \(V\). The coefficient of restitution between the spheres is \(\frac { 3 } { 4 }\).

1. Find, in terms of \(V\), the speed of \(S\)
   1. immediately before the collision,
   2. immediately after the collision.
2. Find the angle through which the direction of motion of \(S\) is deflected as a result of the collision.
   

7

1. Find Dominic's speed at the point when the cord initially becomes taut.
   7
2. Determine whether or not Dominic enters the river and gets wet.
   7
3. One limitation of this model is that Dominic is not a particle.
   Explain the effect of revising this assumption on your answer to part (b). \includegraphics[max width=\textwidth, alt={}, center]{1b79a789-c003-46c9-9235-254c1d8a0501-12_2492_1721_217_150} Question number Additional page, if required.
   Write the question numbers in the left-hand margin. Question number Additional page, if required.
   Write the question numbers in the left-hand margin. Additional page, if required.
   Write the question numbers in the left-hand margin.

3 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A ball of mass of 0.75 kg is thrown vertically upwards with an initial speed of \(12 \mathrm {~ms} ^ { - 1 }\) The ball is thrown from ground level. 3

1. Calculate the initial kinetic energy of the ball. 3
2. The maximum height of the ball above the ground is \(h\) metres.
   Jeff and Gurjas use an energy method to find \(h\) Jeff concludes that \(h = 7.3\) Gurjas concludes that \(h < 7.3\) Explain the reasoning that they have used, showing any calculations that you make.

20 J
25 J
50 J
100 J 2 Two particles of equal mass are moving on a horizontal surface when they collide.
Immediately before the collision, their velocities are \(\left[ \begin{array} { l } 2 \\ 4 \end{array} \right] \mathrm { ms } ^ { - 1 }\) and \(\left[ \begin{array} { c } 6 \\ - 2 \end{array} \right] \mathrm { ms } ^ { - 1 }\) As a result of the collision the particles coalesce to become a single particle.
Find the velocity of the single particle, immediately after the collision.
Circle your answer.
[0pt] [1 mark] \(\left[ \begin{array} { l } 4 \\ 1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) \(\left[ \begin{array} { l } 4 \\ 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) \(\left[ \begin{array} { l } 8 \\ 2 \end{array} \right] \mathrm { ms } ^ { - 1 }\) \(\left[ \begin{array} { l } 8 \\ 6 \end{array} \right] \mathrm { ms } ^ { - 1 }\) 3 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A ball of mass of 0.75 kg is thrown vertically upwards with an initial speed of \(12 \mathrm {~ms} ^ { - 1 }\) The ball is thrown from ground level. 3

1. Calculate the initial kinetic energy of the ball. 3
2. The maximum height of the ball above the ground is \(h\) metres.
   Jeff and Gurjas use an energy method to find \(h\) Jeff concludes that \(h = 7.3\) Gurjas concludes that \(h < 7.3\) Explain the reasoning that they have used, showing any calculations that you make.
   4 Wavelength is defined as the distance from the highest point on one wave to the highest point on the next wave. Surfers classify waves into one of several types related to their wavelengths.
   Two of these classifications are deep water waves and shallow water waves.
   4
   1. The wavelength \(w\) of a deep water wave is given by $$w = \frac { g t ^ { 2 } } { k }$$ where \(g\) is the acceleration due to gravity and \(t\) is the time period between consecutive waves. Given that the formula for a deep water wave is dimensionally consistent, show that \(k\) is a dimensionless constant. 4
   2. The wavelength \(w\) of a shallow water wave is given by $$w = ( g d ) ^ { \alpha } t ^ { \beta }$$ where \(g\) is the acceleration due to gravity, \(d\) is the depth of water and \(t\) is the time period between consecutive waves. Use dimensional analysis to find the values of \(\alpha\) and \(\beta\) 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.