6.03b Conservation of momentum: 1D two particles

4. At time \(t = 0\) a rocket is launched from rest vertically upwards. The rocket propels itself upwards by expelling burnt fuel vertically downwards with constant speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the rocket. The initial mass of the rocket is \(M _ { 0 } \mathrm {~kg}\). At time \(t\) seconds, where \(t < 2\), its mass is \(M _ { 0 } \left( 1 - \frac { 1 } { 2 } t \right) \mathrm { kg }\), and it is moving upwards with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { U } { ( 2 - t ) } - 9.8 .$$
2. Hence show that \(U > 19.6\).
3. Find, in terms of \(U\), the speed of the rocket one second after its launch.

A spaceship is moving in a straight line in deep space and needs to increase its speed. This is done by ejecting fuel backwards from the spaceship at a constant speed \(c\) relative to the spaceship. When the speed of the spaceship is \(v\), its mass is \(m\).

1. Show that, while the spaceship is ejecting fuel, $$\frac { \mathrm { d } v } { \mathrm {~d} m } = - \frac { c } { m } .$$ The initial mass of the spaceship is \(m _ { 0 }\) and at time \(t\) the mass of the spaceship is given by \(m = m _ { 0 } ( 1 - k t )\), where \(k\) is a positive constant.
2. Find the acceleration of the spaceship at time \(t\).

A uniform circular disc has mass \(m\), centre \(O\) and radius \(2 a\). It is free to rotate about a fixed smooth horizontal axis \(L\) which lies in the same plane as the disc and which is tangential to the disc at the point \(A\). The disc is hanging at rest in equilibrium with \(O\) vertically below \(A\) when it is struck at \(O\) by a particle of mass \(m\). Immediately before the impact the particle is moving perpendicular to the plane of the disc with speed \(3 \sqrt { } ( a g )\). The particle adheres to the disc at \(O\).

1. Find the angular speed of the disc immediately after the impact.
2. Find the magnitude of the force exerted on the disc by the axis immediately after the impact.

A spacecraft is moving in a straight line in deep space. The spacecraft moves by ejecting burnt fuel backwards at a constant speed of \(2000 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the spacecraft. The burnt fuel is ejected at a constant rate of \(c \mathrm {~kg} \mathrm {~s} ^ { - 1 }\). At time \(t\) seconds the total mass of the spacecraft, including fuel, is \(m \mathrm {~kg}\) and the speed of the spacecraft is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that, while the spacecraft is ejecting burnt fuel, $$m \frac { \mathrm {~d} v } { \mathrm {~d} t } = 2000 c$$ At time \(t = 0\), the mass of the spacecraft is \(M _ { 0 } \mathrm {~kg}\) and the speed of the spacecraft is \(2000 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(t = 50\), the spacecraft is still ejecting burnt fuel and its speed is \(6000 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find \(c\) in terms of \(M _ { 0 }\).
   

3. A raindrop falls vertically under gravity through a stationary cloud. At time \(t = 0\), the raindrop is at rest and has mass \(m _ { 0 }\). As the raindrop falls, water condenses onto it from the cloud so that the mass of the raindrop increases at a constant rate \(c\). At time \(t\), the mass of the raindrop is \(m\) and the speed of the raindrop is \(v\). The resistance to the motion of the raindrop has magnitude \(m k v\), where \(k\) is a constant. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } + v \left( k + \frac { c } { m _ { 0 } + c t } \right) = g$$

7. A raindrop absorbs water as it falls vertically under gravity through a cloud. In a model of the motion the cloud is assumed to consist of stationary water particles. At time \(t\), the mass of the raindrop is \(m\) and the speed of the raindrop is \(v\). At time \(t = 0\), the raindrop is at rest. The rate of increase of the mass of the raindrop with respect to time is modelled as being \(m k v\), where \(k\) is a positive constant.

1. Ignoring air resistance, show from first principles, that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = g - k v ^ { 2 }$$
2. Find the time taken for the raindrop to reach a speed of \(\frac { 1 } { 2 } \sqrt { } \left( \frac { g } { k } \right)\)

2. A rod \(A B\) has mass \(m\) and length \(4 a\). It is free to rotate about a fixed smooth horizontal axis through the point \(O\) of the rod, where \(A O = a\). The rod is hanging in equilibrium with \(B\) below \(O\) when it is struck by a particle \(P\), of mass \(3 m\), moving horizontally with speed \(v\). When \(P\) strikes the rod, it adheres to it. Immediately after striking the rod, \(P\) has speed \(\frac { 2 } { 3 } v\). Find the distance from \(O\) of the point where \(P\) strikes the rod.
(7 marks)

5. A spaceship is moving in deep space with no external forces acting on it. Initially it has total mass \(M\) and is moving with speed \(V\). The spaceship reduces its speed to \(\frac { 2 } { 3 } V\) by ejecting fuel from its front end with a speed of \(c\) relative to itself and in the same direction as its own motion. Find the mass of fuel ejected.
(11 marks)

6 Three particles, A, B and C are in a straight line on a smooth horizontal surface.
The particles have masses \(5 \mathrm {~kg} , 3 \mathrm {~kg}\) and 1 kg respectively. Particles B and C are at rest. Particle A is projected towards B with a speed of \(u \mathrm {~ms} ^ { - 1 }\) and collides with B . The coefficient of restitution between A and B is \(\frac { 1 } { 3 }\). Particle B subsequently collides with C. The coefficient of restitution between B and C is \(\frac { 1 } { 3 }\).

1. Determine whether any further collisions occur.
2. Given that the loss of kinetic energy during the initial collision between A and B is 4.8 J , find the value of \(u\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6b27d322-417e-4cea-85cc-65d3728173c8-5_607_501_294_301} \captionsetup{labelformat=empty} \caption{Fig. 7}
   \end{figure} Fig. 7 shows a uniform rod AB of length \(4 a\) and mass \(m\).
   The end A rests against a rough vertical wall. A light inextensible string is attached to the rod at B and to a point C on the wall vertically above A , where \(\mathrm { AC } = 4 a\). The plane ABC is perpendicular to the wall and the angle ABC is \(30 ^ { \circ }\). The system is in limiting equilibrium. Find the coefficient of friction between the wall and the rod. \section*{END OF QUESTION PAPER}

7 The diagram shows a cannon fixed to a trolley. The trolley runs on a smooth horizontal track. \includegraphics[max width=\textwidth, alt={}, center]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-8_310_1086_296_520} A driver boards the trolley with two cannon balls. The combined mass of the trolley, driver, cannon and cannon balls is 320 kg . Each cannon ball has a mass of 5 kg . Initially the trolley is at rest. A force of 480 N acts on the trolley in the forward direction for 4 seconds.

1. 1. Calculate the magnitude of the impulse of the force on the trolley.
   2. Calculate the speed of the trolley after the force stops acting. The driver now fires a cannon ball horizontally in the backward direction. The cannon ball and cannon separate at a rate of \(90 \mathrm {~ms} ^ { - 1 }\).
2. Show that, after the firing of the cannon ball, the trolley moves with a speed of \(7.41 \mathrm {~ms} ^ { - 1 }\), correct to \(\mathbf { 3 }\) significant figures. The driver now reverses the direction of the cannon and fires the second cannon ball horizontally in the forward direction. Again, the cannon ball and cannon separate at a rate of \(90 \mathrm {~ms} ^ { - 1 }\).
3. Calculate the overall percentage change in the kinetic energy of the trolley (alone) from before the first cannon ball is fired to after the second is fired, giving your answer correct to \(\mathbf { 2 }\) decimal places. You should make clear whether the change in kinetic energy is a gain or a loss.
4. Give a reason why one of the modelling assumptions that was required in answering parts (a), (b) and (c) may not have been appropriate. \section*{END OF QUESTION PAPER}

4 The diagram shows three beads, A, B and C, of masses \(0.3 \mathrm {~kg} , 0.5 \mathrm {~kg}\) and 0.7 kg respectively, threaded onto a smooth wire circuit consisting of two straight and two semi-circular sections. The circuit occupies a vertical plane, with the two straight sections horizontal and the upper section 0.45 m directly above the lower section. \includegraphics[max width=\textwidth, alt={}, center]{a87d62b8-406d-44cd-9ffa-384005329566-5_361_961_450_248} Initially, the beads are at rest. A and B are each given an impulse so that they move towards each other, A with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and B with a speed of \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In the subsequent collision between A and \(\mathrm { B } , \mathrm { A }\) is brought to rest.

1. Show that the coefficient of restitution between A and B is \(\frac { 1 } { 3 }\). Bead B next collides with C.
2. Show that the speed of B before this collision is \(4.37 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures. In this collision between B and C , B is brought to rest.
3. Determine whether C next collides with A or with B .
4. Explain why, if B has a greater mass than C , B could not be brought to rest in their collision.

4 Three spheres A, B, and C, of equal radius are in the same straight line on a smooth horizontal surface. The masses of \(\mathrm { A } , \mathrm { B }\) and C are \(2 \mathrm {~kg} , 4 \mathrm {~kg}\) and 1 kg respectively. Initially the three spheres are at rest.
Spheres \(A\) and \(C\) are each given impulses so that \(A\) moves towards \(B\) with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and C moves towards B with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{a96a0ebe-8f4f-4d79-9d11-9d348ef72314-5_325_1591_603_239} The coefficient of restitution between \(A\) and \(B\) is \(\frac { 4 } { 5 }\).
It is given that the first collision occurs between A and B .

1. State how you can tell from the information given above that kinetic energy is lost when A collides with B .
2. Show that the combined kinetic energy of A and B decreases by \(24 \%\) during their collision. Sphere B next collides with C. The coefficient of restitution between B and C is \(\frac { 2 } { 3 }\).
3. Given that a third collision occurs, determine the range of possible values for \(u\).
4. State one limitation of the model used in this question.

5 Throughout this question it may be assumed that there are no resistances to motion.
Model trucks A and B, with masses 5 kg and 3 kg respectively, rest on a set of straight, horizontal rails. Truck A is given an impulse of 3.8 Ns towards B .

1. Calculate the initial speed of A. Truck A collides directly with B. After the collision, B moves with a speed of \(0.6 \mathrm {~ms} ^ { - 1 }\).
2. Determine
   1. the velocity of A after the collision,
   2. the kinetic energy lost due to the collision.
3. B continues to move with a speed of \(0.6 \mathrm {~ms} ^ { - 1 }\) and collides with a model truck C, of mass 4 kg , which is travelling at a speed of \(0.2 \mathrm {~ms} ^ { - 1 }\) towards B on the same set of rails. After the collision between B and C , the speeds of B and C are in the ratio 1 to 2 . Determine the two possible values of the coefficient of restitution between B and C .

3 Three small uniform spheres A, B and C have masses \(2 \mathrm {~kg} , 3 \mathrm {~kg}\) and 5 kg respectively. The spheres move in the same straight line on a smooth horizontal table, with B between A and C . Sphere A moves towards B with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 } , \mathrm {~B}\) is at rest and C moves towards B with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{5c1cfe41-d7a2-4f69-ae79-67d9f023c246-3_181_1291_461_251} Spheres A and B collide. Collisions between A and B can be modelled as perfectly elastic.

1. Determine the magnitude of the impulse of A on B in this collision.
2. Use this collision to verify that in a perfectly elastic collision no kinetic energy is lost. After the collision between A and B, sphere B subsequently collides with C. The coefficient of restitution between B and C is \(\frac { 1 } { 4 }\).
3. Show that, after the collision between B and C , B has a speed of \(( 1.225 - 0.78125 \mathrm { u } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) towards C.
4. Determine the range of values for \(u\) for there to be a second collision between A and B .

7 Rose and Steve collide while sitting firmly on trays that are sliding on smooth horizontal ice. There are no external driving forces. Fig. 7 shows the masses of Rose and of Steve with their trays, their velocities just before their collision and the line of their motion and of their impact. Immediately after the collision, Rose has a velocity of \(0.28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of her motion before the collision. \begin{figure}[h] \end{figure}

1. Find Steve's velocity after the collision.
2. Find the coefficient of restitution between Rose and Steve on their trays. Shortly after the collision, Steve catches Rose's hand, pulls her towards him with a horizontal impulse of 4.48 Ns and then lets go of her hand.
3. Calculate Rose's velocity after the pull. When they collide again they hold one another and move together with a common speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
4. Calculate \(V\).
5. Why did you need to know that there are no driving forces and that the ice is smooth? {www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
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4 Two uniform discs, A of mass 0.2 kg and B of mass 0.5 kg , collide with smooth contact while moving on a smooth horizontal surface.
Immediately before the collision, A is moving with speed \(0.5 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) with the line of centres, where \(\sin \alpha = 0.6\), and B is moving with speed \(0.3 \mathrm {~ms} ^ { - 1 }\) at right angles to the line of centres. A straight smooth vertical wall is situated to the right of B , perpendicular to the line of centres, as shown in Fig. 4. The coefficient of restitution between A and B is 0.75 . \begin{figure}[h] \end{figure}

1. Find the speeds of A and B immediately after the collision.
2. Explain why there could be a second collision between A and B if B rebounds from the wall with sufficient speed.
3. Find the range of values of the coefficient of restitution between B and the wall for which there will be a second collision between A and B .
4. How does your answer to part (b) change if the contact between B and the wall is not smooth?

5 Two small uniform discs, A of mass \(2 m \mathrm {~kg}\) and B of mass \(3 m \mathrm {~kg}\), slide on a smooth horizontal surface and collide obliquely with smooth contact. Immediately before the collision, A is moving towards B along the line of centres with speed \(2 \mathrm {~ms} ^ { - 1 }\) and B is moving towards A with speed \(\sqrt { 3 } \mathrm {~ms} ^ { - 1 }\) in a direction making an angle of \(30 ^ { \circ }\) with the line of centres, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{feb9a438-26b0-41d3-b044-6acd6efccde0-5_366_976_539_244}

1. Explain how you know that the motion of A will be along the line of centres after the collision.
2. - Determine the maximum possible speed of A after the collision.
   When the speed of B after the collision is a minimum, the loss of kinetic energy in the collision is 1.4625 J .
3. Determine the value of \(m\).

4 Two small smooth discs, A of mass 0.5 kg and B of mass 0.4 kg , collide while sliding on a smooth horizontal plane. Immediately before the collision A and B are moving towards each other, A with speed \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) with speed \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Before the collision the direction of motion of A makes an angle \(\alpha\) with the line of centres, where \(\tan \alpha = 0.75\), and the direction of motion of B makes an angle of \(60 ^ { \circ }\) with the line of centres, as shown in Fig. 4. \begin{figure}[h] \end{figure} After the collision, one of the discs moves in a direction perpendicular to the line of centres, and the other disc moves in a direction making an angle \(\beta\) with the line of centres.

1. Explain why the disc which moves perpendicular to the line of centres must be A .
2. Determine the value of \(\beta\).
3. Determine the kinetic energy lost in the collision.
4. Determine the value of the coefficient of restitution between A and B .

4 Two uniform circular discs with the same radius, A of mass 1 kg and B of mass 5.25 kg , slide on a smooth horizontal surface and collide obliquely with smooth contact. Fig. 4 gives information about the velocities of the discs just before and just after the collision.

• The line XY passes through the centres of the discs at the moment of collision
• The components parallel and perpendicular to XY of the velocities of A are shown
• Before the collision, B is at rest and after it is moving at \(2 \mathrm {~ms} ^ { - 1 }\) in the direction XY

\begin{figure}[h] \end{figure} The coefficient of restitution between the two discs is \(\frac { 2 } { 3 }\).

1. Find the values of \(U\) and \(u\).
2. What information in the question tells you that \(v = V\) ? The speed of disc A before the collision is \(8.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the speed of disc A after the collision. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-5_398_396_397_475} \captionsetup{labelformat=empty} \caption{Fig. 5.1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-5_399_332_399_945} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-5_305_326_493_1354} \captionsetup{labelformat=empty} \caption{Fig. 5.3}
   \end{figure} Fig. 5.1 shows a vertical light elastic spring. It is fixed to a horizontal table at one end. Fig 5.2 shows the spring with a particle of mass \(m \mathrm {~kg}\) attached to it at the other end. The system is in equilibrium when the spring is compressed by a distance \(h \mathrm {~m}\).

4 Two model railway trucks, A of mass 0.1 kg and B of mass 0.2 kg , are constrained to move on a smooth straight level track.
Initially B is stationary and A is moving towards B with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before they collide. The coefficient of restitution between A and B is \(e\).

1. Find the speed of A and the speed of B after the collision, giving your answers in terms of \(e\) and \(u\).
2. Show that the loss of kinetic energy in the collision is \(\frac { 1 } { 30 } u ^ { 2 } \left( 1 - e ^ { 2 } \right)\).
3. For the case in which the loss of kinetic energy is least

5 Point A lies 20 m vertically below a point B . A particle P of mass 4 m kg is projected upwards from A , at a speed of \(17.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At the same time, a particle Q of mass \(m \mathrm {~kg}\) is released from rest at point B . The particles collide directly, and it is given that the coefficient of restitution in the collision between P and Q is 0.6 .

1. Show that, immediately after the collision, P continues to travel upwards at \(0.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and determine, at this time, the corresponding velocity of Q . In another situation, a particle of mass \(3 m \mathrm {~kg}\) is released from rest and falls vertically. After it has fallen 10 m , it explodes into two fragments. Immediately after the explosion, the lower fragment, of mass \(2 m \mathrm {~kg}\), moves vertically downwards with speed \(v _ { 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and the upper fragment, of mass \(m \mathrm {~kg}\), moves vertically upwards with speed \(v _ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Given that, in the explosion, the kinetic energy of the system increases by \(72 \%\), show that \(2 v _ { 1 } ^ { 2 } + v _ { 2 } ^ { 2 } = 1011.36\).
3. By finding another equation connecting \(v _ { 1 }\) and \(v _ { 2 }\), determine the speeds of the fragments immediately after the explosion.

3 The diagram shows two blocks P and Q of masses 0.5 kg and 2 kg respectively, on a horizontal surface. The points \(\mathrm { A } , \mathrm { B }\) and C lie on the surface in a straight line. There is a wall at C . The surface between B and C is smooth, and the surface between A and B is rough, such that the coefficient of friction between P and AB is \(\frac { 2 } { 3 }\). \includegraphics[max width=\textwidth, alt={}, center]{cad8805d-59f6-4ed2-81f4-9e8c749461f5-3_229_1271_1601_278} P is projected with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) directly towards Q , which is at rest. As a result of the collision between P and Q, P changes direction and subsequently comes to rest at A. You may assume that P only collides with Q once.

1. Determine the coefficient of restitution between P and Q .
2. Calculate the impulse exerted on P by Q during their collision. After colliding with P , Q strikes the wall, which is perpendicular to the direction of the motion of Q , and comes to rest exactly halfway between A and B . The collision between Q and the wall is perfectly elastic.
3. Determine the coefficient of friction between Q and AB .

4 Fig. 4.1 shows two spheres, A and B, on a smooth horizontal surface. Their masses are 3 kg and 1 kg respectively. \begin{figure}[h] \end{figure} Initially, sphere A travels at a speed of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line towards B , which is at rest. The spheres collide and the coefficient of restitution between A and B is \(e\).

1. Show that, after the collision, A has a speed of \(\frac { 1 } { 4 } ( 3 - e ) \mathrm { m } \mathrm { s } ^ { - 1 }\), and find an expression for the speed of B in terms of \(e\). During the collision, the kinetic energy of the system decreases by \(21 \%\).
2. Determine the value of \(e\).
3. State why in part (a) it was necessary to assume that A and B have equal radii. Fig. 4.2 shows two spheres, C and D , of equal radii on a smooth horizontal surface. Their masses are 1 kg and 2 kg respectively. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 4.2} \includegraphics[alt={},max width=\textwidth]{0a790ad0-7eda-40f1-9894-f156766ae46f-4_158_1155_1544_244}
   \end{figure} Spheres C and D travel towards each other along the same straight line, C with a speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and D with a speed of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The spheres collide and during the collision C exerts an impulse on D of magnitude \(\frac { 2 } { 3 } ( u + 1 ) \mathrm { Ns }\).
4. Show that C and D have the same velocity after the collision.
5. Determine the fraction of kinetic energy lost due to the collision between C and D as \(u \rightarrow \infty\).

6 Stones A and B have masses \(m \mathrm {~kg}\) and \(3 m \mathrm {~kg}\) respectively. They lie at rest on a large area of smooth horizontal ice and may move freely over the ice. Stone A is given a horizontal impulse of magnitude \(m u \mathrm {~N} s\) towards B so that the stones collide directly. After the collision the direction of motion of A is reversed. The coefficient of restitution between A and B is denoted by \(e\).

1. Find the range of possible values of \(e\). After the collision, B subsequently collides with a vertical smooth wall perpendicular to its path and rebounds. The coefficient of restitution between \(B\) and the wall is the same as the coefficient of restitution between A and B .
2. Show that A and B will collide again unless the collision between B and the wall is perfectly elastic.
3. Explain why modelling the collision between B and the wall as perfectly elastic is possibly unrealistic.
4. Given that the kinetic energy lost in the first collision between A and B is \(\frac { 5 } { 24 } m u ^ { 2 }\), determine the value of \(e\).
5. Given that B was 2 metres from the wall when the stones first collided, determine the distance of the stones from the wall when they next collide.

6 A block rests on a horizontal surface. The coefficient of friction between the block and the surface is \(\mu\).

1. Show that if the block is given an initial speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it will move a distance of \(\frac { \mathrm { v } ^ { 2 } } { 2 \mu \mathrm {~g} }\) before coming to rest. Block B rests on the same horizontal surface as a sphere S . On the other side of S is a vertical wall, as shown below. The mass of \(B\) is 8 times the mass of \(S\). \includegraphics[max width=\textwidth, alt={}, center]{b3e369f4-13f7-457b-9a43-04ed2e2a2bba-8_211_1013_662_244} S is projected directly towards B with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and hits B . It is given that
   Furthermore, you should model the contact between B and the surface as rough and model the contact between S and the surface as smooth.
2. Determine, in terms of \(u\), expressions for
   It is given that B has sufficient time to come to rest before each subsequent collision with S .
   Let \(\mathrm { X } _ { \mathrm { n } }\) be the distance B moves after the \(n\)th impact between S and B .
3. Explain why \(\mathrm { x } _ { \mathrm { n } + 1 } = \frac { 9 } { 25 } \mathrm { x } _ { \mathrm { n } }\).
4. Given that \(u = 11.2\) and the coefficient of friction between B and the surface is \(\frac { 1 } { 7 }\), show that B will travel a total distance that cannot exceed 2.8 m . \section*{END OF QUESTION PAPER} \section*{OCR
   Oxford Cambridge and RSA}