6.03b Conservation of momentum: 1D two particles

6 My cat Jeoffry has a mass of 4 kg and is sitting on rough ground near a sledge of mass 8 kg . The sledge is on a large area of smooth horizontal ice. Initially, the sledge is at rest and Jeoffry jumps and lands on it with a horizontal velocity of \(2.25 \mathrm {~ms} ^ { - 1 }\) parallel to the runners of the sledge. On landing, Jeoffry grips the sledge with his claws so that he does not move relative to the sledge in the subsequent motion.

1. Show that the sledge with Jeoffry on it moves off with a speed of \(0.75 \mathrm {~ms} ^ { - 1 }\). With the sledge and Jeoffry moving at \(0.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the sledge collides directly with a stationary stone of mass 3 kg . The stone may move freely over the ice. The coefficient of restitution in the collision is \(\frac { 4 } { 15 }\).
2. Calculate the velocity of the sledge and Jeoffry immediately after the collision. In a new situation, Jeoffry is initially sitting at rest on the sledge when it is stationary on the ice. He then walks from the back to the front of the sledge.
3. Giving a brief reason for your answer, describe what happens to the sledge during his walk. Jeoffry is again sitting on the sledge when it is stationary on the ice. He jumps off and, after he has lost contact with the sledge, has a horizontal speed relative to the sledge of \(3 \mathrm {~ms} ^ { - 1 }\).
4. Determine the speed of the sledge after Jeoffry loses contact with it. Fig. 7 shows a container for flowers which is a vertical cylindrical shell with a closed horizontal base. Its radius and its height are both \(\frac { 1 } { 2 } \mathrm {~m}\). Both the curved surface and the base are made of the same thin uniform material. The mass of the container is \(M \mathrm {~kg}\). \begin{figure}[h]
   \includegraphics[width=0.8\textwidth]{54711a46-83ce-4fb9-b6d3-53b264725c74-6_323_709_447_767} \caption{Fig. 7}
   \end{figure}

1. Find, as a fraction, the height above the base of the centre of mass of the container. The container would hold \(\frac { 3 } { 2 } M \mathrm {~kg}\) of soil when full to the top. Some soil is put into the empty container and levelled with its top surface \(y \mathrm {~m}\) above the base. The centre of mass of the container with this much soil is zm above the base.
2. Show that \(z = \frac { 1 + 9 y ^ { 2 } } { 6 ( 1 + 3 y ) }\).
3. It is given that \(\frac { \mathrm { d } z } { \mathrm {~d} y } = 0\) when \(y = 0.14\) (to 2 significant figures) and that \(\frac { \mathrm { d } ^ { 2 } z } { \mathrm {~d} y ^ { 2 } } > 0\) at this value of \(y\). When putting in the soil, how might you use this information if the container is to be placed on slopes without it tipping over? \section*{END OF QUESTION PAPER} }{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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5 Two small uniform smooth spheres A and B , of equal radius, have masses 2 kg and 4 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, A has speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving along the line of centres, and B has speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving along a line which is perpendicular to the line of centres (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-03_389_764_1592_244} The direction of motion of B after the collision makes an angle of \(45 ^ { \circ }\) with the line of centres. Determine the coefficient of restitution between A and B .

11 Two small uniform smooth spheres A and B , of equal radius, have masses 4 kg and 3 kg respectively. The spheres are placed in a smooth horizontal circular groove. The coefficient of restitution between the spheres is \(e\), where \(e > \frac { 2 } { 5 }\). At a given instant B is at rest and A is set moving along the groove with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It may be assumed that in the subsequent motion the two spheres do not leave the groove.

1. Determine, in terms of \(e\) and \(V\), the speeds of A and B immediately after the first collision.
2. Show that the arc through which A moves between the first and second collisions subtends an angle at the centre of the circular groove of $$\frac { 2 \pi ( 4 - 3 e ) } { 7 e } \text { radians. }$$
3. 1. Determine, in terms of \(e\) and \(V\), the speed of B immediately after the second collision.
   2. What can be said about the motion of A and B if the collisions between A and B are perfectly elastic?

7. Three spheres \(A , B , C\), of equal radii and each of mass \(m \mathrm {~kg}\), lie at rest on a smooth horizontal surface such that their centres are in a straight line with \(B\) between \(A\) and \(C\). The coefficient of restitution between \(A\) and \(B\) is \(e\). Sphere \(A\) is projected towards \(B\) with speed \(u \mathrm {~ms} ^ { - 1 }\) so that it collides with \(B\).

1. Find expressions, in terms of \(e\) and \(u\), for the speed of \(A\) and the speed of \(B\) after they collide. You are now given that \(e = \frac { 1 } { 2 }\).
2. Find, in terms of \(m\) and \(u\), the loss in kinetic energy due to the collision between \(A\) and \(B\).
3. After the collision between \(A\) and \(B\), sphere \(B\) then collides with \(C\). The coefficient of restitution between \(B\) and \(C\) is \(e _ { 1 }\). Show that there will be no further collisions if \(e _ { 1 } \leqslant \frac { 1 } { 3 }\).

1. Two particles \(A\) and \(B\), of masses 2 kg and 5 kg respectively, are moving in the same direction along a smooth horizontal surface when they collide directly. Before the collision, \(B\) is moving with speed \(1.2 \mathrm {~ms} ^ { - 1 }\) and, immediately after the collision, its speed is \(3.8 \mathrm {~ms} ^ { - 1 }\). The coefficient of restitution between the particles \(A\) and \(B\) is 0.3 .

1. 1. Find the impulse exerted by \(A\) on \(B\).
   2. Given that the particles \(A\) and \(B\) were in contact for 0.08 seconds, find the average force between \(A\) and \(B\).
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2. Calculate the speed of \(A\) before and after the collision.
3. After the collision between \(A\) and \(B\), particle \(B\) continues to move with speed \(3.8 \mathrm {~ms} ^ { - 1 }\) until it collides directly with a stationary particle \(C\) of mass 4 kg . When \(B\) and \(C\) collide, they coalesce to form a single particle.
   1. Write down the coefficient of restitution between \(B\) and \(C\).
   2. Determine the speed of the combined particle after the collision.
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4. Ryan is playing a game of snooker. The horizontal table is modelled as the horizontal \(x - y\) plane with the point \(O\) as the origin and unit vectors parallel to the \(x\)-axis and the \(y\)-axis denoted by \(\mathbf { i }\) and \(\mathbf { j }\) respectively. All balls on the table have a common mass \(m \mathrm {~kg}\). The table and the four sides, called cushions, are modelled as smooth surfaces. The dimensions of the table, in metres, are as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{3578a810-46da-4d9e-a98f-248be72a517a-5_663_1138_667_482} Initially, all balls are stationary. Ryan strikes ball \(A\) so that it collides with ball \(B\). Before the collision, \(A\) has velocity \(( - \mathbf { i } + 8 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and, after the collision, it has velocity \(( 2 \mathbf { i } + \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).

1. Show that the velocity of ball \(B\) after the collision is \(( - 3 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). After the collision with ball \(A\), ball \(B\) hits the cushion at point \(C\) before rebounding and moving towards the pocket at \(P\). The cushion is parallel to the vector \(\mathbf { i }\) and the coefficient of restitution between the cushion and ball \(B\) is \(\frac { 5 } { 7 }\).
2. Calculate the velocity of ball \(B\) after impact with the cushion.
3. Find, in terms of \(m\), the magnitude of the impulse exerted on ball \(B\) by the cushion at \(C\), stating your units clearly.
4. Given that \(C\) has position vector \(( x \mathbf { i } + 1 \cdot 75 \mathbf { j } ) \mathrm { m }\),
   1. determine the time taken between the ball hitting the cushion at \(C\) and entering the pocket at \(P\),
   2. find the value of \(x\).
5. Describe one way in which the model used could be refined. Explain how your refinement would affect your answer to (d)(i).

5. Two smooth spheres \(A\) and \(B\), of equal radii, are moving on a smooth horizontal plane when they collide. Immediately after the collision sphere \(A\) has velocity ( \(- 2 \mathbf { i } - 5 \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\) and sphere \(B\) has velocity \(( \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). When the spheres collide, their line of centres is parallel to the vector \(\mathbf { i }\) and the coefficient of restitution between the spheres is \(\frac { 2 } { 5 }\). Sphere \(A\) has mass 4 kg and sphere \(B\) has mass 2 kg .

1. Find the velocity of \(A\) and the velocity of \(B\) immediately before the collision. After the collision, sphere \(A\) continues to move with velocity ( \(- 2 \mathbf { i } - 5 \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\) until it collides with a smooth vertical wall. The impulse exerted by the wall on \(A\) is \(32 \mathbf { j }\) Ns.
2. State whether the wall is parallel to the vector \(\mathbf { i }\) or to the vector \(\mathbf { j }\). Give a reason for your answer.
3. Find the speed of \(A\) after the collision with the wall.
4. Calculate the loss of kinetic energy caused by the collision of sphere \(A\) with the wall.

5. In this question, \(\mathbf { i }\) and \(\mathbf { j }\) represent unit vectors due east and due north respectively. Two smooth spheres \(P\) and \(Q\), of equal radii, are moving on a smooth horizontal surface. The mass of \(P\) is 2 kg and the mass of \(Q\) is 6 kg . The velocity of \(P\) is \(( 8 \mathbf { i } - 6 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and the velocity of \(Q\) is \(( 4 \mathbf { i } + 10 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). At a particular instant, \(Q\) is positioned 12 m east and 48 m south of \(P\).

1. Prove that \(P\) and \(Q\) will collide. At the instant the spheres collide, the line joining their centres is parallel to the vector \(\mathbf { j }\). Immediately after the collision, sphere \(Q\) has speed \(5 \mathrm {~ms} ^ { - 1 }\).
2. Determine the coefficient of restitution between the spheres and hence calculate the velocity of \(P\) immediately after the collision.
3. Find the magnitude of the impulse required to stop sphere \(P\) after the collision.

5. A particle \(A\), of mass \(m \mathrm {~kg}\), has position vector \(11 \mathbf { i } + 6 \mathbf { j }\) and a velocity \(2 \mathbf { i } + 7 \mathbf { j }\). At the same moment, second particle \(B\), of mass \(2 m \mathrm {~kg}\), has position vector \(7 \mathbf { i } + 10 \mathbf { j }\) and a velocity \(5 \mathbf { i } + 4 \mathbf { j }\).

1. If the particles continue to move with these velocities, prove that the particles will collide. Given that the particles coalesce after collision, find the common velocity of the particles after collision.
2. Determine the impulse exerted by \(A\) on \(B\).
3. Calculate the loss of kinetic energy caused by the collision.

4 Two discs, \(A\) and \(B\), have equal radii and masses 0.8 kg and 0.4 kg respectively. The discs are placed on a horizontal surface. The discs are set in motion when they are 3 metres apart, so that they move directly towards each other, each travelling at a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The discs collide directly with each other. After the collision \(A\) moves in the opposite direction with a speed of \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The coefficient of restitution between the two discs is \(e\). 4

1. Assuming that the surface is smooth, show that \(e = 0.8\) 4
2. Describe one way in which the model you have used could be refined. Turn over for the next question

1. A particle \(P\) of mass \(3 m\) is moving in a straight line on a smooth horizontal floor. A particle \(Q\) of mass \(5 m\) is moving in the opposite direction to \(P\) along the same straight line.

The particles collide directly.
Immediately before the collision, the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(u\). The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Show that the speed of \(Q\) immediately after the collision is \(\frac { u } { 8 } ( 9 e + 1 )\)
2. Find the range of values of \(e\) for which the direction of motion of \(P\) is not changed as a result of the collision. When \(P\) and \(Q\) collide they are at a distance \(d\) from a smooth fixed vertical wall, which is perpendicular to their direction of motion. After the collision with \(P\), particle \(Q\) collides directly with the wall and rebounds so that there is a second collision between \(P\) and \(Q\). This second collision takes place at a distance \(x\) from the wall. Given that \(e = \frac { 1 } { 18 }\) and the coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 3 }\)
3. find \(x\) in terms of \(d\).

Two particles, \(A\) and \(B\), of masses \(2 m\) and \(3 m\) respectively, are moving on a smooth horizontal plane. The particles are moving in opposite directions towards each other along the same straight line when they collide directly. Immediately before the collision the speed of \(A\) is \(2 u\) and the speed of \(B\) is \(u\). In the collision the impulse of \(A\) on \(B\) has magnitude 5 mu .

1. Find the coefficient of restitution between \(A\) and \(B\).
2. Find the total loss in kinetic energy due to the collision.

1. Three particles, \(P , Q\) and \(R\), are at rest on a smooth horizontal plane. The particles lie along a straight line with \(Q\) between \(P\) and \(R\). The particles \(Q\) and \(R\) have masses \(m\) and \(k m\) respectively, where \(k\) is a constant.

Particle \(Q\) is projected towards \(R\) with speed \(u\) and the particles collide directly.
The coefficient of restitution between each pair of particles is \(e\).

1. Find, in terms of \(e\), the range of values of \(k\) for which there is a second collision. Given that the mass of \(P\) is \(k m\) and that there is a second collision,
2. write down, in terms of \(u , k\) and \(e\), the speed of \(Q\) after this second collision.

1. Three particles \(A , B\) and \(C\) are at rest on a smooth horizontal plane. The particles lie along a straight line with \(B\) between \(A\) and \(C\).

Particle \(B\) has mass \(4 m\) and particle \(C\) has mass \(k m\), where \(k\) is a positive constant. Particle \(B\) is projected with speed \(u\) along the plane towards \(C\) and they collide directly. The coefficient of restitution between \(B\) and \(C\) is \(\frac { 1 } { 4 }\)

1. Find the range of values of \(k\) for which there would be no further collisions. The magnitude of the impulse on \(B\) in the collision between \(B\) and \(C\) is \(3 m u\)
2. Find the value of \(k\).

1. Two particles, \(P\) and \(Q\), have masses \(m\) and \(e m\) respectively. The particles are moving on a smooth horizontal plane in the same direction along the same straight line when they collide directly. The coefficient of restitution between \(P\) and \(Q\) is \(e\), where \(0 < e < 1\)

Immediately before the collision the speed of \(P\) is \(u\) and the speed of \(Q\) is \(e u\).

1. Show that the speed of \(Q\) immediately after the collision is \(u\).
2. Show that the direction of motion of \(P\) is unchanged by the collision. The magnitude of the impulse on \(Q\) in the collision is \(\frac { 2 } { 9 } m u\)
3. Find the possible values of \(e\).

1. Two particles, \(A\) and \(B\), have masses \(m\) and \(3 m\) respectively. The particles are moving in opposite directions along the same straight line on a smooth horizontal plane when they collide directly.

Immediately before they collide, \(A\) is moving with speed \(2 u\) and \(B\) is moving with speed \(u\). The direction of motion of each particle is reversed by the collision.
In the collision, the magnitude of the impulse exerted on \(A\) by \(B\) is \(\frac { 9 m u } { 2 }\)

1. Find the value of the coefficient of restitution between \(A\) and \(B\).
2. Hence, write down the total loss in kinetic energy due to the collision, giving a reason for your answer.

A particle \(P\) of mass \(2 m \mathrm {~kg}\) is moving with speed \(2 u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a smooth horizontal plane. Particle \(P\) collides with a particle \(Q\) of mass \(3 m \mathrm {~kg}\) which is at rest on the plane. The coefficient of restitution between \(P\) and \(Q\) is \(e\). Immediately after the collision the speed of \(Q\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

1. Show that \(v = \frac { 4 u ( 1 + e ) } { 5 }\)
2. Show that \(\frac { 4 u } { 5 } \leqslant v \leqslant \frac { 8 u } { 5 }\) Given that the direction of motion of \(P\) is reversed by the collision,
3. find, in terms of \(u\) and \(e\), the speed of \(P\) immediately after the collision. After the collision, \(Q\) hits a wall, that is fixed at right angles to the direction of motion of \(Q\), and rebounds. The coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 6 }\) Given that \(P\) and \(Q\) collide again,
4. find the full range of possible values of \(e\).

1. Two particles, \(P\) and \(Q\), of masses \(3 m\) and \(2 m\) respectively, are moving on a smooth horizontal plane. They are moving in opposite directions along the same straight line when they collide directly.

Immediately before the collision, \(P\) is moving with speed \(2 u\).
The magnitude of the impulse exerted on \(P\) by \(Q\) in the collision is \(\frac { 9 m u } { 2 }\)

1. Find the speed of \(P\) immediately after the collision. The coefficient of restitution between \(P\) and \(Q\) is \(e\).
   Given that the speed of \(Q\) immediately before the collision is \(u\),
2. find the value of \(e\).

4. \begin{figure}[h] \end{figure} Three particles, \(P , Q\) and \(R\), lie at rest on a smooth horizontal plane. The particles are in a straight line with \(Q\) between \(P\) and \(R\), as shown in Figure 1 . Particle \(P\) is projected towards \(Q\) with speed \(u\). At the same time, \(R\) is projected with speed \(\frac { 1 } { 2 } u\) away from \(Q\), in the direction \(Q R\). Particle \(P\) has mass \(m\) and particle \(Q\) has mass \(2 m\).
The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Show that the speed of \(Q\) immediately after the collision between \(P\) and \(Q\) is $$\frac { u ( 1 + e ) } { 3 }$$ It is given that \(e > \frac { 1 } { 2 }\)
2. Determine whether there is a collision between \(Q\) and \(R\).
3. Determine the direction of motion of \(P\) immediately after the collision between \(P\) and \(Q\).
4. Find, in terms of \(m , u\) and \(e\), the total kinetic energy lost in the collision between \(P\) and \(Q\), simplifying your answer.
5. Explain how using \(e = 1\) could be used to check your answer to part (d).

1. A particle \(A\) has mass \(2 m\) and a particle \(B\) has mass \(3 m\). The particles are moving in opposite directions along the same straight line and collide directly.

Immediately before the collision, the speed of \(A\) is \(2 u\) and the speed of \(B\) is \(u\). Immediately after the collision, the speed of \(A\) is \(0.5 u\) and the speed of \(B\) is \(w\). Given that the direction of motion of each particle is reversed by the collision,

1. find \(w\) in terms of \(u\)
2. find the coefficient of restitution between the particles,
3. find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(A\) in the collision.

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) and a particle \(Q\) of mass \(4 m\) are at rest on a smooth horizontal plane, as shown in Figure 2. Particle \(P\) is projected with speed \(u\) along the plane towards \(Q\) and the particles collide.
The coefficient of restitution between the particles is \(e\), where \(e > \frac { 1 } { 4 }\) As a result of the collision, the direction of motion of \(P\) is reversed and \(P\) has speed \(\frac { u } { 5 } ( 4 e - 1 )\).

1. Find, in terms of \(u\) and \(e\), the speed of \(Q\) after the collision. After the collision, \(P\) goes on to hit a vertical wall which is fixed at right angles to the direction of motion of \(P\). The coefficient of restitution between \(P\) and the wall is \(f\), where \(f > 0\) Given that \(e = \frac { 3 } { 4 }\)
2. find, in terms of \(m , u\) and \(f\), the kinetic energy lost by \(P\) as a result of its impact with the wall. Give your answer in its simplest form. After its impact with the wall, \(P\) goes on to collide with \(Q\) again.
3. Find the complete range of possible values of \(f\).

A particle P of mass 3 m is moving in a straight line on a smooth horizontal table. A particle \(Q\) of mass \(m\) is moving in the opposite direction to \(P\) along the same straight line. The particles collide directly. Immediately before the collision the speed of P is u and the speed of Q is 2 u . The velocities of P and Q immediately after the collision, measured in the direction of motion of P before the collision, are V and W respectively. The coefficient of restitution between P and Q is e .

1. Find an expression for v in terms of u and e . Given that the direction of motion of P is changed by the collision,
2. find the range of possible values of e.
3. Show that \(\mathrm { w } = \frac { \mathrm { u } } { 4 } ( 1 + 9 \mathrm { e } )\). Following the collision with P , the particle Q then collides with and rebounds from a fixed vertical wall which is perpendicular to the direction of motion of \(Q\). The coefficient of restitution between \(Q\) and the wall is \(f\).
   Given that \(\mathrm { e } = \frac { 5 } { 9 }\), and that P and Q collide again in the subsequent motion,
4. find the range of possible values of f .

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   \section*{Q uestion 4 continued}

1. \begin{figure}[h] \end{figure} Figure 1 represents the plan of part of a smooth horizontal floor, where \(W _ { 1 }\) and \(W _ { 2 }\) are two fixed parallel vertical walls. The walls are 3 metres apart. A particle lies at rest at a point \(O\) on the floor between the two walls, where the point \(O\) is \(d\) metres, \(0 < d \leqslant 3\), from \(W _ { 1 }\) At time \(t = 0\), the particle is projected from \(O\) towards \(W _ { 1 }\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction perpendicular to the walls. The coefficient of restitution between the particle and each wall is \(\frac { 2 } { 3 }\) The particle returns to \(O\) at time \(t = T\) seconds, having bounced off each wall once.

1. Show that \(T = \frac { 45 - 5 d } { 4 u }\) The value of \(u\) is fixed, the particle still hits each wall once but the value of \(d\) can now vary.
2. Find the least possible value of \(T\), giving your answer in terms of \(u\). You must give a reason for your answer.

1. A particle \(P\) of mass \(3 m\) and a particle \(Q\) of mass \(2 m\) are moving along the same straight line on a smooth horizontal plane. The particles are moving in opposite directions towards each other and collide directly.

Immediately before the collision the speed of \(P\) is \(u\) and the speed of \(Q\) is \(2 u\).
Immediately after the collision \(P\) and \(Q\) are moving in opposite directions.
The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Find the range of possible values of \(e\), justifying your answer. Given that \(Q\) loses 75\% of its kinetic energy as a result of the collision,
2. find the value of \(e\).

1. Two particles, \(A\) and \(B\), have masses \(3 m\) and \(4 m\) respectively. The particles are moving in the same direction along the same straight line on a smooth horizontal surface when they collide directly. Immediately before the collision the speed of \(A\) is \(2 u\) and the speed of \(B\) is \(u\).

The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. Show that the direction of motion of each of the particles is unchanged by the collision.
   (8) After the collision with \(A\), particle \(B\) collides directly with a third particle, \(C\), of mass \(2 m\), which is at rest on the surface. The coefficient of restitution between \(B\) and \(C\) is also \(e\).
2. Show that there will be a second collision between \(A\) and \(B\).