6.03i Coefficient of restitution: e

1

1. An object P , with mass 6 kg and speed \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is sliding on a smooth horizontal table. Object P explodes into two small parts, Q and \(\mathrm { R } . \mathrm { Q }\) has mass 4 kg and R has mass 2 kg and speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of motion of P before the explosion. This information is shown in Fig. 1.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f2aaae62-a5f3-47da-afa5-1dd4b37ea2d6-2_346_1267_429_479} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure}
   1. Calculate the velocity of Q . Just as object R reaches the edge of the table, it collides directly with a small object S of mass 3 kg that is travelling horizontally towards R with a speed of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). This information is shown in Fig. 1.2. The coefficient of restitution in this collision is 0.1 . \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{f2aaae62-a5f3-47da-afa5-1dd4b37ea2d6-2_506_647_1215_790} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
      \end{figure}
   2. Calculate the velocities of R and S immediately after the collision. The table is 0.4 m above a horizontal floor. After the collision, R and S have no contact with the table.
   3. Calculate the distance apart of R and S when they reach the floor.
2. A particle of mass \(m \mathrm {~kg}\) bounces off a smooth horizontal plane. The components of velocity of the particle just before the impact are \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) parallel to the plane and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) perpendicular to the plane. The coefficient of restitution is \(e\). Show that the mechanical energy lost in the impact is \(\frac { 1 } { 2 } m v ^ { 2 } \left( 1 - e ^ { 2 } \right) \mathrm { J }\).

6. Particle \(S\) of mass \(2 M\) is moving with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a smooth horizontal plane when it collides directly with a particle \(T\) of mass \(5 M\) which is lying at rest on the plane. The coefficient of restitution between \(S\) and \(T\) is \(\frac { 3 } { 4 }\). Given that the speed of \(T\) after the collision is \(4 \mathrm {~ms} ^ { - 1 }\),

1. find \(U\). As a result of the collision, \(T\) is projected horizontally from the top of a building of height 19.6 m and falls freely under gravity. \(T\) strikes the ground at the point \(X\) as shown in Figure 3. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ef2dd10c-5a3c-4868-af00-aaf7f2937d7e-4_663_928_740_523} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure}
2. Find the time taken for \(T\) to reach \(X\).
3. Show that the angle between the horizontal and the direction of motion of \(T\), just before it strikes the ground at \(X\), is \(78.5 ^ { \circ }\) correct to 3 significant figures.
   (4 marks)

7. Particle \(A\) of mass 7 kg is moving with speed \(u _ { 1 }\) on a smooth horizontal surface when it collides directly with particle \(B\) of mass 4 kg moving in the same direction as \(A\) with speed \(u _ { 2 }\). After the impact, \(A\) continues to move in the same direction but its speed has been halved. Given that the coefficient of restitution between the particles is \(e\),

1. show that \(8 u _ { 2 } ( e + 1 ) = u _ { 1 } ( 8 e - 3 )\). Given also that \(u _ { 1 } = 14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(u _ { 2 } = 3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
2. find \(e\),
3. show that the percentage of the kinetic energy of the particles lost as a result of the impact is \(9.6 \%\), correct to 2 significant figures.

6 Two smooth billiard balls \(A\) and \(B\), of identical size and equal mass, move towards each other on a horizontal surface and collide. Just before the collision, \(A\) has velocity \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at \(30 ^ { \circ }\) to the line of centres of the balls, and \(B\) has velocity \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at \(60 ^ { \circ }\) to the line of centres, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fc5bfc4b-68bb-4a23-874b-87e9558dc990-05_508_1420_532_294} The coefficient of restitution between the balls is \(\frac { 1 } { 2 }\).

1. Find the speed of \(B\) immediately after the collision.
2. Find the angle between the velocity of \(B\) and the line of centres of the balls immediately after the collision.

7 A projectile is fired from a point \(O\) on the slope of a hill which is inclined at an angle \(\alpha\) to the horizontal. The projectile is fired up the hill with velocity \(U\) at an angle \(\theta\) above the hill and first strikes it at a point \(A\). The projectile is modelled as a particle and the hill is modelled as a plane with \(O A\) as a line of greatest slope.

1. 1. Find, in terms of \(U , g , \alpha\) and \(\theta\), the time taken by the projectile to travel from \(O\) to \(A\).
   2. Hence, or otherwise, show that the magnitude of the component of the velocity of the projectile perpendicular to the hill, when it strikes the hill at the point \(A\), is the same as it was initially at \(O\).
2. The projectile rebounds and strikes the hill again at a point \(B\). The hill is smooth and the coefficient of restitution between the projectile and the hill is \(e\). \includegraphics[max width=\textwidth, alt={}, center]{fc5bfc4b-68bb-4a23-874b-87e9558dc990-06_428_1332_1023_338} Find the ratio of the time of flight from \(O\) to \(A\) to the time of flight from \(A\) to \(B\). Give your answer in its simplest form.

2 Three smooth spheres \(A , B\) and \(C\) of equal radii and masses \(m , m\) and \(2 m\) respectively lie at rest on a smooth horizontal table. The centres of the spheres lie in a straight line with \(B\) between \(A\) and \(C\). The coefficient of restitution between any two spheres is \(e\). The sphere \(A\) is projected directly towards \(B\) with speed \(u\) and collides with \(B\).

1. Find, in terms of \(u\) and \(e\), the speed of \(B\) immediately after the impact between \(A\) and \(B\).
2. The sphere \(B\) subsequently collides with \(C\). The speed of \(C\) immediately after this collision is \(\frac { 3 } { 8 } u\). Find the value of \(e\).

6 Two smooth billiard balls \(A\) and \(B\), of identical size and equal mass, move towards each other on a horizontal surface and collide. Just before the collision, \(A\) has velocity \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at \(30 ^ { \circ }\) to the line of centres of the balls, and \(B\) has velocity \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at \(60 ^ { \circ }\) to the line of centres, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{f8c04360-f54b-4d08-aee9-fe28612918d0-4_508_1420_532_294} The coefficient of restitution between the balls is \(\frac { 1 } { 2 }\).

1. Find the speed of \(B\) immediately after the collision.
2. Find the angle between the velocity of \(B\) and the line of centres of the balls immediately after the collision.

7 A projectile is fired from a point \(O\) on the slope of a hill which is inclined at an angle \(\alpha\) to the horizontal. The projectile is fired up the hill with velocity \(U\) at an angle \(\theta\) above the hill and first strikes it at a point \(A\). The projectile is modelled as a particle and the hill is modelled as a plane with \(O A\) as a line of greatest slope.

1. 1. Find, in terms of \(U , g , \alpha\) and \(\theta\), the time taken by the projectile to travel from \(O\) to \(A\).
   2. Hence, or otherwise, show that the magnitude of the component of the velocity of the projectile perpendicular to the hill, when it strikes the hill at the point \(A\), is the same as it was initially at \(O\).
2. The projectile rebounds and strikes the hill again at a point \(B\). The hill is smooth and the coefficient of restitution between the projectile and the hill is \(e\). \includegraphics[max width=\textwidth, alt={}, center]{f8c04360-f54b-4d08-aee9-fe28612918d0-5_428_1332_1023_338} Find the ratio of the time of flight from \(O\) to \(A\) to the time of flight from \(A\) to \(B\). Give your answer in its simplest form.

4 Two small smooth spheres, \(A\) and \(B\), of equal radii have masses 0.3 kg and 0.2 kg respectively. They are moving on a smooth horizontal surface directly towards each other with speeds \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively when they collide. The coefficient of restitution between \(A\) and \(B\) is 0.8 .

1. Find the speeds of \(A\) and \(B\) immediately after the collision.
2. Subsequently, \(B\) collides with a fixed smooth vertical wall which is at right angles to the path of the sphere. The coefficient of restitution between \(B\) and the wall is 0.7 . Show that \(B\) will collide again with \(A\).

6 A smooth spherical ball, \(A\), is moving with speed \(u\) in a straight line on a smooth horizontal table when it hits an identical ball, \(B\), which is at rest on the table. Just before the collision, the direction of motion of \(A\) makes an angle of \(30 ^ { \circ }\) with the line of the centres of the two balls, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{daea0765-041a-4569-a535-f90fe4708313-4_362_1632_621_242} The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. Given that \(\cos 30 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }\), show that the speed of \(B\) immediately after the collision is $$\frac { \sqrt { 3 } } { 4 } u ( 1 + e )$$
2. Find, in terms of \(u\) and \(e\), the components of the velocity of \(A\), parallel and perpendicular to the line of centres, immediately after the collision.
3. Given that \(e = \frac { 2 } { 3 }\), find the angle that the velocity of \(A\) makes with the line of centres immediately after the collision. Give your answer to the nearest degree.
   (3 marks)

6 A small smooth ball of mass \(m\), moving on a smooth horizontal surface, hits a smooth vertical wall and rebounds. The coefficient of restitution between the wall and the ball is \(\frac { 3 } { 4 }\). Immediately before the collision, the ball has velocity \(u\) and the angle between the ball's direction of motion and the wall is \(\alpha\). The ball's direction of motion immediately after the collision is at right angles to its direction of motion before the collision, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{eed9842d-cd89-4eb7-b5ba-9380971be196-4_483_344_657_854}

1. Show that \(\tan \alpha = \frac { 2 } { \sqrt { 3 } }\).
2. Find, in terms of \(u\), the speed of the ball immediately after the collision.
3. The force exerted on the ball by the wall acts for 0.1 seconds. Given that \(m = 0.2 \mathrm {~kg}\) and \(u = 4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the average force exerted by the wall on the ball.

3 Three smooth spheres, \(A , B\) and \(C\), of equal radii have masses \(1 \mathrm {~kg} , 3 \mathrm {~kg}\) and \(x \mathrm {~kg}\) respectively. The spheres lie at rest in a straight line on a smooth horizontal surface with \(B\) between \(A\) and \(C\). The sphere \(A\) is projected with speed \(3 u\) directly towards \(B\) and collides with it. \includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-08_250_835_511_605} The coefficient of restitution between each pair of spheres is \(\frac { 1 } { 3 }\).

1. Show that \(A\) is brought to rest by the impact and find the speed of \(B\) immediately after the collision in terms of \(u\).
2. Subsequently, \(B\) collides with \(C\). Show that the speed of \(C\) immediately after the collision is \(\frac { 4 u } { 3 + x }\).
   Find the speed of \(B\) immediately after the collision in terms of \(u\) and \(x\).
3. Show that \(B\) will collide with \(A\) again if \(x > 9\).
4. Given that \(x = 5\), find the magnitude of the impulse exerted on \(C\) by \(B\) in terms of \(u\).
   \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-09_2484_1709_223_153}
   \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-10_2484_1712_223_153}
   \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-11_2484_1709_223_153}

5 A smooth sphere is moving on a smooth horizontal surface when it strikes a smooth vertical wall and rebounds. Immediately before the impact, the sphere is moving with speed \(4 \mathrm {~ms} ^ { - 1 }\) and the angle between the sphere's direction of motion and the wall is \(\alpha\). Immediately after the impact, the sphere is moving with speed \(v \mathrm {~ms} ^ { - 1 }\) and the angle between the sphere's direction of motion and the wall is \(40 ^ { \circ }\). The coefficient of restitution between the sphere and the wall is \(\frac { 2 } { 3 }\). \includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-14_529_250_831_909}

1. Show that \(\tan \alpha = \frac { 3 } { 2 } \tan 40 ^ { \circ }\).
2. Find the value of \(v\).
   \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-15_2484_1709_223_153}

6 Two smooth spheres, \(A\) and \(B\), have equal radii and masses 1 kg and 2 kg respectively. The sphere \(A\) is moving with velocity \(( 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and the sphere \(B\) is moving with velocity \(( - \mathbf { i } - 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) on the same smooth horizontal surface. The spheres collide when their line of centres is parallel to the unit vector \(\mathbf { i }\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-16_456_1052_721_550}

1. Briefly state why the components of the velocities of \(A\) and \(B\) parallel to the unit vector \(\mathbf { j }\) are not changed by the collision.
2. The coefficient of restitution between the spheres is 0.5 . Find the velocities of \(A\) and \(B\) immediately after the collision. \includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-17_2484_1709_223_153} \(7 \quad\) A ball is projected from a point \(O\) on a smooth plane which is inclined at an angle of \(35 ^ { \circ }\) above the horizontal. The ball is projected with velocity \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the plane, as shown in the diagram. The motion of the ball is in a vertical plane containing a line of greatest slope of the inclined plane. The ball strikes the inclined plane at the point \(A\). \includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-18_321_838_605_577}
   1. Find the components of the velocity of the ball, parallel and perpendicular to the plane, as it strikes the inclined plane at \(A\).
   2. On striking the plane at \(A\), the ball rebounds. The coefficient of restitution between the plane and the ball is \(\frac { 4 } { 5 }\). Show that the ball next strikes the plane at a point lower down than \(A\).
      \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-19_2484_1709_223_153}

5 A ball is dropped from a height of 2.5 m above a horizontal floor. The ball bounces repeatedly on the floor.

1. Find the speed of the ball when it first hits the floor.
2. The coefficient of restitution between the ball and the floor is \(e\).
   1. Show that the time taken between the first contact of the ball with the floor and the second contact of the ball with the floor is \(\frac { 10 e } { 7 }\) seconds.
   2. Find, in terms of \(e\), the time taken between the second contact and the third contact of the ball with the floor.
3. Find, in terms of \(e\), the total vertical distance travelled by the ball from when it is dropped until its third contact with the floor.
4. State a modelling assumption for answering this question, other than the ball being a particle.

4 \includegraphics[max width=\textwidth, alt={}, center]{af1f9f1b-f6c0-4044-9864-5b9ce309d3fa-02_283_711_1754_722} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 5 kg and 2 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, \(A\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving perpendicular to the line of centres, and \(B\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres (see diagram). The coefficient of restitution is 0.75 . Find the speed and direction of motion of each sphere immediately after the collision.

3 \includegraphics[max width=\textwidth, alt={}, center]{3243c326-a51c-462f-a57c-a150d0044ea9-2_403_951_1247_559} Two uniform smooth spheres \(A\) and \(B\) of equal radius are moving on a horizontal surface when they collide. \(A\) has mass 0.1 kg and \(B\) has mass 0.4 kg . Immediately before the collision \(A\) is moving with speed \(2.8 \mathrm {~ms} ^ { - 1 }\) along the line of centres, and \(B\) is moving with speed \(1 \mathrm {~ms} ^ { - 1 }\) at an angle \(\theta\) to the line of centres, where \(\cos \theta = 0.8\) (see diagram). Immediately after the collision \(A\) is stationary. Find

1. the coefficient of restitution between \(A\) and \(B\),
2. the angle turned through by the direction of motion of \(B\) as a result of the collision. \section*{\(\mathrm { OCR } ^ { \text {勾 } }\)}

6 \includegraphics[max width=\textwidth, alt={}, center]{3243c326-a51c-462f-a57c-a150d0044ea9-4_547_515_267_772} A hollow cylinder is fixed with its axis horizontal. \(O\) is the centre of a vertical cross-section of the cylinder and \(D\) is the highest point on the cross-section. \(A\) and \(C\) are points on the circumference of the cross-section such that \(A O\) and \(C O\) are both inclined at an angle of \(30 ^ { \circ }\) below the horizontal diameter through \(O\). The inner surface of the cylinder is smooth and has radius 0.8 m (see diagram). A particle \(P\), of mass \(m \mathrm {~kg}\), and a particle \(Q\), of mass \(5 m \mathrm {~kg}\), are simultaneously released from rest from \(A\) and \(C\), respectively, inside the cylinder. \(P\) and \(Q\) collide; the coefficient of restitution between them is 0.95 .

1. Show that, immediately after the collision, \(P\) moves with speed \(6.3 \mathrm {~ms} ^ { - 1 }\), and find the speed and direction of motion of \(Q\).
2. Find, in terms of \(m\), an expression for the normal reaction acting on \(P\) when it subsequently passes through \(D\).

2 \includegraphics[max width=\textwidth, alt={}, center]{bfa6d51d-0992-4f43-adab-77ce893c1ca9-2_296_798_461_694} A sphere \(A\) of mass \(m\), moving on a horizontal surface, collides with another sphere \(B\) of mass \(2 m\), which is at rest on the surface. The spheres are smooth and uniform, and have equal radius. Immediately before the collision, \(A\) has velocity \(u\) at an angle \(\theta ^ { \circ }\) to the line of centres of the spheres (see diagram). Immediately after the collision, the spheres move in directions that are perpendicular to each other.

1. Find the coefficient of restitution between the spheres.
2. Given that the spheres have equal speeds after the collision, find \(\theta\).

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}

5 Fig. 5.1 shows a small smooth sphere A at rest on a smooth horizontal surface. At both ends of the surface is a smooth vertical wall. \begin{figure}[h] \end{figure} Sphere A is projected directly towards the left-hand wall at a speed of \(5 \mathrm {~ms} ^ { - 1 }\). Sphere A collides directly with the left-hand wall, rebounds, then collides directly with the right-hand wall. After this second collision A has a speed of \(3.2 \mathrm {~ms} ^ { - 1 }\).

1. Explain how it can be deduced that the collision between A and the left-hand wall was not inelastic. The coefficient of restitution between A and each wall is \(e\).
2. Calculate the value of \(e\). Sphere A is now brought to rest and a second identical sphere B is placed on the surface. The surface is 1 m long, and A and B are positioned so that they are both 0.5 m from each wall, as shown in Fig. 5.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-6_241_1307_1322_242} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure} Sphere A is projected directly towards the left-hand wall at a speed of \(0.2 \mathrm {~ms} ^ { - 1 }\). At the same time, B is projected directly towards the right-hand wall at a speed of \(0.3 \mathrm {~ms} ^ { - 1 }\). You may assume that the duration of impact of a sphere and a wall is negligible.
3. Calculate the distance of A and B from the left-hand wall when they meet again.

2 A ball P of mass \(m \mathrm {~kg}\) is held at a height of 12.8 m above a horizontal floor. P is released from rest and rebounds from the floor. After the first bounce, P reaches a maximum height of 5 m above the floor. Two models, A and B , are suggested for the motion of P .
Model A assumes that air resistance may be neglected.

1. Determine, according to model A , the coefficient of restitution between P and the floor. Model B assumes that the collision between P and the floor is perfectly elastic, but that work is done against air resistance at a constant rate of \(E\) joules per metre.
2. Show that, according to model \(\mathrm { B } , \mathrm { E } = \frac { 39 } { 89 } \mathrm { mg }\).
3. Show that both models predict that P will attain the same maximum height after the second bounce.

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.

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.