Momentum and Collisions 1

5 Two uniform small smooth spheres \(A\) and \(B\), of equal radii, have masses \(2 m\) and \(m\) respectively. They lie at rest on a smooth horizontal plane. Sphere \(A\) is projected directly towards \(B\) with speed \(u\). After the collision \(B\) goes on to collide directly with a fixed smooth vertical barrier, before colliding with \(A\) again. The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\) and the coefficient of restitution between \(B\) and the barrier is \(e\). After the second collision between \(A\) and \(B\), the speed of \(B\) is five times the speed of \(A\). Find the two possible values of \(e\).

4. A particle \(A\) of mass \(2 m\) is moving with speed \(3 u\) in a straight line on a smooth horizontal table. The particle collides directly with a particle \(B\) of mass \(m\) moving with speed \(2 u\) in the opposite direction to \(A\). Immediately after the collision the speed of \(B\) is \(\frac { 8 } { 3 } u\) and the direction of motion of \(B\) is reversed.

1. Calculate the coefficient of restitution between \(A\) and \(B\).
2. Show that the kinetic energy lost in the collision is \(7 m u ^ { 2 }\). After the collision \(B\) strikes a fixed vertical wall that is perpendicular to the direction of motion of \(B\). The magnitude of the impulse of the wall on \(B\) is \(\frac { 14 } { 3 } m u\).
3. Calculate the coefficient of restitution between \(B\) and the wall.
   (4)

5 Two uniform small smooth spheres \(A\) and \(B\), of equal radii, have masses \(2 m\) and \(m\) respectively. They lie at rest on a smooth horizontal plane. Sphere \(A\) is projected directly towards \(B\) with speed \(u\). After the collision \(B\) goes on to collide directly with a fixed smooth vertical barrier, before colliding with \(A\) again. The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\) and the coefficient of restitution between \(B\) and the barrier is \(e\). After the second collision between \(A\) and \(B\), the speed of \(B\) is five times the speed of \(A\). Find the two possible values of \(e\).

3 Three particles, \(A , B\) and \(C\), of masses \(2 \mathrm {~kg} , 3 \mathrm {~kg}\) and 5 kg respectively, are at rest in a straight line on a smooth horizontal plane with \(B\) between \(A\) and \(C\). Collisions between \(A\) and \(B\) are perfectly elastic. The coefficient of restitution for collisions between \(B\) and \(C\) is \(e\).
\(A\) is projected towards \(B\) with a speed of \(5 u \mathrm {~ms} ^ { - 1 }\) (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{493f11f4-e25c-4eeb-a0ab-20ec6d7a7a7d-2_186_903_2330_251} Show that only two collisions occur.

7. Three particles \(A , B\) and \(C\), each of mass \(m\), lie at rest in a straight line \(L\) on a smooth horizontal surface, with \(B\) between \(A\) and \(C\). Particles \(A\) and \(B\) are projected directly towards each other with speeds \(5 u\) and \(4 u\) respectively. Particle \(C\) is projected directly away from \(B\) with speed \(3 u\). In the subsequent motion, \(A , B\) and \(C\) move along \(L\). Particles \(A\) and \(B\) collide directly. The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. Find (i) the speed of \(A\) immediately after the collision,
   (ii) the speed of \(B\) immediately after the collision. Given that the direction of motion of \(A\) is reversed in the collision between \(A\) and \(B\), and that there is no collision between \(B\) and \(C\),
2. find the set of possible values of \(e\).

3 Three small smooth spheres \(A , B\) and \(C\) have equal radii and have masses \(m , 9 m\) and \(k m\) respectively. They are at rest on a smooth horizontal table and lie in a straight line with \(B\) between \(A\) and \(C\). The coefficient of restitution between any pair of the spheres is \(e\). Sphere \(A\) is projected directly towards \(B\) with speed \(u\). Given that half of the total kinetic energy is lost as result of the collision between \(A\) and \(B\), find the value of \(e\). After \(B\) and \(C\) collide they move in the same direction and the speed of \(C\) is twice the speed of \(B\). Find the value of \(k\).

2 Two particles are moving directly towards each other when they collide.
Given that the collision is perfectly elastic, state the value of the coefficient of restitution. Circle your answer.
\(e = - 1\)
\(e = 0\)
\(e = \frac { 1 } { 2 }\)
\(e = 1\)

2 Two particles are moving directly towards each other when they collide.
Given that the collision is perfectly elastic, state the value of the coefficient of restitution. Circle your answer.
\(e = - 1\)
\(e = 0\)
\(e = \frac { 1 } { 2 }\)
\(e = 1\)

8 A point \(O\) is situated a distance \(h\) above a smooth horizontal plane, and a particle \(A\) of mass \(m\) is attached to \(O\) by a light inextensible string of length \(h\). A particle \(B\) of mass \(2 m\) is at rest on the plane, directly below \(O\), and is attached to a point \(C\) on the plane, where \(B C = l\), by a light inextensible string of length \(l . A\) is released from rest with the string \(O A\) taut and making an acute angle \(\theta\) with the downward vertical (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{19c3a9d0-15b6-4dd0-a00b-577c3fd2cf52-5_604_1137_486_552}
\(A\) moves in a vertical plane perpendicular to \(C B\) and collides directly with \(B\). As a result of this collision, \(A\) is brought to rest and \(B\) moves on the plane in a horizontal circle with centre \(C\). After \(B\) has made one complete revolution the particles collide again.

1. Show that, on the next occasion that \(A\) comes to rest, the string \(O A\) makes an angle \(\phi\) with the downward vertical through \(O\), where \(\cos \phi = \frac { 3 + \cos \theta } { 4 }\).
   \(A\) and \(B\) collide again when \(A O\) is next vertical.
2. Find the percentage of the original energy of the system that remains immediately after this collision.
3. Explain why the total momentum of the particles immediately before the first collision is the same as the total momentum of the particles immediately after the second collision.
4. Explain why the total momentum of the particles immediately before the first collision is different from the total momentum of the particles immediately after the third collision. \section*{OCR} \section*{Oxford Cambridge and RSA}

1 Two smooth spheres \(A\) and \(B\), of equal radii and of masses \(3 m\) and \(6 m\) respectively, are at rest on a smooth horizontal surface. Sphere \(A\) is projected directly towards \(B\) with speed \(u\). The coefficient of restitution between \(A\) and \(B\) is \(e\). Show that the kinetic energy lost in the collision between \(A\) and \(B\) is \(m u ^ { 2 } \left( 1 - e ^ { 2 } \right)\).

1 Two stones, A and B , are sliding along the same straight line on a horizontal sheet of ice. Stone A, of mass 50 kg , is moving with a constant velocity of \(2.1 \mathrm {~ms} ^ { - 1 }\) towards stone B. Stone B, of mass 70 kg , is moving with a constant velocity of \(0.8 \mathrm {~ms} ^ { - 1 }\) towards stone A. A and B collide directly. Immediately after their collision stone A's velocity is \(0.35 \mathrm {~ms} ^ { - 1 }\) in the same direction as its velocity before the collision.

1. Find the speed of stone B immediately after the collision.
2. Find the coefficient of restitution for the collision.
3. Find the total loss of kinetic energy caused by the collision.
4. Explain whether the collision was perfectly elastic.

3 Two uniform small smooth spheres \(A\) and \(B\), of masses \(m\) and \(2 m\) respectively, and with equal radii, are at rest on a smooth horizontal surface. Sphere \(A\) is projected directly towards \(B\) with speed \(u\), and collides with \(B\). After this collision, sphere \(B\) collides directly with a fixed smooth vertical barrier. The total kinetic energy of the spheres after this second collision is equal to one-ninth of its value before the first collision. Given that the coefficient of restitution between \(B\) and the barrier is 0.5 , find the coefficient of restitution between \(A\) and \(B\).

2 Two particles, of masses \(3 m\) and \(m\), are moving in the same straight line towards each other with speeds \(2 u\) and \(u\) respectively. When they collide, the impulse acting on each particle has magnitude \(4 m u\). Show that the total loss in kinetic energy is \(\frac { 4 } { 3 } m u ^ { 2 }\).

1

1. Disc A of mass 6 kg and disc B of mass 0.5 kg are moving in the same straight line. The relative positions of the discs and the \(\mathbf { i }\) direction are shown in Fig. 1.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-2_282_1325_402_450} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure} The discs collide directly. The impulse on A in the collision is \(- 12 \mathbf { i }\) Ns and after the collision A has velocity \(3 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and B has velocity \(11 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   1. Show that the velocity of A just before the collision is \(5 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and find the velocity of B at this time.
   2. Calculate the coefficient of restitution in the collision.
   3. After the collision, a force of \(- 2 \mathbf { i } \mathrm {~N}\) acts on B for 7 seconds. Find the velocity of B after this time.
2. A ball bounces off a smooth plane. The angles its path makes with the plane before and after the impact are \(\alpha\) and \(\beta\), as shown in Fig. 1.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-2_317_1082_1468_575} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{figure} The velocity of the ball before the impact is \(u \mathbf { i } - v \mathbf { j }\) and the coefficient of restitution in the impact is \(e\). Write down an expression in terms of \(u , v , e , \mathbf { i }\) and \(\mathbf { j }\) for the velocity of the ball immediately after the impact. Hence show that \(\tan \beta = e \tan \alpha\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-3_581_486_274_383} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-3_593_392_264_1370} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure} A uniform wire is bent to form a bracket OABCD . The sections \(\mathrm { OA } , \mathrm { AB }\) and BC lie on three sides of a square and CD is parallel to AB . This is shown in Fig. 2.1 where the dimensions, in centimetres, are also given.
   1. Show that, referred to the axes shown in Fig. 2.1, the \(x\)-coordinate of the centre of mass of the bracket is 3.6 . Find also the \(y\)-coordinate of its centre of mass.
   2. The bracket is now freely suspended from D and hangs in equilibrium. Draw a diagram showing the position of the centre of mass and calculate the angle of CD to the vertical.
   3. The bracket is now hung by means of vertical, light strings BP and DQ attached to B and to D , as shown in Fig. 2.2. The bracket has weight 5 N and is in equilibrium with OA horizontal. Calculate the tensions in the strings BP and DQ . The original bracket shown in Fig. 2.1 is now changed by adding the section OE, where AOE is a straight line. This section is made of the same type of wire and has length \(L \mathrm {~cm}\), as shown in Fig. 2.3.
      \(\begin{array} { l l l l } \begin{array} { l } \text { not to } \\ \text { scale } \end{array} & 2 & 6 & \\ \mathrm {~L} \longrightarrow & \mathrm {~L} & & \\ \mathrm {~L} & \mathrm { O } & 6 & \mathrm {~A} \end{array}\) Fig. 2.3 The value of \(L\) is chosen so that the centre of mass is now on the \(y\)-axis.
   4. Calculate \(L\).

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 \(e\).

2. A particle \(A\) of mass \(4 m\) is moving with speed \(3 u\) in a straight line on a smooth horizontal table. The particle \(A\) collides directly with a particle \(B\) of mass \(3 m\) moving with speed \(2 u\) in the same direction as \(A\). The coefficient of restitution between \(A\) and \(B\) is \(e\). Immediately after the collision the speed of \(B\) is \(4 e u\).

1. Show that \(e = \frac { 3 } { 4 }\).
2. Find the total kinetic energy lost in the collision.

2. A particle \(P\) of mass 0.7 kg is moving in a straight line on a smooth horizontal surface. The particle \(P\) collides with a particle \(Q\) of mass 1.2 kg which is at rest on the surface. Immediately before the collision the speed of \(P\) is \(6 \mathrm {~ms} ^ { - 1 }\). Immediately after the collision both particles are moving in the same direction. The coefficient of restitution between the particles is \(e\).

1. Show that \(e < \frac { 7 } { 12 }\) Given that \(e = \frac { 1 } { 4 }\)
2. find the magnitude of the impulse exerted on \(Q\) in the collision.
   

6
\includegraphics[max width=\textwidth, alt={}, center]{74eaa61a-1507-4cef-8f97-5c1860bdc36a-4_243_1179_1580_443} The masses of two particles \(A\) and \(B\) are 0.2 kg and \(m \mathrm {~kg}\) respectively. The particles are moving with constant speeds \(4 \mathrm {~ms} ^ { - 1 }\) and \(u \mathrm {~ms} ^ { - 1 }\) in the same horizontal line and in the same direction (see diagram). The two particles collide and the coefficient of restitution between the particles is \(e\). After the collision, \(A\) and \(B\) continue in the same direction with speeds \(4 \left( 1 - e + e ^ { 2 } \right) \mathrm { ms } ^ { - 1 }\) and \(4 \mathrm {~ms} ^ { - 1 }\) respectively.

1. Find \(u\) and \(m\) in terms of \(e\).
2. Find the value of \(e\) for which the speed of \(A\) after the collision is least and find, in this case, the total loss in kinetic energy due to the collision.
3. Find the possible values of \(e\) for which the magnitude of the impulse that \(B\) exerts on \(A\) is 0.192 Ns .
   \includegraphics[max width=\textwidth, alt={}, center]{74eaa61a-1507-4cef-8f97-5c1860bdc36a-5_744_887_264_589} The diagram shows a surface consisting of a horizontal part \(O A\) and a plane \(A B\) inclined at an angle of \(70 ^ { \circ }\) to the horizontal. A particle is projected from the point \(O\) with speed \(u \mathrm {~ms} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal \(O A\). The particle hits the plane \(A B\) at the point \(P\), with speed \(14 \mathrm {~ms} ^ { - 1 }\) and at right angles to the plane, 1.4 s after projection.
4. Show that the value of \(u\) is 15.9 , correct to 3 significant figures, and find the value of \(\theta\).
5. Find the height of \(P\) above the level of \(A\). The particle rebounds with speed \(v \mathrm {~ms} ^ { - 1 }\). The particle next lands at \(A\).
6. Find the value of \(v\).
7. Find the coefficient of restitution between the particle and the plane at \(P\).

2 Two smooth spheres \(A\) and \(B\), of equal radius and of masses 0.2 kg and 0.1 kg respectively, are free to move on a smooth horizontal table. \(A\) is moving with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it collides directly with \(B\), which is stationary. The collision is perfectly elastic. Calculate the speed of \(A\) after the impact. [4]

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]
\(\left[ \begin{array} { l } 4 \\ 1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\)
\(\left[ \begin{array} { l } 4 \\ 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\)
\(\left[ \begin{array} { l } 8 \\ 2 \end{array} \right] \mathrm { ms } ^ { - 1 }\)
\(\left[ \begin{array} { l } 8 \\ 6 \end{array} \right] \mathrm { ms } ^ { - 1 }\)

1 A sledge and a child sitting on it have a combined mass of 29.5 kg . The sledge slides on horizontal ice with negligible resistance to its movement.

1. While at rest, the sledge is hit directly from behind by a ball of mass 0.5 kg travelling horizontally at \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution in the collision is 0.8 . After the impact the speeds of the sledge and the ball are \(V _ { 1 } \mathrm {~ms} ^ { - 1 }\) and \(V _ { 2 } \mathrm {~ms} ^ { - 1 }\) respectively. Calculate \(V _ { 1 }\) and \(V _ { 2 }\) and state the direction in which the ball is travelling after the impact. [7]
2. While at rest, the sledge is hit directly from behind by a snowball of mass 0.5 kg travelling horizontally at \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The snowball sticks to the sledge.
   (A) Calculate the velocity with which the combined sledge and snowball start to move.
   (B) The child scoops up the 0.5 kg of snow and drops it over the back of the sledge. What happens to the velocity of the sledge? Give a reason for your answer.
3. In another situation, the sledge is travelling over the ice at \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) with 10.5 kg of snow on it (giving a total mass of 40 kg ). The child throws a snowball of mass 0.5 kg from the sledge, parallel to the ground and in the positive direction of the motion of the sledge. Immediately after the snowball is thrown, the sledge has a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the snowball and sledge are separating at a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Draw a diagram showing the velocities of the sledge and snowball before and after the snowball is thrown. Calculate \(V\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{26d30179-589e-462a-a38f-f9e4e5dec4f4-3_527_720_335_667} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} Fig. 2 shows a framework in a vertical plane made from the equal, light, rigid rods \(\mathrm { AB } , \mathrm { BC } , \mathrm { AD }\), \(\mathrm { BD } , \mathrm { BE } , \mathrm { CE }\) and DE . [The triangles \(\mathrm { ABD } , \mathrm { BDE }\) and BCE are all equilateral.] The rods \(\mathrm { AB } , \mathrm { BC }\) and DE are horizontal.
   The rods are freely pin-jointed to each other at A, B, C, D and E.
   The pin-joint at A is also fixed to an inclined plane. The plane is smooth and parallel to the rod AD . The pin-joint at D rests on this plane. The following external forces act on the framework: a vertical load of \(L \mathrm {~N}\) at C ; the normal reaction force \(R \mathrm {~N}\) of the plane on the framework at D ; the horizontal and vertical forces \(X \mathrm {~N}\) and \(Y \mathrm {~N}\), respectively, acting at A .
4. Write down equations for the horizontal and vertical equilibrium of the framework.
5. By considering moments, find the relationship between \(R\) and \(L\). Hence show that \(X = \sqrt { 3 } L\) and \(Y = 0\).
6. Draw a diagram showing all the forces acting on the pin-joints, including the forces internal to the rods.
7. Show that the internal force in the rod AD is zero.
8. Find the forces internal to \(\mathrm { AB } , \mathrm { CE }\) and BC in terms of \(L\) and state whether each is a tension or a thrust (compression). [You may leave your answers in surd form.]
9. Without calculating their values in terms of \(L\), show that the forces internal to the rods BD and BE have equal magnitude but one is a tension and the other a thrust.

2 A small sphere of mass 0.3 kg is dropped from rest at a height of 2 m above horizontal ground. It falls vertically, hits the ground and rebounds vertically upwards, coming to instantaneous rest at a height of 1.4 m above the ground. Ignoring air resistance, calculate the magnitude of the impulse which the ground exerts on the sphere when it rebounds.

2 A small sphere of mass 0.2 kg is dropped from rest at a height of 3 m above horizontal ground. It falls vertically, hits the ground and rebounds vertically upwards, coming to instantaneous rest at a height of 1.8 m above the ground.

1. Calculate the magnitude of the impulse which the ground exerts on the sphere.
2. Calculate the coefficient of restitution between the sphere and the ground.

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.

1. A smooth sphere is moving with speed \(U\) in a straight line on a smooth horizontal plane. It strikes a fixed smooth vertical wall at right angles. The coefficient of restitution between the sphere and the wall is \(\frac { 1 } { 2 }\).

Find the fraction of the kinetic energy of the sphere that is lost as a result of the impact.
(5 marks)

1. A smooth sphere is moving with speed \(U\) in a straight line on a smooth horizontal plane. It strikes a fixed smooth vertical wall at right angles. The coefficient of restitution between the sphere and the wall is \(\frac { 1 } { 2 }\).

Find the fraction of the kinetic energy of the sphere that is lost as a result of the impact.
(5 marks)

1. A small ball, of mass \(m\), is thrown vertically upwards with speed \(\sqrt { 8 g H }\) from a point \(O\) on a smooth horizontal floor. The ball moves towards a smooth horizontal ceiling that is a vertical distance \(H\) above \(O\). The coefficient of restitution between the ball and the ceiling is \(\frac { 1 } { 2 }\)
   In a model of the motion of the ball, it is assumed that the ball, as it moves up or down, is subject to air resistance of constant magnitude \(\frac { 1 } { 2 } \mathrm { mg }\).
   Using this model,
   1. use the work-energy principle to find, in terms of \(g\) and \(H\), the speed of the ball immediately before it strikes the ceiling,
   2. find, in terms of \(g\) and \(H\), the speed of the ball immediately before it strikes the floor at \(O\) for the first time.
   In a simplified model of the motion of the ball, it is assumed that the ball, as it moves up or down, is subject to no air resistance. Using this simplified model,
2. explain, without any detailed calculation, why the speed of the ball, immediately before it strikes the floor at \(O\) for the first time, would still be less than \(\sqrt { 8 g H }\)

2 A particle of mass 5 kg is moving with velocity \(2 \mathbf { i } + 5 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It receives an impulse of magnitude 15 Ns in the direction \(\mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k }\). Find the velocity of the particle immediately afterwards.

3 Two smooth spheres \(A\) and \(B\), of equal radius, are moving in the same direction in the same straight line on a smooth horizontal table. Sphere \(A\) has mass \(m\) and speed \(u\) and sphere \(B\) has mass \(\alpha m\) and speed \(\frac { 1 } { 4 } u\). The spheres collide and \(A\) is brought to rest by the collision. Find the coefficient of restitution in terms of \(\alpha\). Deduce that \(\alpha \geqslant 2\).

2. Two small smooth spheres \(P\) and \(Q\) are moving along a straight line in opposite directions, with equal speeds, and collide directly. Immediately after the impact, the direction of \(P\) 's motion has been reversed and its speed has been halved. The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Express the speed of \(Q\) after the impact in the form \(a u ( b e + c )\), where \(a , b\) and \(c\) are constants to be found.
2. Deduce the range of values of \(e\) for which the direction of motion of \(Q\) remains unaltered.

5. A particle of mass \(m \mathrm {~kg}\) lies on a smooth horizontal surface. Initially the particle is at rest at a point \(O\) midway between a pair of fixed parallel vertical walls. The walls are 2 m apart. At time \(t = 0\) the particle is projected from \(O\) 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 magnitude of the impulse on the particle due to the first impact with a wall is \(\lambda m u \mathrm {~N} \mathrm {~s}\).

1. Find the value of \(\lambda\). The particle returns to \(O\), having bounced off each wall once, at time \(t = 3\) seconds.
2. Find the value of \(u\).

5
\includegraphics[max width=\textwidth, alt={}, center]{5cc14ffc-e957-4582-b9d0-182fd89b3df5-08_561_1068_255_500} Two uniform smooth spheres \(A\) and \(B\) of equal radii each have mass \(m\). The two spheres are each moving with speed \(u\) on a horizontal surface when they collide. Immediately before the collision A's direction of motion makes an angle of \(\alpha ^ { \circ }\) with the line of centres, and \(B\) 's direction of motion is perpendicular to that of \(A\) (see diagram). The coefficient of restitution between the spheres is \(e\). Immediately after the collision, \(B\) moves in a direction at right angles to the line of centres.

1. Show that \(\tan \alpha = \frac { 1 + e } { 1 - e }\).
2. Given that \(\tan \alpha = 2\), find the speed of \(A\) after the collision.

2 The graph shows how a force, \(F\), varies with time during a period of 0.8 seconds.
\includegraphics[max width=\textwidth, alt={}, center]{18522f4c-4aa2-4ef5-898f-5ad2b06e287c-03_440_960_568_516} Find the magnitude of the impulse of \(F\) during the 0.8 seconds.
Circle your answer.
[0pt] [1 mark]
1.0 Ns
1.6 Ns
2.2 Ns
3.2 Ns Turn over for the next question

1 Two smooth spheres \(A\) and \(B\), of equal radii and masses \(2 m\) and \(m\) respectively, lie at rest on a smooth horizontal table. The spheres \(A\) and \(B\) are projected directly towards each other with speeds \(4 u\) and \(3 u\) respectively. The coefficient of restitution between the spheres is \(e\). Find the set of values of \(e\) for which the direction of motion of \(A\) is reversed in the collision.

8 Two uniform smooth spheres, \(A\) and \(B\), have the same radius. The mass of \(A\) is 2 kg and the mass of \(B\) is \(m \mathrm {~kg}\). Sphere \(A\) is travelling in a straight line on a smooth horizontal surface, with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when it collides directly with sphere \(B\), which is at rest. As a result of the collision, sphere \(A\) continues in the same direction with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the greatest possible value of \(m\). It is given that \(m = 1\).
2. Find the coefficient of restitution between \(A\) and \(B\). On another occasion \(A\) and \(B\) are travelling towards each other, each with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when they collide directly.
3. Find the kinetic energy lost due to the collision.

3. A smooth uniform sphere \(P\) of mass \(m\) is falling vertically and strikes a fixed smooth inclined plane with speed \(u\). The plane is inclined at an angle \(\theta , \theta < 45 ^ { \circ }\), to the horizontal. The coefficient of restitution between \(P\) and the inclined plane is \(e\). Immediately after \(P\) strikes the plane, \(P\) moves horizontally.

1. Show that \(e = \tan ^ { 2 } \theta\).
2. Show that the magnitude of the impulse exerted by \(P\) on the plane is \(m u \sec \theta\).