Sphere rebounds off fixed wall obliquely

5 Two spheres \(A\) and \(B\), of equal radius, have masses \(m _ { 1 }\) and \(m _ { 2 }\) respectively. They lie at rest on a smooth horizontal plane. Sphere \(A\) is projected directly towards sphere \(B\) with speed \(u\) and, as a result of the collision, \(A\) is brought to rest. Show that

1. the speed of \(B\) immediately after the collision cannot exceed \(u\),
2. \(m _ { 1 } \leqslant m _ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{15ed1dfc-8188-4e20-9c0b-ce31af35f0b6-3_273_611_1745_767} After the collision, \(B\) hits a smooth vertical wall which is at an angle of \(60 ^ { \circ }\) to the direction of motion of \(B\) (see diagram). In the impact with the wall \(B\) loses \(\frac { 2 } { 3 }\) of its kinetic energy. Find the coefficient of restitution between \(B\) and the wall and show that the direction of motion of \(B\) turns through \(90 ^ { \circ }\).

2
A uniform sphere \(P\) of mass \(m\) is at rest on a smooth horizontal table. The sphere is projected along the table with speed \(u\) and strikes a smooth vertical barrier \(A\) at an acute angle \(\alpha\). It then strikes another smooth vertical barrier \(B\) which is at right angles to \(A\) (see diagram). The coefficient of restitution between \(P\) and each of the barriers is \(e\). Show that the final direction of motion of \(P\) makes an angle \(\frac { 1 } { 2 } \pi - \alpha\) with the barrier \(B\) and find the total loss in kinetic energy as a result of the two impacts. [7]

2 \includegraphics[max width=\textwidth, alt={}, center]{2c6b6722-ebba-4ade-9a9d-dd70e61cf52b-2_312_409_525_868} A small smooth ball \(P\) is moving on a smooth horizontal plane with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It strikes a smooth vertical barrier at an angle \(\alpha\) (see diagram). The coefficient of restitution between \(P\) and the barrier is 0.4 . Given that the speed of \(P\) is halved as a result of the collision, find the value of \(\alpha\).

2 \includegraphics[max width=\textwidth, alt={}, center]{5d40f5b4-e3d4-482c-8d8d-05a01bd3b43f-2_312_409_525_868} A small smooth ball \(P\) is moving on a smooth horizontal plane with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It strikes a smooth vertical barrier at an angle \(\alpha\) (see diagram). The coefficient of restitution between \(P\) and the barrier is 0.4 . Given that the speed of \(P\) is halved as a result of the collision, find the value of \(\alpha\).

1 A particle \(P\) of mass 12.5 kg is moving on a smooth horizontal plane when it collides obliquely with a fixed vertical wall. At the instant before the collision, the velocity of \(P\) is \(- 5 \mathbf { i } + 12 \mathbf { j } \mathrm {~ms} ^ { - 1 }\).
At the instant after the collision, the velocity of \(P\) is \(\mathbf { i } + 4 \mathbf { j } \mathrm {~ms} ^ { - 1 }\).

1. Find the magnitude of the momentum of \(P\) before the collision.
2. Find, in vector form, the impulse that the wall exerts on \(P\).
3. State, in vector form, the impulse that \(P\) exerts on the wall.
4. Find in either order.

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.

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}

4 The diagram shows part of a horizontal snooker table of width 1.69 m . A player strikes the ball \(B\) directly, and it moves in a straight line. The ball hits the cushion of the table at \(C\) before rebounding and moving to the pocket at \(P\) at the corner of the table, as shown in the diagram. The point \(C\) is 1.20 m from the corner \(A\) of the table. The ball has mass 0.15 kg and, immediately before the collision with the cushion, it has velocity \(u\) in a direction inclined at \(60 ^ { \circ }\) to the cushion. The table and the cushion are modelled as smooth. \includegraphics[max width=\textwidth, alt={}, center]{a90a2de3-5cc0-4e87-b29a-2562f86eee17-08_517_963_719_511}

1. Find the coefficient of restitution between the ball and the cushion.
2. Show that the magnitude of the impulse on the cushion at \(C\) is approximately \(0.236 u\).
3. Find, in terms of \(u\), the time taken between the ball hitting the cushion at \(C\) and entering the pocket at \(P\).
4. Explain how you have used the assumption that the cushion is smooth in your answers.

1 A particle \(P\) of mass 0.05 kg is moving on a smooth horizontal surface with speed \(2 \mathrm {~ms} ^ { - 1 }\), when it is struck by a horizontal blow in a direction perpendicular to its direction of motion. The magnitude of the impulse of the blow is \(I\) Ns. The speed of \(P\) after the blow is \(2.5 \mathrm {~ms} ^ { - 1 }\).

1. Find the value of \(I\). Immediately before the blow \(P\) is moving parallel to a smooth vertical wall. After the blow \(P\) hits the wall and rebounds from the wall with speed \(\sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the coefficient of restitution between \(P\) and the wall.

4. \begin{figure}[h] \end{figure} A small smooth ball \(B\), moving on a horizontal plane, collides with a fixed vertical wall. Immediately before the collision the angle between the direction of motion of \(B\) and the wall is \(2 \theta\), where \(0 ^ { \circ } < \theta < 45 ^ { \circ }\). Immediately after the collision the angle between the direction of motion of \(B\) and the wall is \(\theta\), as shown in Figure 1. Given that the coefficient of restitution between \(B\) and the wall is \(\frac { 3 } { 8 }\), find the value of \(\tan \theta\).
(8)

2. \begin{figure}[h] \end{figure} Figure 2 represents part of the smooth rectangular floor of a sports hall. A ball is at \(B\), 4 m from one wall of the hall and 5 m from an adjacent wall. These two walls are smooth and meet at the corner \(C\). The ball is kicked so that it travels along the floor, bounces off the first wall at the point \(X\) and hits the second wall at the point \(Y\). The point \(Y\) is 7.5 m from the corner \(C\).
The coefficient of restitution between the ball and the first wall is \(\frac { 3 } { 4 }\).
Modelling the ball as a particle, find the distance \(C X\).

2. A small ball \(B\), moving on a smooth horizontal plane, collides with a fixed smooth vertical wall. Immediately before the collision the angle between the direction of motion of \(B\) and the wall is \(\alpha\). The coefficient of restitution between \(B\) and the wall is \(\frac { 3 } { 4 }\). The kinetic energy of \(B\) immediately after the collision is \(60 \%\) of its kinetic energy immediately before the collision. Find, in degrees, the size of angle \(\alpha\).

1. A smooth sphere \(S\) is moving on a smooth horizontal plane with speed \(u\) when it collides with a smooth fixed vertical wall. At the instant of collision the direction of motion of \(S\) makes an angle of \(30 ^ { \circ }\) with the wall. The coefficient of restitution between \(S\) and the wall is \(\frac { 1 } { 3 }\).

Find the speed of \(S\) immediately after the collision.

2. \begin{figure}[h] \end{figure} Figure 2 represents the plan view of part of a horizontal floor, where \(A B\) and \(B C\) are fixed vertical walls with \(A B\) perpendicular to \(B C\). A small ball is projected along the floor towards \(A B\) with speed \(6 \mathrm {~ms} ^ { - 1 }\) on a path that makes an angle \(\alpha\) with \(A B\), where \(\tan \alpha = \frac { 4 } { 3 }\). The ball hits \(A B\) and then hits \(B C\).
Immediately after hitting \(A B\), the ball is moving at an angle \(\beta\) to \(A B\), where \(\tan \beta = \frac { 1 } { 3 }\) The coefficient of restitution between the ball and \(A B\) is \(e\).
The coefficient of restitution between the ball and \(B C\) is \(\frac { 1 } { 2 }\) By modelling the ball as a particle and the floor and walls as being smooth,

1. show that the value of \(e = \frac { 1 } { 4 }\)
2. find the speed of the ball immediately after it hits \(B C\).
3. Suggest two ways in which the model could be refined to make it more realistic.

1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]

\begin{figure}[h] \end{figure} Figure 1 represents the plan view of part of a smooth horizontal floor, where \(A B\) represents a fixed smooth vertical wall. A small ball of mass 0.5 kg is moving on the floor when it strikes the wall.
Immediately before the impact the velocity of the ball is \(( 7 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
Immediately after the impact the velocity of the ball is \(( \mathbf { i } + 6 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
The coefficient of restitution between the ball and the wall is \(e\).

1. Show that \(A B\) is parallel to \(( 2 \mathbf { i } + 3 \mathbf { j } )\).
2. Find the value of \(e\).

1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]

\begin{figure}[h] \end{figure} Figure 3 represents the plan view of part of a smooth horizontal floor, where \(A B\) is a fixed smooth vertical wall. The direction of \(\overrightarrow { A B }\) is in the direction of the vector \(( \mathbf { i } + \mathbf { j } )\) A small ball of mass 0.25 kg is moving on the floor when it strikes the wall \(A B\).
Immediately before its impact with the wall \(A B\), the velocity of the ball is \(( 8 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) Immediately after its impact with the wall \(A B\), the velocity of the ball is \(\mathbf { v m s } ^ { - 1 }\) The coefficient of restitution between the ball and the wall is \(\frac { 1 } { 3 }\) By modelling the ball as a particle,

1. show that \(\mathbf { v } = 4 \mathbf { i } + 6 \mathbf { j }\)
2. Find the magnitude of the impulse received by the ball in the impact.

8. \begin{figure}[h] \end{figure} Figure 5 represents the plan view of part of a smooth horizontal floor, where \(R S\) and \(S T\) are smooth fixed vertical walls. The vector \(\overrightarrow { R S }\) is in the direction of \(\mathbf { i }\) and the vector \(\overrightarrow { S T }\) is in the direction of \(( 2 \mathbf { i } + \mathbf { j } )\). A small ball \(B\) is projected across the floor towards \(R S\). Immediately before the impact with \(R S\), the velocity of \(B\) is \(( 6 \mathbf { i } - 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The ball bounces off \(R S\) and then hits \(S T\). The ball is modelled as a particle.
Given that the coefficient of restitution between \(B\) and \(R S\) is \(e\),

1. find the full range of possible values of \(e\). It is now given that \(e = \frac { 1 } { 4 }\) and that the coefficient of restitution between \(B\) and \(S T\) is \(\frac { 1 } { 2 }\)
2. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(B\) immediately after its impact with \(S T\).

1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors.]

A particle \(P\) is moving with velocity ( \(4 \mathbf { i } - \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) on a smooth horizontal plane. The particle collides with a smooth vertical wall and rebounds with velocity \(( \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) The coefficient of restitution between \(P\) and the wall is \(e\).

1. Find the value of \(e\). After the collision, \(P\) goes on to hit a second smooth vertical wall, which is parallel to \(\mathbf { i }\).
   The coefficient of restitution between \(P\) and this second wall is \(\frac { 1 } { 3 }\) The angle through which the direction of motion of \(P\) has been deflected by its collision with this second wall is \(\alpha ^ { \circ }\).
2. Find the value of \(\alpha\), giving your answer to the nearest whole number.

6 A ball moving on a smooth horizontal surface collides with a fixed vertical wall. Before the collision, the ball moves with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at an angle of \(40 ^ { \circ }\) to the wall. After the collision, the ball moves with speed \(5 \mathrm {~ms} ^ { - 1 }\) and at an angle of \(26 ^ { \circ }\) to the wall. Model the ball as a particle.
6

1. Find the coefficient of restitution between the ball and the wall, giving your answer correct to two significant figures.
   6
2. Determine whether or not the wall is smooth. Fully justify your answer.

5 A ball, of mass 0.3 kg , is moving on a smooth horizontal surface. The ball collides with a smooth fixed vertical wall and rebounds.
Before the ball hits the wall, the ball is moving at \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the wall as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{b0d0c552-71cb-4e5a-b545-de8a9052def0-06_634_268_584_886} The magnitude of the force, \(F\) newtons, exerted on the ball by the wall at time \(t\) seconds is modelled by $$F = k t ^ { 2 } ( 0.1 - t ) ^ { 2 } \quad \text { for } \quad 0 \leq t \leq 0.1$$ where \(k\) is a constant. The ball is in contact with the wall for 0.1 seconds. 5 (b) Explain why \(1800000 < k \leq 3600000\) Fully justify your answer.
5 (c) Given that \(k = 2400000\) Find the speed of the ball after the collision with the wall.
[0pt] [4 marks]

12 \includegraphics[max width=\textwidth, alt={}, center]{742ef62b-bd72-45b4-88e3-70399632e9d6-4_247_801_1535_671} A small smooth sphere is projected from a point \(A\) across a smooth horizontal surface. The sphere strikes a smooth vertical wall at the point \(P\). The acute angle between the direction of motion of the sphere and the wall is \(\theta\). After the impact, the sphere passes through the point \(B\), where angle \(A P B = \phi\) (see diagram). The coefficient of restitution between the sphere and the wall is \(e\).

1. Given that \(\theta = \tan ^ { - 1 } 3\) and \(\phi = 90 ^ { \circ }\), find the exact value of \(e\).
2. Given instead that \(e = \frac { 2 } { 3 }\) and \(\phi = 45 ^ { \circ }\), show that \(\theta = \tan ^ { - 1 } 3\).