Ball between two walls, successive rebounds

5 \includegraphics[max width=\textwidth, alt={}, center]{f6887893-66c5-40df-ba8d-9439a5c268eb-3_531_908_1674_616} A rectangular pool table \(K L M N\) has \(K L = a\) and \(K N = 2 a\). A ball lies at rest on the table just outside the pocket at \(L\) and is projected along the table with speed \(u\) in a direction making an angle \(\theta\) with the edge \(L M\). The ball hits the edge \(K N\) at \(Y\), rebounds to hit the edge \(L M\) at \(X\) and then rebounds into the pocket at \(N\). Angle \(L X Y\) is denoted by \(\phi\) (see diagram). The coefficient of restitution between the ball and an edge is \(\frac { 3 } { 4 }\), and all resistances to motion may be neglected. Show that \(\tan \phi = \frac { 3 } { 4 } \tan \theta\). [3] Show that \(X M = \left( 2 - \frac { 7 } { 3 } \cot \theta \right) a\), and find the value of \(\theta\). Find the speed with which the ball reaches \(N\), giving the answer in the form \(k u\), where \(k\) is correct to 3 significant figures.

2 \includegraphics[max width=\textwidth, alt={}, center]{58728f93-bfdb-4f76-a9b9-3a1d1592bfc9-2_531_760_927_696} Two smooth vertical walls each with their base on a smooth horizontal surface intersect at an angle of \(60 ^ { \circ }\). A small smooth sphere \(P\) is moving on the horizontal surface with speed \(u\) when it collides with the first vertical wall at the point \(D\). The angle between the direction of motion of \(P\) and the wall is \(\alpha ^ { \circ }\) before the collision and \(75 ^ { \circ }\) after the collision. The speed of \(P\) after this collision is \(v\) and the coefficient of restitution between \(P\) and the first wall is \(e\). Sphere \(P\) then collides with the second vertical wall at the point \(E\). The speed of \(P\) after this second collision is \(\frac { 1 } { 4 } u\) (see diagram). The coefficient of restitution between \(P\) and the second wall is \(\frac { 3 } { 4 }\).

1. By considering the collision at \(E\), show that \(v = \frac { \sqrt { } 2 } { 5 } u\).
2. Find the value of \(\alpha\) and the value of \(e\).

2 \includegraphics[max width=\textwidth, alt={}, center]{62d0d8cb-8f8c-4298-9705-71a735a9a4e7-2_531_760_927_696} Two smooth vertical walls each with their base on a smooth horizontal surface intersect at an angle of \(60 ^ { \circ }\). A small smooth sphere \(P\) is moving on the horizontal surface with speed \(u\) when it collides with the first vertical wall at the point \(D\). The angle between the direction of motion of \(P\) and the wall is \(\alpha ^ { \circ }\) before the collision and \(75 ^ { \circ }\) after the collision. The speed of \(P\) after this collision is \(v\) and the coefficient of restitution between \(P\) and the first wall is \(e\). Sphere \(P\) then collides with the second vertical wall at the point \(E\). The speed of \(P\) after this second collision is \(\frac { 1 } { 4 } u\) (see diagram). The coefficient of restitution between \(P\) and the second wall is \(\frac { 3 } { 4 }\).

1. By considering the collision at \(E\), show that \(v = \frac { \sqrt { } 2 } { 5 } u\).
2. Find the value of \(\alpha\) and the value of \(e\).

2 \includegraphics[max width=\textwidth, alt={}, center]{184020e1-7ff2-4172-8d33-baff963afa76-2_531_760_927_696} Two smooth vertical walls each with their base on a smooth horizontal surface intersect at an angle of \(60 ^ { \circ }\). A small smooth sphere \(P\) is moving on the horizontal surface with speed \(u\) when it collides with the first vertical wall at the point \(D\). The angle between the direction of motion of \(P\) and the wall is \(\alpha ^ { \circ }\) before the collision and \(75 ^ { \circ }\) after the collision. The speed of \(P\) after this collision is \(v\) and the coefficient of restitution between \(P\) and the first wall is \(e\). Sphere \(P\) then collides with the second vertical wall at the point \(E\). The speed of \(P\) after this second collision is \(\frac { 1 } { 4 } u\) (see diagram). The coefficient of restitution between \(P\) and the second wall is \(\frac { 3 } { 4 }\).

1. By considering the collision at \(E\), show that \(v = \frac { \sqrt { } 2 } { 5 } u\).
2. Find the value of \(\alpha\) and the value of \(e\).

7. \begin{figure}[h] \end{figure} Figure 4 represents the plan view of part of a smooth horizontal floor, where \(A B\) and \(B C\) are smooth vertical walls. The angle between \(A B\) and \(B C\) is \(120 ^ { \circ }\). A ball is projected along the floor towards \(A B\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a path at an angle of \(60 ^ { \circ }\) to \(A B\). The ball hits \(A B\) and then hits \(B C\). The ball is modelled as a particle. The coefficient of restitution between the ball and each wall is \(\frac { 1 } { 2 }\)

1. Show that the speed of the ball immediately after it has hit \(A B\) is \(\frac { \sqrt { 7 } } { 4 } u\). The speed of the ball immediately after it has hit \(B C\) is \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
2. Find \(w\) in terms of \(u\).

2. Figure 2 A small spherical ball \(P\) is at rest at the point \(A\) on a smooth horizontal floor. The ball is struck and travels along the floor until it hits a fixed smooth vertical wall at the point \(X\). The angle between \(A X\) and this wall is \(\alpha\), where \(\alpha\) is acute. A second fixed smooth vertical wall is perpendicular to the first wall and meets it in a vertical line through the point \(C\) on the floor. The ball rebounds from the first wall and hits the second wall at the point \(Y\). After \(P\) rebounds from the second wall, \(P\) is travelling in a direction parallel to \(X A\), as shown in Figure 2. The coefficient of restitution between the ball and the first wall is \(e\). The coefficient of restitution between the ball and the second wall is ke. Find the value of \(k\).
2. \includegraphics[max width=\textwidth, alt={}, center]{b01b3a41-3ed1-4104-b20d-4cfb845df4a1-03_582_645_118_648}

7. \begin{figure}[h] \end{figure} Figure 2 represents the plan view of part of a horizontal floor, where \(A B\) and \(C D\) represent fixed vertical walls, with \(A B\) parallel to \(C D\). A small ball is projected along the floor towards wall \(A B\). Immediately before hitting wall \(A B\), the ball is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) to \(A B\), where \(0 < \alpha < \frac { \pi } { 2 }\) The ball hits wall \(A B\) and then hits wall \(C D\).
After the impact with wall \(C D\), the ball is moving at angle \(\frac { 1 } { 2 } \alpha\) to \(C D\).
The coefficient of restitution between the ball and wall \(A B\) is \(\frac { 2 } { 3 }\) The coefficient of restitution between the ball and wall \(C D\) is also \(\frac { 2 } { 3 }\) The floor and the walls are modelled as being smooth. The ball is modelled as a particle.

1. Show that \(\tan \left( \frac { 1 } { 2 } \alpha \right) = \frac { 1 } { 3 }\)
2. Find the percentage of the initial kinetic energy of the ball that is lost as a result of the two impacts.

5. \begin{figure}[h] \end{figure} Figure 1 represents the plan view of part of a horizontal floor, where \(A B\) and \(B C\) represent fixed vertical walls, with \(A B\) perpendicular to \(B C\). A small ball is projected along the floor towards the wall \(A B\). Immediately before hitting the wall \(A B\) the ball is moving with speed \(v \mathrm {~ms} ^ { - 1 }\) at an angle \(\theta\) to \(A B\). The ball hits the wall \(A B\) and then hits the wall \(B C\).
The coefficient of restitution between the ball and the wall \(A B\) is \(\frac { 1 } { 3 }\) The coefficient of restitution between the ball and the wall \(B C\) is \(e\).
The floor and the walls are modelled as being smooth.
The ball is modelled as a particle.
The ball loses half of its kinetic energy in the impact with the wall \(A B\).

1. Find the exact value of \(\cos \theta\). The ball loses half of its remaining kinetic energy in the impact with the wall \(B C\).
2. Find the exact value of \(e\).

7. \begin{figure}[h] \end{figure} A small smooth snooker ball is projected from the corner \(A\) of a horizontal rectangular snooker table \(A B C D\). The ball is projected so it first hits the side \(D C\) at the point \(P\), then hits the side \(C B\) at the point \(Q\) and then returns to \(A\). Angle \(A P D = \alpha\), Angle \(Q P C = \beta\), Angle \(A Q B = \gamma\) The ball moves along \(A P\) with speed \(U\), along \(P Q\) with speed \(V\) and along \(Q A\) with speed \(W\), as shown in Figure 2. The coefficient of restitution between the ball and side \(D C\) is \(e _ { 1 }\) The coefficient of restitution between the ball and side \(C B\) is \(e _ { 2 }\) The ball is modelled as a particle. \section*{Use the model to answer all parts of this question.}

1. Show that \(\tan \beta = e _ { 1 } \tan \alpha\)
2. Hence show that \(e _ { 1 } \tan \alpha = e _ { 2 } \cot \gamma\)
3. By considering (angle \(A P Q\) + angle \(A Q P\) ) or otherwise, show that it would be possible for the ball to return to \(A\) only if \(e _ { 2 } > e _ { 1 }\) If instead \(e _ { 1 } = e _ { 2 }\), the ball would not return to \(A\).
   Given that \(e _ { 1 } = e _ { 2 }\)
4. use the result from part (b) to describe the path of the ball after it hits \(C B\) at \(Q\), explaining your answer.

4. \begin{figure}[h] \end{figure} Figure 1 represents the plan view of part of a horizontal floor, where \(A B\) and \(B C\) are perpendicular vertical walls. The floor and the walls are modelled as smooth.
A ball is projected along the floor towards \(A B\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a path at an angle of \(60 ^ { \circ }\) to \(A B\). The ball hits \(A B\) and then hits \(B C\). The ball is modelled as a particle.
The coefficient of restitution between the ball and wall \(A B\) is \(\frac { 1 } { \sqrt { 3 } }\) The coefficient of restitution between the ball and wall \(B C\) is \(\sqrt { \frac { 2 } { 5 } }\)

1. Show that, using this model, the final kinetic energy of the ball is \(35 \%\) of the initial kinetic energy of the ball.
2. In reality the floor and the walls may not be smooth. What effect will the model have had on the calculation of the percentage of kinetic energy remaining?

\includegraphics{figure_5} \(AB\) and \(BC\) are two fixed smooth vertical barriers on a smooth horizontal surface, with angle \(ABC = 60°\). A particle of mass \(m\) is moving with speed \(u\) on the surface. The particle strikes \(AB\) at an angle \(\theta\) with \(AB\). It then strikes \(BC\) and rebounds at an angle \(\beta\) with \(BC\) (see diagram). The coefficient of restitution between the particle and each barrier is \(e\) and \(\tan \theta = 2\). The kinetic energy of the particle after the first collision is 40% of its kinetic energy before the first collision.

1. Find the value of \(e\). [4]
2. Find the size of angle \(\beta\). [4]