Two wall collisions sequence

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]{833c338f-53c1-436e-a772-0cdaf17fa72d-2_547_811_625_667} 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]{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 \(e\).

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 \(C X\).

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}

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.

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 \(e\).

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 \(S T\).

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.

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.

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?