Multiple wall bounces or returns

2. \begin{figure}[h] \end{figure} The point \(A\) lies on a smooth horizontal floor between two fixed smooth parallel vertical walls \(W X\) and \(Y Z\), as shown in the plan view in Figure 1.
The distance between \(W X\) and \(Y Z\) is \(3 d\).
The distance of \(A\) from \(Y Z\) is \(d\).
A particle is projected from \(A\) along the floor with speed \(u\) towards \(Y Z\) in a direction perpendicular to \(Y Z\). The coefficient of restitution between the particle and each wall is \(\frac { 2 } { 3 }\)
The time taken for the particle to move from \(A\), bounce off each wall once and return to A for the first time is \(T _ { 1 }\)

1. Find \(T _ { 1 }\) in terms of \(d\) and \(u\). The ball returns to \(A\) for the first time after bouncing off each wall once. The further time taken for the particle to move from \(A\), bounce off each wall once and return to \(A\) for the second time is \(T _ { 2 }\)
2. Find \(T _ { 2 }\) in terms of \(d\) and \(u\).

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 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. \begin{figure}[h] \end{figure} A particle of mass em is at rest on a smooth horizontal plane between two smooth fixed parallel vertical walls, as shown in the plan view in Figure 2. The particle is projected along the plane with speed \(u\) towards one of the walls and strikes the wall at right angles. The coefficient of restitution between the particle and each wall is \(e\) and air resistance is modelled as being negligible. Using the model,

1. find, in terms of \(m , u\) and \(e\), an expression for the total loss in the kinetic energy of the particle as a result of the first two impacts. Given that \(e\) can vary such that \(0 < e < 1\) and using the model,
2. find the value of \(e\) for which the total loss in the kinetic energy of the particle as a result of the first two impacts is a maximum,
3. describe the subsequent motion of the particle.

1. \begin{figure}[h] \end{figure} Figure 1 represents the plan of part of a smooth horizontal floor, where \(W _ { 1 }\) and \(W _ { 2 }\) are two fixed parallel vertical walls. The walls are 3 metres apart. A particle lies at rest at a point \(O\) on the floor between the two walls, where the point \(O\) is \(d\) metres, \(0 < d \leqslant 3\), from \(W _ { 1 }\) At time \(t = 0\), the particle is projected from \(O\) towards \(W _ { 1 }\) 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 particle returns to \(O\) at time \(t = T\) seconds, having bounced off each wall once.

1. Show that \(T = \frac { 45 - 5 d } { 4 u }\) The value of \(u\) is fixed, the particle still hits each wall once but the value of \(d\) can now vary.
2. Find the least possible value of \(T\), giving your answer in terms of \(u\). You must give a reason for your answer.