Successive collisions with wall rebound

3 \includegraphics[max width=\textwidth, alt={}, center]{f8dd2aee-4ed5-4588-aa03-5dd56d9e7529-2_159_707_1443_721} Two perfectly elastic small smooth spheres \(A\) and \(B\) have masses \(3 m\) and \(m\) respectively. They lie at rest on a smooth horizontal plane with \(B\) at a distance \(a\) from a smooth vertical barrier. The line of centres of the spheres is perpendicular to the barrier, and \(B\) is between \(A\) and the barrier (see diagram). Sphere \(A\) is projected towards sphere \(B\) with speed \(u\) and, after the collision between the spheres, \(B\) hits the barrier. The coefficient of restitution between \(B\) and the barrier is \(\frac { 1 } { 2 }\). Find the speeds of \(A\) and \(B\) immediately after they first collide, and the distance from the barrier of the point where they collide for the second time.

4 Two uniform spheres \(A\) and \(B\), of equal radius, are at rest on a smooth horizontal table. Sphere \(A\) has mass \(3 m\) and sphere \(B\) has mass \(m\). Sphere \(A\) is projected directly towards \(B\), with speed \(u\). The coefficient of restitution between the spheres is 0.6 . Find the speeds of \(A\) and \(B\) after they collide. Sphere \(B\) now strikes a wall that is perpendicular to its path, rebounds and collides with \(A\) again. The coefficient of restitution between \(B\) and the wall is \(e\). Given that the second collision between \(A\) and \(B\) brings \(A\) to rest, find \(e\).

2 Two uniform small spheres \(A\) and \(B\) have equal radii and masses \(4 m\) and \(m\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is at rest. The coefficient of restitution between the spheres is \(e\).

1. Show that after the collision \(A\) moves with speed \(\frac { 1 } { 5 } u ( 4 - e )\) and find the speed of \(B\).
   Sphere \(B\) continues to move until it collides with a fixed smooth vertical barrier which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the barrier is \(\frac { 3 } { 4 } e\). After this collision, the speeds of \(A\) and \(B\) are equal.
2. Find the value of \(e\).
   The spheres \(A\) and \(B\) now collide directly again.
3. Determine whether sphere \(B\) collides with the barrier for a second time.

6 Particles \(A\) of mass \(2 m\) and \(B\) of mass \(m\) are on a smooth horizontal floor. \(A\) is moving with speed \(u\) directly towards a vertical wall, and \(B\) is at rest between \(A\) and the wall (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{74bada9e-60cf-4ed4-abd0-4be155b7cf81-5_224_828_354_244} A collides directly with \(B\). The coefficient of restitution in this collision is \(\frac { 1 } { 2 }\). \(B\) then collides with the wall, rebounds, and collides with \(A\) for a second time.

1. Show that the speed of \(B\) after its second collision with \(A\) is \(\frac { 1 } { 2 } u\). The first collision between \(A\) and \(B\) occurs at a distance \(d\) from the wall. The second collision between \(A\) and \(B\) occurs at a distance \(\frac { 1 } { 5 } d\) from the wall.
2. Find the coefficient of restitution for the collision between \(B\) and the wall.

7 Two particles, \(P\) and \(Q\), are on a smooth horizontal floor. \(P\), of mass 1 kg , is moving with speed \(1.79 \mathrm {~ms} ^ { - 1 }\) directly towards a vertical wall. \(Q\), of mass 2.74 kg , is between \(P\) and the wall, moving directly towards \(P\) with speed \(0.08 \mathrm {~ms} ^ { - 1 }\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{98053e88-1aec-4b0d-ae5f-ece4ad340266-4_232_830_1370_246} \(P\) and \(Q\) collide directly and the coefficient of restitution for this collision is denoted by \(e\).

1. Show that after this collision the speed of \(Q\) is given by \(0.42 + 0.5 e \mathrm {~ms} ^ { - 1 }\). After this collision, \(Q\) then goes on to collide directly with the wall. The coefficient of restitution for the collision between \(Q\) and the wall is also \(e\). There is then no subsequent collision between \(P\) and \(Q\).
2. Determine the range of possible values of \(e\).

6 Two identical spheres, \(A\) and \(B\), each of mass \(m \mathrm {~kg}\), are moving directly towards each other along the same straight line on a smooth horizontal surface until they collide. Just before they collide, the speeds of \(A\) and \(B\) are \(20 \mathrm {~ms} ^ { - 1 }\) and \(10 \mathrm {~ms} ^ { - 1 }\) respectively. The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. By finding, in terms of \(e\), an expression for the velocity of \(B\) after the collision, show that the direction of motion of \(B\) is reversed by the collision. After the collision between \(A\) and \(B\), which is not perfectly elastic, \(B\) goes on to collide directly with a fixed, vertical wall. The coefficient of restitution between \(B\) and the wall is \(\frac { 2 } { 5 } e\). After the collision between \(B\) and the wall, there are no further collisions between \(A\) and \(B\).
2. Determine the range of possible values of \(e\). \(7 \quad\) A body \(B\) of mass 1.5 kg is moving along the \(x\)-axis. At the instant that it is at the origin, \(O\), its velocity is \(u \mathrm {~ms} ^ { - 1 }\) in the positive \(x\)-direction. At any instant, the resistance to the motion of \(B\) is modelled as being directly proportional to \(v ^ { 2 }\) where \(v \mathrm {~ms} ^ { - 1 }\) is the velocity of \(B\) at that instant. The resistance to motion is the only horizontal force acting on \(B\). At an instant when \(B\) 's velocity is \(2 \mathrm {~ms} ^ { - 1 }\), the resistance to its motion is 24 N .

7 Two small spheres \(A\) and \(B\), with masses 0.3 kg and \(m \mathrm {~kg}\) respectively, lie at rest on a smooth horizontal surface. \(A\) is projected directly towards \(B\) with speed \(6 \mathrm {~ms} ^ { - 1 }\) and hits \(B\). The direction of motion of \(A\) is reversed in the collision. The speeds of \(A\) and \(B\) after the collision are \(1 \mathrm {~ms} ^ { - 1 }\) and \(3 \mathrm {~ms} ^ { - 1 }\) respectively. The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. Show that \(m = 0.7\).
2. Find \(e\). B continues to move at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and strikes a vertical wall at right angles. The coefficient of restitution between \(B\) and the wall is \(f\).
3. Find the range of values of \(f\) for which there will be a second collision between \(A\) and \(B\).
4. Find, in terms of \(f\), the magnitude of the impulse that the wall exerts on \(B\).
5. Given that \(f = \frac { 3 } { 4 }\), calculate the final speeds of \(A\) and \(B\), correct to 1 decimal place.

7 Two small smooth spheres, \(A\) and \(B\), are the same size and have masses \(2 m\) and \(m\) respectively. Initially, the spheres are at rest on a smooth horizontal surface. The sphere \(A\) receives an impulse of magnitude \(J\) and moves with speed \(2 u\) directly towards \(B\).

1. \(\quad\) Find \(J\) in terms of \(m\) and \(u\).
2. The sphere \(A\) collides directly with \(B\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\). Find, in terms of \(u\), the speeds of \(A\) and \(B\) immediately after the collision.
3. At the instant of collision, the centre of \(B\) is at a distance \(s\) from a fixed smooth vertical wall which is at right angles to the direction of motion of \(A\) and \(B\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-20_280_1114_1048_497} Subsequently, \(B\) collides with the wall. The radius of each sphere is \(r\).
   Show that the distance of the centre of \(A\) from the wall at the instant that \(B\) hits the wall is \(\frac { 3 s + 12 r } { 5 }\).
4. The diagram below shows the positions of \(A\) and \(B\) when \(B\) hits the wall. \includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-20_330_1109_1822_493} The sphere \(B\) collides with \(A\) again after rebounding from the wall. The coefficient of restitution between \(B\) and the wall is \(\frac { 2 } { 5 }\). Find the distance of the centre of \(\boldsymbol { B }\) from the wall at the instant when \(A\) and \(B\) collide again.
   [0pt] [4 marks] \includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-24_2488_1728_219_141}

4 Two uniform discs, A of mass 0.2 kg and B of mass 0.5 kg , collide with smooth contact while moving on a smooth horizontal surface.
Immediately before the collision, A is moving with speed \(0.5 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) with the line of centres, where \(\sin \alpha = 0.6\), and B is moving with speed \(0.3 \mathrm {~ms} ^ { - 1 }\) at right angles to the line of centres. A straight smooth vertical wall is situated to the right of B , perpendicular to the line of centres, as shown in Fig. 4. The coefficient of restitution between A and B is 0.75 . \begin{figure}[h] \end{figure}

1. Find the speeds of A and B immediately after the collision.
2. Explain why there could be a second collision between A and B if B rebounds from the wall with sufficient speed.
3. Find the range of values of the coefficient of restitution between B and the wall for which there will be a second collision between A and B .
4. How does your answer to part (b) change if the contact between B and the wall is not smooth?