Collision with coefficient of restitution

6 Two smooth spheres \(A\) and \(B\) have equal radii and masses \(m\) and \(2 m\) respectively. Sphere \(B\) is at rest on a smooth horizontal floor. Sphere \(A\) is moving on the floor with velocity \(u\) and collides directly with \(B\). The coefficient of restitution between the spheres is \(e\).

1. Find, in terms of \(u\) and \(e\), the velocities of \(A\) and \(B\) after the collision.
   Subsequently, \(B\) collides with a fixed vertical wall which makes an angle \(\theta\) with the direction of motion of \(B\), where \(\tan \theta = \frac { 3 } { 4 }\). The coefficient of restitution between \(B\) and the wall is \(\frac { 2 } { 3 }\). Immediately after \(B\) collides with the wall, the kinetic energy of \(A\) is \(\frac { 5 } { 32 }\) of the kinetic energy of \(B\).
2. Find the possible values of \(e\).

7 Rose and Steve collide while sitting firmly on trays that are sliding on smooth horizontal ice. There are no external driving forces. Fig. 7 shows the masses of Rose and of Steve with their trays, their velocities just before their collision and the line of their motion and of their impact. Immediately after the collision, Rose has a velocity of \(0.28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of her motion before the collision. \begin{figure}[h] \end{figure}

1. Find Steve's velocity after the collision.
2. Find the coefficient of restitution between Rose and Steve on their trays. Shortly after the collision, Steve catches Rose's hand, pulls her towards him with a horizontal impulse of 4.48 Ns and then lets go of her hand.
3. Calculate Rose's velocity after the pull. When they collide again they hold one another and move together with a common speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
4. Calculate \(V\).
5. Why did you need to know that there are no driving forces and that the ice is smooth? {www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
   OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

6 My cat Jeoffry has a mass of 4 kg and is sitting on rough ground near a sledge of mass 8 kg . The sledge is on a large area of smooth horizontal ice. Initially, the sledge is at rest and Jeoffry jumps and lands on it with a horizontal velocity of \(2.25 \mathrm {~ms} ^ { - 1 }\) parallel to the runners of the sledge. On landing, Jeoffry grips the sledge with his claws so that he does not move relative to the sledge in the subsequent motion.

1. Show that the sledge with Jeoffry on it moves off with a speed of \(0.75 \mathrm {~ms} ^ { - 1 }\). With the sledge and Jeoffry moving at \(0.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the sledge collides directly with a stationary stone of mass 3 kg . The stone may move freely over the ice. The coefficient of restitution in the collision is \(\frac { 4 } { 15 }\).
2. Calculate the velocity of the sledge and Jeoffry immediately after the collision. In a new situation, Jeoffry is initially sitting at rest on the sledge when it is stationary on the ice. He then walks from the back to the front of the sledge.
3. Giving a brief reason for your answer, describe what happens to the sledge during his walk. Jeoffry is again sitting on the sledge when it is stationary on the ice. He jumps off and, after he has lost contact with the sledge, has a horizontal speed relative to the sledge of \(3 \mathrm {~ms} ^ { - 1 }\).
4. Determine the speed of the sledge after Jeoffry loses contact with it. Fig. 7 shows a container for flowers which is a vertical cylindrical shell with a closed horizontal base. Its radius and its height are both \(\frac { 1 } { 2 } \mathrm {~m}\). Both the curved surface and the base are made of the same thin uniform material. The mass of the container is \(M \mathrm {~kg}\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{54711a46-83ce-4fb9-b6d3-53b264725c74-6_323_709_447_767} \captionsetup{labelformat=empty} \caption{Fig. 7}
   \end{figure}
5. Find, as a fraction, the height above the base of the centre of mass of the container. The container would hold \(\frac { 3 } { 2 } M \mathrm {~kg}\) of soil when full to the top. Some soil is put into the empty container and levelled with its top surface \(y \mathrm {~m}\) above the base. The centre of mass of the container with this much soil is zm above the base.
6. Show that \(z = \frac { 1 + 9 y ^ { 2 } } { 6 ( 1 + 3 y ) }\).
7. It is given that \(\frac { \mathrm { d } z } { \mathrm {~d} y } = 0\) when \(y = 0.14\) (to 2 significant figures) and that \(\frac { \mathrm { d } ^ { 2 } z } { \mathrm {~d} y ^ { 2 } } > 0\) at this value of \(y\). When putting in the soil, how might you use this information if the container is to be placed on slopes without it tipping over? \section*{END OF QUESTION PAPER} OCR is committed to seeking permission to reproduce all third-party content that it uses in the assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
   OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }\section*{}

1. A particle of mass \(m \mathrm {~kg}\) lies on a smooth horizontal surface.

Initially the particle is at rest at a point \(O\) between two fixed parallel vertical walls.
The point \(O\) is equidistant from the two walls and the walls are 4 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 \(\frac { 3 } { 4 }\)
The magnitude of the impulse on the particle due to the first impact with a wall is \(\lambda m u\) Ns.

1. Find the value of \(\lambda\). The particle returns to \(O\), having bounced off each wall once, at time \(t = 7\) seconds.
2. Find the value of \(u\).