4 Fig. 4.1 shows two spheres, A and B, on a smooth horizontal surface. Their masses are 3 kg and 1 kg respectively. \begin{figure}[h] \end{figure} Initially, sphere A travels at a speed of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line towards B , which is at rest. The spheres collide and the coefficient of restitution between A and B is \(e\).

1. Show that, after the collision, A has a speed of \(\frac { 1 } { 4 } ( 3 - e ) \mathrm { m } \mathrm { s } ^ { - 1 }\), and find an expression for the speed of B in terms of \(e\). During the collision, the kinetic energy of the system decreases by \(21 \%\).
2. Determine the value of \(e\).
3. State why in part (a) it was necessary to assume that A and B have equal radii. Fig. 4.2 shows two spheres, C and D , of equal radii on a smooth horizontal surface. Their masses are 1 kg and 2 kg respectively. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 4.2} \includegraphics[alt={},max width=\textwidth]{0a790ad0-7eda-40f1-9894-f156766ae46f-4_158_1155_1544_244}
   \end{figure} Spheres C and D travel towards each other along the same straight line, C with a speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and D with a speed of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The spheres collide and during the collision C exerts an impulse on D of magnitude \(\frac { 2 } { 3 } ( u + 1 ) \mathrm { Ns }\).
4. Show that C and D have the same velocity after the collision.
5. Determine the fraction of kinetic energy lost due to the collision between C and D as \(u \rightarrow \infty\).