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]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Fig. 4.1}
  \includegraphics[alt={},max width=\textwidth]{0a790ad0-7eda-40f1-9894-f156766ae46f-4_158_1153_436_246}
\end{center}
\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 $e$.
\begin{enumerate}[label=(\alph*)]
\item 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 $21 \%$.
\item Determine the value of $e$.
\item 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]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Fig. 4.2}
  \includegraphics[alt={},max width=\textwidth]{0a790ad0-7eda-40f1-9894-f156766ae46f-4_158_1155_1544_244}
\end{center}
\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 $\frac { 2 } { 3 } ( u + 1 ) \mathrm { Ns }$.
\item Show that C and D have the same velocity after the collision.
\item Determine the fraction of kinetic energy lost due to the collision between C and D as $u \rightarrow \infty$.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Mechanics Minor 2024 Q4 [15]}}