\includegraphics{figure_4}

Three uniform spheres $L$, $M$ and $N$ have masses $0.8$ kg, $0.6$ kg and $0.7$ kg respectively. The spheres are moving in a straight line on a smooth horizontal table, with $M$ between $L$ and $N$. The sphere $L$ is moving towards $M$ with speed $4$ m s$^{-1}$ and the spheres $M$ and $N$ are moving towards $L$ with speeds $2$ m s$^{-1}$ and $0.5$ m s$^{-1}$ respectively (see diagram).

\begin{enumerate}[label=(\roman*)]
\item $L$ collides with $M$. As a result of this collision the direction of motion of $M$ is reversed, and its speed remains $2$ m s$^{-1}$. Find the speed of $L$ after the collision. [4]
\item $M$ then collides with $N$.
\begin{enumerate}[label=(\alph*)]
\item Find the total momentum of $M$ and $N$ in the direction of $M$'s motion before this collision takes place, and deduce that the direction of motion of $N$ is reversed as a result of this collision. [4]
\item Given that $M$ is at rest immediately after this collision, find the speed of $N$ immediately after this collision. [2]
\end{enumerate}
\end{enumerate}

\hfill \mbox{\textit{OCR M1  Q4 [10]}}