2. Two smooth uniform spheres $S$ and $T$ have equal radii. The mass of $S$ is 0.3 kg and the mass of $T$ is 0.6 kg . The spheres are moving on a smooth horizontal plane and collide obliquely. Immediately before the collision the velocity of $S$ is $\mathbf { u } _ { 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the velocity of $T$ is $\mathbf { u } _ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The coefficient of restitution between the spheres is 0.5 . Immediately after the collision the velocity of $S$ is $( - \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ and the velocity of $T$ is $( \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. Given that when the spheres collide the line joining their centres is parallel to $\mathbf { i }$,
\begin{enumerate}[label=(\alph*)]
\item find
\begin{enumerate}[label=(\roman*)]
\item $\mathbf { u } _ { 1 }$,
\item $\mathbf { u } _ { 2 }$.

After the collision, $T$ goes on to collide with a smooth vertical wall which is parallel to $\mathbf { j }$. Given that the coefficient of restitution between $T$ and the wall is also 0.5 , find
\end{enumerate}\item the angle through which the direction of motion of $T$ is deflected as a result of the collision with the wall,
\item the loss in kinetic energy of $T$ caused by the collision with the wall.
\end{enumerate}

\hfill \mbox{\textit{Edexcel M4 2010 Q2 [14]}}