A uniform smooth sphere $A$ moves on a smooth horizontal surface towards a smooth vertical wall. Immediately before the sphere hits the wall it has components of velocity parallel and perpendicular to the wall each of magnitude $4$ m s$^{-1}$. Immediately after hitting the wall the components have magnitudes $u$ m s$^{-1}$ and $v$ m s$^{-1}$, respectively (see Fig. 1).

\includegraphics{figure_1}

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\item Given that the coefficient of restitution between the sphere and the wall is $\frac{1}{4}$, state the values of $u$ and $v$. [2]
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Shortly after hitting the wall the sphere $A$ comes into contact with another uniform smooth sphere $B$, which has the same mass and radius as $A$. The sphere $B$ is stationary and at the instant of contact the line of centres of the spheres is parallel to the wall (see Fig. 2). The contact between the spheres is perfectly elastic.

\includegraphics{figure_2}

\begin{enumerate}[label=(\roman*)]
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\item Find, for each sphere, its speed and its direction of motion immediately after the contact. [6]
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\hfill \mbox{\textit{OCR M3 2010 Q3 [8]}}