3\\
\includegraphics[max width=\textwidth, alt={}, center]{7e0f600a-18f1-458b-8549-27fca592b19c-3_419_921_267_612}

Two uniform smooth spheres $A$ and $B$, of equal radius, have masses 6 kg and 3 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision the velocity of $A$ has components $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ along the line of centres towards $B$, and $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ perpendicular to the line of centres. $B$ is moving with speed $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ along the line of centres towards $A$ (see diagram). The coefficient of restitution between the spheres is $e$.\\
(i) Find, in terms of $e$, the component of the velocity of $A$ along the line of centres immediately after the collision.\\
(ii) Given that the speeds of $A$ and $B$ are the same immediately after the collision, and that $3 e ^ { 2 } = 1$, find $v$.

\hfill \mbox{\textit{OCR M3 2008 Q3 [9]}}