5.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{904c44f8-bd97-4a1d-8eb1-73cb52ddc8c5-07_478_1185_267_376}
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\caption{Figure 1}
\end{figure}
Two smooth uniform spheres \(A\) and \(B\) have equal radii. The mass of \(A\) is \(m\) and the mass of \(B\) is \(3 m\). The spheres are moving on a smooth horizontal plane when they collide obliquely. Immediately before the collision, \(A\) is moving with speed \(3 u\) at angle \(\alpha\) to the line of centres and \(B\) is moving with speed \(u\) at angle \(\beta\) to the line of centres, as shown in Figure 1. The coefficient of restitution between the two spheres is \(\frac { 1 } { 5 }\). It is given that \(\cos \alpha = \frac { 1 } { 3 }\) and \(\cos \beta = \frac { 2 } { 3 }\) and that \(\alpha\) and \(\beta\) are both acute angles.
- Find the magnitude of the impulse on \(A\) due to the collision in terms of \(m\) and \(u\).
- Express the kinetic energy lost by \(A\) in the collision as a fraction of its initial kinetic energy.