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 \(\mathbf{u}_1\) m s\(^{-1}\) and the velocity of \(T\) is \(\mathbf{u}_2\) m s\(^{-1}\). The coefficient of restitution between the spheres is 0.5. Immediately after the collision the velocity of \(S\) is \((-\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\) and the velocity of \(T\) is \((\mathbf{i} + \mathbf{j})\) m s\(^{-1}\). Given that when the spheres collide the line joining their centres is parallel to \(\mathbf{i}\),
(a) find
(i) \(\mathbf{u}_1\),
(ii) \(\mathbf{u}_2\).
(6) 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
(b) the angle through which the direction of motion of \(T\) is deflected as a result of the collision with the wall, (5)
(c) the loss in kinetic energy of \(T\) caused by the collision with the wall. (3)
