A small ball is moving on a horizontal plane when it strikes a smooth vertical wall. The coefficient of restitution between the ball and the wall is \(e\). Immediately before the impact the direction of motion of the ball makes an angle of \(60 ^ { \circ }\) with the wall. Immediately after the impact the direction of motion of the ball makes an angle of \(30 ^ { \circ }\) with the wall.
Find the fraction of the kinetic energy of the ball which is lost in the impact.
(6)
Find the value of \(e\).
(4)
A lorry of mass \(M\) moves along a straight horizontal road against a constant resistance of magnitude \(R\). The engine of the lorry works at a constant rate \(R U\), where \(U\) is a constant. At time \(t\), the lorry is moving with speed \(v\).
Show that \(M v \frac { \mathrm {~d} v } { \mathrm {~d} t } = R ( U - v )\).
(3)
At time \(t = 0\), the lorry has speed \(\frac { 1 } { 4 } U\) and the time taken by the lorry to attain a speed of \(\frac { 1 } { 3 } U\) is \(\frac { k M U } { R }\), where \(k\) is a constant.