A small smooth ball of mass \(m\) is falling vertically when it strikes a fixed smooth plane which is inclined to the horizontal at an angle \(\alpha\), where \(0 ^ { \circ } < \alpha < 45 ^ { \circ }\). Immediately before striking the plane the ball has speed \(u\). Immediately after striking the plane the ball moves in a direction which makes an angle of \(45 ^ { \circ }\) with the plane. The coefficient of restitution between the ball and the plane is \(e\). Find, in terms of \(m , u\) and \(e\), the magnitude of the impulse of the plane on the ball.
A ship \(A\) is travelling at a constant speed of \(30 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on a bearing of \(050 ^ { \circ }\). Another ship \(B\) is travelling at a constant speed of \(v \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and sets a course to intercept \(A\). At 1400 hours \(B\) is 20 km from \(A\) and the bearing of \(A\) from \(B\) is \(290 ^ { \circ }\).