A small smooth ball of mass \(\frac { 1 } { 2 } \mathrm {~kg}\) is falling vertically. The ball strikes a smooth plane which is inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac { 3 } { 4 }\). Immediately before striking the plane the ball has speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution between ball and plane is \(\frac { 1 } { 2 }\). Find
the speed, to 3 significant figures, of the ball immediately after the impact,
the magnitude of the impulse received by the ball as it strikes the plane.
A cyclist \(P\) is cycling due north at a constant speed of \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). At 12 noon another cyclist \(Q\) is due west of \(P\). The speed of \(Q\) is constant at \(10 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Find the course which \(Q\) should set in order to pass as close to \(P\) as possible, giving your answer as a bearing.
(5)
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A smooth sphere \(P\) lies at rest on a smooth horizontal plane. A second identical sphere \(Q\), moving on the plane, collides with the sphere \(P\). Immediately before the collision the direction of motion of \(Q\) makes an angle \(\alpha\) with the line joining the centres of the spheres. Immediately after the collision the direction of motion of \(Q\) makes an angle \(\beta\) with the line joining the centres of spheres, as shown in Figure 1. The coefficient of restitution between the spheres is \(e\).
Show that \(( 1 - e ) \tan \beta = 2 \tan \alpha\).