At noon, a boat \(P\) is on a bearing of \(120 ^ { \circ }\) from boat \(Q\). Boat \(P\) is moving due east at a constant speed of \(12 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Boat \(Q\) is moving in a straight line with a constant speed of \(15 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on a course to intercept \(P\). Find the direction of motion of \(Q\), giving your answer as a bearing.
A smooth uniform sphere \(S\) of mass \(m\) is moving on a smooth horizontal plane when it collides with a fixed smooth vertical wall. Immediately before the collision, the speed of \(S\) is \(U\) and its direction of motion makes an angle \(\alpha\) with the wall. The coefficient of restitution between \(S\) and the wall is \(e\). Find the kinetic energy of \(S\) immediately after the collision.
(6)
A cyclist \(C\) is moving with a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) due south. Cyclist \(D\) is moving with a constant speed of \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a bearing of \(240 ^ { \circ }\).
Show that the magnitude of the velocity of \(C\) relative to \(D\) is \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
At \(2 \mathrm { pm } , D\) is 4 km due east of \(C\).
Find
the shortest distance between \(C\) and \(D\) during the subsequent motion,
the time, to the nearest minute, at which this shortest distance occurs.