A particle \(P\) of mass \(2 m\) is moving in a straight line with speed \(u\) on a smooth horizontal plane. The particle \(P\) collides directly with a particle \(Q\), of mass \(m\), which is moving on the plane along the same straight line as \(P\) but in the opposite direction to \(P\). Immediately before the collision the speed of \(Q\) is \(3 u\). The coefficient of restitution between \(P\) and \(Q\) is \(e\), where \(e > \frac { 1 } { 8 }\)
Find, in terms of \(u\) and \(e\),
the speed of \(P\) immediately after the collision,
the speed of \(Q\) immediately after the collision.
Show that, for all possible values of \(e\), the direction of motion of \(P\) is reversed by the collision.
After the collision, \(Q\) strikes a smooth fixed vertical wall, which is perpendicular to the direction of motion of \(Q\), and rebounds. The coefficient of restitution between \(Q\) and the wall is \(f\).
Given that \(e = \frac { 3 } { 4 }\) and that there is a second collision between \(Q\) and \(P\),