A particle \(P\) of mass \(2 m \mathrm {~kg}\) is moving with speed \(2 u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a smooth horizontal plane. Particle \(P\) collides with a particle \(Q\) of mass \(3 m \mathrm {~kg}\) which is at rest on the plane. The coefficient of restitution between \(P\) and \(Q\) is \(e\). Immediately after the collision the speed of \(Q\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
Show that \(v = \frac { 4 u ( 1 + e ) } { 5 }\)
Show that \(\frac { 4 u } { 5 } \leqslant v \leqslant \frac { 8 u } { 5 }\)
Given that the direction of motion of \(P\) is reversed by the collision,
find, in terms of \(u\) and \(e\), the speed of \(P\) immediately after the collision.
After the collision, \(Q\) hits a wall, that is fixed at right angles to the direction of motion of \(Q\), and rebounds.
The coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 6 }\)
Given that \(P\) and \(Q\) collide again,