A particle P of mass 3 m is moving in a straight line on a smooth horizontal table. A particle \(Q\) of mass \(m\) is moving in the opposite direction to \(P\) along the same straight line. The particles collide directly. Immediately before the collision the speed of P is u and the speed of Q is 2 u . The velocities of P and Q immediately after the collision, measured in the direction of motion of P before the collision, are V and W respectively. The coefficient of restitution between P and Q is e .
Find an expression for v in terms of u and e .
Given that the direction of motion of P is changed by the collision,
find the range of possible values of e.
Show that \(\mathrm { w } = \frac { \mathrm { u } } { 4 } ( 1 + 9 \mathrm { e } )\).
Following the collision with P , the particle Q then collides with and rebounds from a fixed vertical wall which is perpendicular to the direction of motion of \(Q\). The coefficient of restitution between \(Q\) and the wall is \(f\).
Given that \(\mathrm { e } = \frac { 5 } { 9 }\), and that P and Q collide again in the subsequent motion,