1. A particle \(P\) of mass \(3 m\) is moving in a straight line on a smooth horizontal floor. A particle \(Q\) of mass \(5 m\) 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 \(2 u\) and the speed of \(Q\) is \(u\). The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Show that the speed of \(Q\) immediately after the collision is \(\frac { u } { 8 } ( 9 e + 1 )\)
2. Find the range of values of \(e\) for which the direction of motion of \(P\) is not changed as a result of the collision. When \(P\) and \(Q\) collide they are at a distance \(d\) from a smooth fixed vertical wall, which is perpendicular to their direction of motion. After the collision with \(P\), particle \(Q\) collides directly with the wall and rebounds so that there is a second collision between \(P\) and \(Q\). This second collision takes place at a distance \(x\) from the wall. Given that \(e = \frac { 1 } { 18 }\) and the coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 3 }\)
3. find \(x\) in terms of \(d\).