1. Particle \(P\) has mass \(4 m\) and particle \(Q\) has mass \(2 m\).

The particles are moving in opposite directions along the same straight line on a smooth horizontal surface. Particle \(P\) collides directly with particle \(Q\).
Immediately before the collision, the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(3 u\).
Immediately after the collision, the speed of \(P\) is \(x\) and the speed of \(Q\) is \(y\).
The direction of motion of each particle is reversed as a result of the collision.
The total kinetic energy of \(P\) and \(Q\) after the collision is half of the total kinetic energy of \(P\) and \(Q\) before the collision.

1. Show that \(y = \frac { 8 } { 3 } u\) The coefficient of restitution between \(P\) and \(Q\) is \(e\).
2. Find the value of \(e\). After the collision, \(Q\) hits a smooth fixed vertical wall that is perpendicular to the direction of motion of \(Q\). Particle \(Q\) rebounds.
   The coefficient of restitution between \(Q\) and the wall is \(f\).
   Given that there is no second collision between \(P\) and \(Q\),
3. find the range of possible values of \(f\). Given that \(f = \frac { 1 } { 4 }\)
4. find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(Q\) as a result of its impact with the wall.