Two particles \(P\) and \(Q\) have masses \(m\) and \(4 m\) respectively. The particles are at rest on a smooth horizontal plane. Particle \(P\) is given a horizontal impulse, of magnitude \(I\), in the direction \(P Q\). Particle \(P\) then collides directly with \(Q\). Immediately after this collision, \(P\) is at rest and \(Q\) has speed \(w\). The coefficient of restitution between the particles is \(e\).
Find \(I\) in terms of \(m\) and \(w\).
Show that \(e = \frac { 1 } { 4 }\)
Find, in terms of \(m\) and \(w\), the total kinetic energy lost in the collision between \(P\) and \(Q\).