7. Two smooth spheres, \(A\) and \(B\), of equal radius but of masses \(3 m\) and \(4 m\) respectively, are free to move in a straight horizontal groove. The coefficient of restitution between them is \(e\). \(A\) is projected with speed \(u\) to hit \(B\), which is initially at rest.

1. Show that \(B\) begins to move with speed \(\frac { 3 } { 7 } u ( 1 + e )\).
2. Given that \(A\) is brought to rest by the collision, show that \(e = 0.75\). Having been brought to rest, \(A\) is now set in motion again by being given an impulse of magnitude \(k m u \mathrm { Ns }\), where \(k > 2 \cdot 25\). A then collides again with \(B\).
3. Show that the speed of \(A\) after this second impact is independent of \(k\).