2 Two uniform small spheres \(A\) and \(B\) have equal radii and masses \(4 m\) and \(m\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is at rest. The coefficient of restitution between the spheres is \(e\).
- Show that after the collision \(A\) moves with speed \(\frac { 1 } { 5 } u ( 4 - e )\) and find the speed of \(B\).
Sphere \(B\) continues to move until it collides with a fixed smooth vertical barrier which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the barrier is \(\frac { 3 } { 4 } e\). After this collision, the speeds of \(A\) and \(B\) are equal. - Find the value of \(e\).
The spheres \(A\) and \(B\) now collide directly again. - Determine whether sphere \(B\) collides with the barrier for a second time.