3 Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(3 m\) 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 \(e\).
- Find, in terms of \(u\) and \(e\), expressions for the velocities of \(A\) and \(B\) after the collision.
Sphere \(B\) continues to move until it strikes 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 \(\frac { 3 } { 4 }\). When the spheres subsequently collide, \(A\) is brought to rest. - Find the value of \(e\).
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Three identical uniform discs, \(A , B\) and \(C\), each have mass \(m\) and radius \(a\). They are joined together by uniform rods, each of which has mass \(\frac { 1 } { 3 } m\) and length \(2 a\). The discs lie in the same plane and their centres form the vertices of an equilateral triangle of side \(4 a\). Each rod has one end rigidly attached to the circumference of a disc and the other end rigidly attached to the circumference of an adjacent disc, so that the rod lies along the line joining the centres of the two discs (see diagram).