2 Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(2 m\) 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 \(\frac { 2 } { 3 }\).
- Find, in terms of \(u\), the speeds of \(A\) and \(B\) after this collision.
Sphere \(B\) is initially at a distance \(d\) from a fixed smooth vertical wall which is perpendicular to the direction of motion of \(A\). The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 2 }\). - Find, in terms of \(d\) and \(u\), the time that elapses between the first and second collisions between \(A\) and \(B\).
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A particle of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The point \(A\) is such that \(O A = a\) and \(O A\) makes an angle \(\alpha\) with the upward vertical, where \(\tan \alpha = \frac { 12 } { 5 }\). The particle is projected downwards from \(A\) with speed \(u\) perpendicular to the string and moves in a vertical plane (see diagram). The string becomes slack after the string has rotated through \(270 ^ { \circ }\) from its initial position, with the particle now at the point \(B\).