A particle slides, from rest at a depth \(r/2\) below the highest point, down the outside of a smooth sphere of radius \(r\); prove that it leaves the sphere at a height \(r/3\) above the centre. Shew further that when the particle is at a distance \(r\sqrt{2}\) from the vertical diameter of the sphere it is at a depth \(4r\) below the centre.
Two particles \(A, B\) of masses \(2m\) and \(m\) respectively are connected by a light rod and lie on a smooth horizontal table. If the mass \(A\) is struck a blow in a direction \(\tan^{-1}\frac{1}{2}\) with \(AB\), prove that the initial velocity of \(A\) is \(\sqrt{5}\) times that of \(B\).