A particle \(P\) is projected with speed \(u\,\text{m s}^{-1}\) at an angle \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity.

During its flight, \(P\) passes through the point which is a horizontal distance \(3a\) from \(O\) and a vertical distance \(\dfrac38a\) above the horizontal plane. It is given that \(\tan\theta=\dfrac13\).

(a) Show that \(u^2=8ag\).

A particle \(Q\) is projected with speed \(V\,\text{m s}^{-1}\) at an angle \(\alpha\) above the horizontal from \(O\) at the instant when \(P\) is at its highest point. Particles \(P\) and \(Q\) both land at the same point on the horizontal plane at the same time.

(b) Find \(V\) in terms of \(a\) and \(g\).