One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(\frac{16}{3} M g\), is attached to a fixed point \(O\). A particle \(P\) of mass \(4 M\) is attached to the other end of the string and hangs vertically in equilibrium. Another particle of mass \(2 M\) is attached to \(P\) and the combined particle is then released from rest. The speed of the combined particle when it has descended a distance \(\frac{1}{4} a\) is \(v\).

Find an expression for \(v\) in terms of \(g\) and \(a\).