Problem

One end of a light elastic string of natural length $a$ and modulus of elasticity $5mg$ is attached to a fixed point $O$. Two particles, $P$ and $Q$, of masses $m$ and $4m$ respectively are attached to the other end of the string and they hang vertically in equilibrium. Particle $Q$ is then detached from the string, hence releasing particle $P$ from rest.

Find, in terms of $a$, the length of the string when the speed of particle $P$ is first equal to $\sqrt{\frac75ag}$.