Problem

3.

A light elastic string has natural length $a$ and modulus of elasticity $mg$.

One end of the elastic string is attached to a fixed point $O$.

The other end is attached to a small smooth ring of mass $m$.

The ring is threaded on a fixed smooth vertical pole which is a distance $\dfrac32a$ from $O$.

Initially, the ring is held at the point $A$ on the pole with the elastic string horizontal.

The ring is then projected vertically downwards with speed $\sqrt{ag}$ and reaches the point $B$, where $AB=2a$, with speed $V$, as shown in Figure 2.

Air resistance is assumed to be negligible.

Find $V$ in terms of $a$ and $g$.