The gravitational potential at P, $V_P=-\frac{GM}{AP}$

$V_P=-\frac{GM}{\sqrt{a^2+a^2}}=-\frac{GM}{\sqrt{2a}}$ and gravitational potential at O, $V_O=-\frac{GM}{a}$

Let v be the velocity of the particle when it reaches O.

$\therefore$ K.E. at $O=\frac{1}{2}m{v}^2$

Gravitational P.E. = Gravitational potential × mass

$\therefore$ P.E. at $P=-\frac{GMm}{\sqrt{2}a}$ and P.E. at $O=-\frac{GMm}{a}$

By the principle of conservation of mechanical energy

$\frac{1}{2}m{v}^2-\frac{GMm}{a}=-\frac{GMm}{\sqrt{2}a}$

$\therefore v^2=2\left[\frac{GMm}{a}-\frac{GMm}{\sqrt{2}a}\right]=\frac{2GM}{a}\left[1-\frac{1}{\sqrt{2}}\right]$

$\therefore v=\sqrt{\frac{2GM}{a}\left(1-\frac{1}{\sqrt{2}}\right)}$