Problem

A fixed smooth sphere with radius $a$ and centre $O$ rests on horizontal ground. A particle is projected horizontally from the highest point, $A$, of the sphere with speed $u$. The particle begins to move in a vertical circle along the surface of the sphere. The particle loses contact with the sphere at the point $B$, where the angle $AOB$ is $\theta$.

After leaving the surface of the sphere, the particle moves freely under gravity before striking the horizontal ground with speed $3u$ at an angle $\beta$ to the horizontal (see diagram).

Find the value of $\beta$.