A smooth sphere with centre $O$ and radius $a$ is fixed with a point $B$ of its surface in contact with a vertical wall. A particle $P$ of mass $m$ rests at the highest point $A$ of the sphere. It is slightly disturbed so that it moves from rest towards the wall in the plane $OAB$. When $P$ makes an angle $θ$ with $OA$ the velocity of $P$ is $\sqrt2ag(1-cosθ)$ and $P$ leaves the sphere when $cos θ = \frac{2}{3}$ when its speed is $\sqrt\frac{2ag}{3}$.
Show that P hits the wall at a height $\frac{1}{8}a(5\sqrt5 -9)$ above $B$.
(treat the particle as a projectile moving freely under gravity).
Once you know the take-off point and the velocity vector there, you can forget about the sphere: as the hint says, after that you have free projectile motion. The $x$ component of velocity is constant, so you can figure out the time until you hit the wall. Then you find the $y$ value for that time.