OME 1971 Problem 6

The velocities of a submerged and surfaced submarine are, respectively, $v$ and $kv$ . It is situated at a point $P$ at $30$ miles from the center $O$ of a circle of $60$ mile radius. The surveillance of an enemy squadron forces him to navigate submerged while inside the circle. Discuss, according to the values of $k$ , the fastest path to move to the opposite end of the diameter that passes through $P$ . (Consider the case particular $k =\sqrt5$ .)