Problem

A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$. Initially $P$ is held with the string taut and making an angle of $60^\circ$ with the upward vertical through $O$. The particle $P$ is projected perpendicular to the string in a downwards direction with speed $\sqrt{17ag}$. It then starts to move along a circular path in a vertical plane with centre $O$ (see diagram). At the lowest point of its path, vertically below $O$, the particle $P$ collides with a stationary particle $Q$.

(a) Find, in terms of $a$ and $g$, an expression for the speed of $P$ immediately before the collision with $Q$.

As a result of the collision, $P$ rebounds and moves back along a circular path with centre $O$. The string becomes slack when $P$ reaches the point on the circle vertically above $O$.

(b) Find, in terms of $a$ and $g$, an expression for the speed of $P$ immediately after the collision with $Q$.

The mass of particle $Q$ is $km$ and the collision between $P$ and $Q$ is perfectly elastic.

(c) Find the value of $k$.