Question 39 of 183

A bead $P$ sliding on a frictionless semi-circular string $(A C B)$ and it is at point $S$ at $t =0$ and at this instant the horizontal component of its velocity is $v$. Another bead $Q$ of the same mass as $P$ is ejected from point $A$ at $t=0$ along the horizontal string $A B$, with the speed $v$, friction between the beads and the respective strings may be neglected in both cases. Let $t_{P}$ and $t_{Q}$ be the respective times taken by beads $P$ and $Q$ to reach the point $B$, then the relation between $t_{P}$ and $t_{Q}$ is