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…

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