Problem

$X$ and $Y$ are two fixed smooth vertical walls on a smooth horizontal surface. The walls are parallel and at a distance $d$ apart. The points $P_1$, $P_2$ and $P_3$ all lie on the surface.

A particle $Q$ is projected horizontally from the point $P_1$ on Wall $X$ with speed $u$, and moves along the surface. The particle $Q$ strikes Wall $Y$ at the point $P_2$. Immediately before the collision, the direction of motion of $Q$ makes an angle $\alpha$ with Wall $Y$, where $\sin\alpha=\frac15$. Immediately after the collision, the direction of motion of $Q$ makes an angle $\theta$ with Wall $Y$. The particle $Q$ then strikes Wall $X$ at the point $P_3$ (see diagram).

The time that it takes $Q$ to travel the distance $P_1P_2$ is $T$. The time that it takes $Q$ to travel the distance $P_2P_3$ is $kT$.

Find, in terms of $k$, the coefficient of restitution between $Q$ and Wall $Y$.