A projectile is shot at $t=0$ on an infinite horizontal plane with a speed $u$ at an angle $\theta$ to the horizontal. The coefficient of restitution between the projectile and the ground is $e \neq 1$. If $L_0$ is the total arc length traversed by the projectile till $t=t_0$ and $\beta$ is the angle of projection such that $L_0$ is maximized , evaluate $\displaystyle \lfloor \beta \rfloor$.

**Details and Assumptions:**

$u$ is a constant while $\theta$ can vary.

$t_0$ is the time beyond which the projectile only has a velocity parallel to the ground i.e., $\displaystyle \forall t \geq t_0$, velocity of projectile has only a horizontal component.

Neglect all forces other than that due to gravity.

$\beta$ is an acute angle measured in degrees.

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