The Elongated Arc

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.