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.