The Elongated Arc

A projectile is shot at t=0t=0 on an infinite horizontal plane with a speed uu at an angle θ\theta to the horizontal. The coefficient of restitution between the projectile and the ground is e1e \neq 1. If L0L_0 is the total arc length traversed by the projectile till t=t0t=t_0 and β\beta is the angle of projection such that L0L_0 is maximized , evaluate β\displaystyle \lfloor \beta \rfloor .

Details and Assumptions:

  • uu is a constant while θ\theta can vary.

  • t0t_0 is the time beyond which the projectile only has a velocity parallel to the ground i.e., tt0 \displaystyle \forall t \geq t_0 , velocity of projectile has only a horizontal component.

  • Neglect all forces other than that due to gravity.

  • β\beta is an acute angle measured in degrees.

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