# 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.

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