Question

A small ball is suspended from a fixed point $O$ by means of a light and inextensible string of length $l$. The ball is first taken aside such that string becomes horizontal and then released from rest. At the bottom it collides with a fixed obstacle. The coefficient of restitution is $e$. Find the maximum angular deflection of the string after $n$th collision

Answer 1

Let $v_1,v_2,v_3,...v_n$ be the velocities acquired by the ball immediately after respective collision
$v_1=e{v}_0$, where $v_0=\sqrt{2gl}$
$v_2=e^2{v}_0$
.......
$v_n=e^n{v}_0$
From law of COE
$\frac{1}{2}m{v}_n^2=mg(1-\cos \theta )\Rightarrow e^{2n{v}_0^2}=2g(1-\cos \theta )\Rightarrow e^{2n}\times 2gl=2gl(1-\cos \theta )\Rightarrow \theta ={\cos }^{-1}(1-e^{2n})$