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 borrom it collides with a fixed obstacle. The co-efficient of restitution is 'e'. Find the maximum angular deflection of the string after `n^(th)` collision.

Explanation

Let `v_(1), v_(2), v_(3)……v_(n)` be the velocities acquired by the ball immediately after respective collision
`v_(1)=ev_(0)`, where `v_(0)=sqrt(2gl)`
`v_(2)=e^(2)v_(0)`
`"….. ….. ….."`
`"….. ….. ….."`
`v_(n)=e^(n)v_(0)`
From law of COE
`(1)/(2)mv_(n)^(2)=mgl(1-cos theta)rArr e^(2n)v_(0)^(2)=2gl(1-cos theta)`
`rArr e^(2n)xx2gl=2gl(1-cos theta)rArr theta = cos^(-1)(1-e^(2n))`.