A steel ball is suspended by a light in extensible string of length l from a fixed point `O`. When the ball is in equilibrium it just touches a vertical wall as shown in the figure. The ball is first taken aside such that string becomes horizontal and then released from rest. If coefficient of restitution is e, then find the maximum deflection of the string after `nth` collision.

Explanation

Let `v_(0)` be the spee do fhte ball just before first collisiion and `v_(1)` be the speed just after first collision.
From the definition of the coefficient of restitution.
`v_(1)=ev_(0)`………..i
Just before second collision seed of the ball will be `v_(1)` only. if `v_(2)` be the speed just after second collision then
`v_(2)=ev_(1)=e^(2)v_(0)`...............ii
Similarly `V_(n)=e^(n)v0`..............iii
If after `nth` collisionmaximum deflection of the strig from veritcal is `theta` then from COE, we get
`=1/2mv_(n)^(2)=mg(1-costheta)`
`implies e^(2n)v_(0)^(2)=2gl(1-costheta)`.........iv
also from COE `v_(0)^(2)=2gl....................v`
frm iv and `v` we get `theta=cos^(-1)(1-e^(2n))`