A ball is projected from the ground with speed u at an angle `alpha` with horizontal. It collides with a wall at a distance .a. from the point of projection and returns to its original position. Find the coefficient of restitution between the ball and the wall.

Explanation

As we have discussed in the theory, the horizontal component of the velocity of ball during the path OAB is `u cos alpha ` while in its return journey BCO it is `eu coa alpha.` fThe time of fight T also remains unchanged. Hence,
`T = t _(OAB)+ t _(BCO) or (2 u sin alpha )/( g) = (a)/( u cos alpha ) + (a)/( eu cos alpha )`
(or) `(a)/( eu cos alpha ) = (2 u sin alpha )/(g ) (a)/( u cos alpha )`
(or )` (a )/( eu cos alpha ) = (2u ^(2) sin alpha cos alpha - ag)/(gu cos alpha)`
`therefore e = (ag)/( 2u ^(2) sin alpha cos alpha - ag ) or e = (1)/(( ( u ^(2) sin 2 alpha )/( ag )-1))`