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A ball is projected from the ground with...

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.

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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))`
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