A ball is given velocity u = sqrt(3 gd) at an angle 45^(@) with horizontal. It strikes a wall at distance d and returns to its original position. Find the coefficient of restitution e.

A ball is given velocity u = sqrt(3 gd) at an angle 45^(@) with horizo

1.	A ball is given velocity u = sqrt(3 gd) at an angle 45^(@) with horizontal. It strikes a wall at distance d and returns to its original position. Find the coefficient of restitution e.
Answer» Solution :The velocity remains same in the vertical direction (along the SURFACE) and the velocity BECOMES `e` times and opposite in the horizontal direction (`_\|_^(ar)` to the surface) Time from O to `A_(1) , t_(1) = (d)/( u cos 45^(@))` (uniform MOTION) Time from A to `O, t_(2) = (d)/( e u cos 45^(@))` TOTAL time `t = t_(1) + t_(2) = (d)/( u cos 45^(@)) (1 + (1)/(e))` Time for vertical motion (O to O) `0 = u sin 45^(@) T - (1)/(2) g T^(2) rArr T = ( 2 u sin 45^(@))/(g)` `t = T` `(d)/( u cos 45^(@)) (1 +(1)/(e)) = ( 2 u sin 45^(@))/(g)` `d(1 + (1)/(e)) = ( 2u^(2) sin 45^(@) cos 45^(@))/(g)` `= (u^(2))/(g) sin 90^(@) = ( 3 gd)/(g) = 3d` `1 + (1)/(e) = 3 rArr (1)/(e ) = 2 rArr e = (1)/(2)`