A ball is projected from a point in a horizontal plane so as to strike a vertical wall at right angle `A`.after rebounding from the wall it strikes the horizontal plane once and returns to the point of projection just before second collision at horizontal surface. Find co-efficient of restitution for the two collisions, assuming it to be same for both the collision.

Explanation

Let the distance of the point of projection be `'a'` from the wall and angle of projection be `theta` and `u` respectively. As the ball strikes the wall at right angle, therefore `a=` half the horizontal range
`rArra=(u^(2)sin2theta)/(2g)`……(`i`)
and time of flight is `(usintheta)/(g)`
After striking the wall, velocity of the ball becomes `eu costheta` along horizontal.
Now `NB=eu costhetaxx(usintheta)/(g)=(eusin2theta)/(2g)=ae`.
`OB=a-ae`.......(`ii`)
After striking at `B` the vertical component becomes `eu sin theta` and the horizontal and the horizontal component remains unchanged.
Now the tme of flight from `B` to `O` is given by
`T=(2eu sin theta)/(g)`
Range of the projectile is `eu cos theta.(2eu sintheta)/(g)=(e^(2)u^(2)sin2theta)/(g)`
Now `(e^(2)u^(2)sin2theta)/(g)=a-ae=a(1-e)=(u^(2)sin2theta)/(2g)(1-e)`(from `1`)
`2e^(2)=1-e`
`2e^(2)+2e-e-1`
`2e(e+1)-1(1+e)=0`
`rArr(e+1)(2e-1)=0`
`e=(1)/(2)`
As `e= -1` is not a admissible.