A ball is projected with a velocity u at an elevation from a point distance d from a smooth vertical wall in a plane perpendicular to it. After rebounding from the wall, it returns to the point of projection, find the maximum distance d for which the ball can return to the point of projection.
The vertical force on the ball is only mg throughout its motion but during impact, it experiences a horizontal force from the wall.
⇒ we can use Uyt−12gt2 = Sy
Let t = total time of flight ⇒ 0 = u sin a t−12gt2 ⇒ t = 2u sin αg
Due to impact with the wall at B, the normal component (i.e. horizontal component) of velocity is reversed and becomes e times.
⇒ horizontal velocity before impact = u cosα and horizontal velocity after impact = eucosα
⇒time taken to reach the wall = t1 = ducosα
and time taken to come back to O from B = t2 = deucosαt1 + t2 = t
⇒ du cosα + deu cosα = 2u sinαg ⇒ u2sin 2α = gd[1 + 1e]
⇒ gdu2[1 + 1e]≤ 1 ⇒ d≤eu2g(1+e)