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Question

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.


A

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B

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C

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D

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Solution

The correct option is C


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)


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