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A cylindrical hall has a horizontal smooth floor. A ball is projected along the floor from A point on the wall in a direction making an angle θwith the radius through the point the ball returns back to the initial point after two impacts with the wall. If the coefficient of restitution is e then tan2θ will be

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a

1+e+e2e3

b

1+ee2

c

e21+e

d

e31+e+e2

answer is D.

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Detailed Solution

tan⁡α=usin⁡θeucos⁡θ=tan⁡θe∴ tan⁡α=1etan⁡θ∴ tan⁡β=1etan⁡θ2θ+2α+2β=180∘α+β=(90−θ)tan⁡(α+β)=cot⁡θ⇒tan⁡α+tan⁡β1−tan⁡αtan⁡β=1tan⁡θ⇒1tan⁡θ+1e2tan⁡θ1−1e3tan2⁡θ=1tan⁡θ⇒tan2⁡θ1e+1e2=1−1e3tan2⁡θ⇒tan2⁡θ1e+1e2+1e3=1tan2⁡θ11e+1e2+1e3=e31+e+e2

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A cylindrical hall has a horizontal smooth floor. A ball is projected along the floor from A point on the wall in a direction making an angle θwith the radius through the point the ball returns back to the initial point after two impacts with the wall. If the coefficient of restitution is e then tan2θ will be