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Question

A ball strikes a wall with a velocity u at angle θ with the normal to the wall surface and rebounds from it at an angle β with the surface. Then:

A
(θ+β)<90, if the wall is smooth
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B
if the wall is rough, coefficient of restitution e=tanβ/cotθ
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C
if the wall is rough, coefficient of restitution e<tanβ/cotθ
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D
(θ+β)>90, if the wall is smooth
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Solution

The correct option is C if the wall is rough, coefficient of restitution e<tanβ/cotθ


If wall is smooth, vcosα=eucosθ ...(i)
& vsinα=usinθ ... (ii)
where α=90β
Dividing the above two equations,
vcotα=eucotθ

Squaring and adding (i) and (ii),
v2=u2+u2(e21)cos2θ
v<u [since e<1]
Therefore, from (ii),
α>θ
α+β>θ+β
90>θ+β

If the wall is rough, then vsinα<usinθ
i.e vcosβ<usinθ ...(iii)
& vsinβ=eucosθ ... (iv)
From (iii) and (iv),
tanβ>ecotθ
or e<tanβ/cotθ

Hence, options (a) and (c) are correct.

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