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Question

A ball of mass 'm' moving with a velocity 'v' undergoes an oblique elastic collision with another ball of the same mass 'm' but at rest. After the collision, if the two balls move with different speeds after the collision, the angle between their directions of motion will be

A
Less than
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B
More than
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C
Exactly
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D
Exactly
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Solution

The correct option is C Exactly
Let v1 and v2 be the speeds of the balls after the collision. Then Equation (ii) above gives
v1sinθ1=v2sinθ2 (v)
Also, conservation of kinetic energy gives 12mu2=12mv21+12mv22
or u2=v21+v22 (vi)
Also Equation (i) above becomes
mu=mv1cosθ1+mv2cosθ2
or v2cosθ2=uv1cosθ1 (vii)
Squaring (v) and (vii) and adding we get v22=u2+v212uv1cosθ1 (viii)
Eliminating v2 between (vi) and (vii), we get cosθ1=v2u (ix)
Eliminating v2 between (v) and (vii) we get tanθ2=sinθ1(uv1cosθ1) (x)
Using (ix) in (x), we get
tanθ2=sinθ1(1cosθ1cosθ1)=sinθ1cosθ1(1cos2θ1) =cotθ1=tan(90θ1)
or θ2=90θ1orθ1+θ2=90.
Hence the correct choice is (c).

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