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Question

Two masses, nm and m, start simultaneously from the intersection of two straight lines with velocities v and nv respectively. It is observed that the path of their centre of mass is a straight line bisecting the angle between the given straight lines. Find the magnitude of the velocity of centre of mass.[Here θ = angle between the lines]


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 D


Vcm=m1v1+m2v2m1+m2
(\vec{v}_{cm} = \frac{mnv\hat{i}+m~nv~cos\theta\hat{J}}{m+nm} \Rightarrow V_{cm}=\frac{\sqrt{(nmv+mnv~cos\theta)^2+(nmv~sin\theta)^2}}{m(1+n)}\)
Vcm=nmv(1+cosθ)2+(sinθ)2m(1+n)=nv1+cos2θ+2cosθ+sin2θ(1+n)=(nvn+1)2cosθ2
Vcm=2nv cosθ2n+1


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