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]

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Solution

The correct option is D

${\to V}_{c m} = \frac{m_{1} {\to v}_{1} + m_{2} {\to v}_{2}}{m_{1} + m_{2}}$
${\to v}_{c m} = \frac{m n v^i + m n v c o s θ^J}{m + n m} \Rightarrow V_{c m} = \frac{\sqrt{(n m v + m n v c o s θ)^{2} + (n m v s i n θ)^{2}}}{m (1 + n)}$
$V_{c m} = \frac{n m v \sqrt{(1 + c o s θ)^{2} + (s i n θ)^{2}}}{m (1 + n)} = \frac{n v \sqrt{1 + c o s^{2} θ + 2 c o s θ + s i n^{2} θ}}{(1 + n)} = (\frac{n v}{n + 1}) \sqrt{2 c o s \frac{θ}{2}}$
$V_{c m} = \frac{2 n v c o s \frac{θ}{2}}{n + 1}$

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