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 $\theta$ = angle between the lines]

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

${\overrightarrow{V}}_{cm}=\frac{m_1{\overrightarrow{v}}_1+m_2{\overrightarrow{v}}_2}{m_1+m_2}$
(\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)}\)
$V_{cm}=\frac{nmv\sqrt{(1+cos\theta {)}^2+(sin\theta {)}^2}}{m(1+n)}=\frac{nv\sqrt{1+co{s}^2\theta +2cos\theta +si{n}^2\theta }}{(1+n)}=\left(\frac{nv}{n+1}\right)\sqrt{2cos\frac{\theta }{2}}$
$V_{cm}=\frac{2nvcos\frac{\theta }{2}}{n+1}$