Question

Two masses, $nm$ and $m$, start simultaneously from the intersection of two straight lines with velocities $\nu$ and $n\nu$ respectively. It is observed that the path of their bisecting the angle between the given straight lines. Find the magnitude of the velocity of centre of mass. [Here $\theta =$ angle between the lines].

Answer 1

${\overrightarrow{\nu }}_{cm}=\frac{m_1{\overrightarrow{\nu }}_2}{m_1+m_2}$
${\overrightarrow{\nu }}_{cm}=\frac{nm\nu \hat{i}+mn\nu cos\theta \hat{i}+nm\nu sin\theta \hat{j}}{m(1+\pi )}$
${\nu }_{cm}=\frac{\sqrt{(nm\nu +mn\nu cos\theta {)}^2+(nm\nu sin\theta {)}^2}}{m(1+\pi )}$
${\nu }_{cm}=\frac{nm\nu \sqrt{(1+cos\theta {)}^2+(sin\theta {)}^2}}{m(1+n)}$
$=\frac{n\nu \sqrt{1+co{s}^2\theta +2cos\theta +si{n}^2\theta }}{(1+n)}$
$\left(\frac{n\nu }{n+1}\right)\sqrt{2cos\frac{\theta }{2}}$
${\nu }_{cm}=\frac{2n\nu cos\frac{\theta }{2}}{n+1}$