Question

A large number of particles are moving with same magnitude of velocity $v$ but having random directions. The average relative velocity between any two particles averaged over all the pairs is

• A. $\frac{\pi }{4}v$
• B. $\frac{\pi }{2}v$
• C. $\frac{3}{\pi }v$
• D. $\frac{4}{\pi }v$

Answer 1

The correct option is D $\frac{4}{\pi }v$

Let $\alpha$ be the angle between velocities of a pair of particles, then relative velocity is given by,
$v_r=\sqrt{v^2+v^2-2v\times v\times \cos \alpha }$
$=2v\sin \frac{\alpha }{2}(\because 1-\cos =2{\sin }^2\frac{\alpha }{2})$
Average relative velocity is given by
average $v_r={\int }_0^{2\pi }\frac{2v\left(\sin \alpha /2\right)d\alpha }{{\int }_0^{2\pi }d\alpha }=\frac{4}{\pi }v$