Question

A large number of particles are moving towards each other with velocity $V$ having directions of motion randomly distributed. What is the average relative velocity between any two particles averaged over all the pairs?

• A. $4V/\pi$
• B. $4\pi V$
• C. $V$
• D. $\pi V/4$

Answer 1

The correct option is A $4V/\pi$

${\overline{V}}_{rel}=\overline{V_1}-\overline{V_2}$
$|{\overline{V}}_{rel}|=\sqrt{V_1^2+V_2^2-2V_1{V}_{2}cos\theta }$
$=\sqrt{2V^2-2V^{2}cos\theta }$
$=\sqrt{2}V\sqrt{1-cos\theta }$
$=\sqrt{2}V\left(\sqrt{2}sin\frac{\theta }{2}\right)=2Vsin\frac{\theta }{2}$
$<V_{rel}>=\frac{{\int }_0^{2\pi }{V}_{rel}d\theta }{{\int }_0^{2\pi }d\theta }$
$=\frac{{\int }_0^{2\pi }2Vsin\frac{\theta }{2}d\theta }{{\int }_0^{2\pi }d\theta }$
$=\frac{4V{[-cos\frac{\theta }{2}]}_0^{2\pi }}{2\pi }$
$=\frac{2V}{\pi }[-cos(\pi )+cos0]$
$=\frac{4V}{\pi }$