Question

Consider a collection of large number of particles, each moving with a speed $v$. The direction of velocity is randomly distributed in the collection. The magnitude of the relative velocity between a pair of particles averaged over all the pairs in the collection :

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

Answer 1

Answer: (D)

$\frac{4v}{\pi }$

$\overrightarrow{V_{rel}}=\overrightarrow{V_A}-\overrightarrow{V_B}$
$|V_{AB}|=\sqrt{v^2+v^2-2v^2\cos \theta }$
$=2v\sin \frac{\theta }{2}$
$\left(v_{rel}\right)average=\frac{{\int }_0^{2\pi }{v}_{rel}d\theta }{{\int }_0^{2\pi }d\theta }=\frac{1}{2\pi }{\int }_0^{2\pi }2v\sin \frac{\theta }{2}d\theta$
$=\frac{2v}{\pi }|-\cos \left(\frac{\theta }{2}\right){|}_0^{2\pi }$
$=\frac{4v}{\pi }$