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Question: Consider a collection of large number of particles, each moving with a speed v. The direction of vel...

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

2v/π

C

πv/4

D

4v/π

Answer

4v/π

Explanation

Solution

V→\overrightarrow{V}rel = V→\overrightarrow{V}A − V→\overrightarrow{V}B

|VAB| = v2+v2−2v2cos⁡θ\sqrt{v^{2} + v^{2} - 2v^{2}\cos\theta}

= 2v sin θ/2

(vrel)average = ∫02πvreldθ∫02πdθ=12π∫02π2vsin⁡θ2dθ\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}

= 2vπ∣−cos⁡(θ2)∣02π=4vπ\frac{2v}{\pi}\left| - \cos\left( \frac{\theta}{2} \right) \right|_{0}^{2\pi} = \frac{4v}{\pi}