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 paris is

• A. π/4 v
• B. π/2 v
• C. 3/π v
• D. 4/π v

Answer 1

Answer: (D)

4/π v

Answer 2

Let α 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 α/2 ( 1 – cos α = 2 sin²α/2)

Average relative velocity is given by

average v_r = $\int_{0}^{2\pi}\frac{2v\left( \sin\alpha/2 \right)d\alpha}{\int_{0}^{2\pi}{d\alpha}} = \frac{4}{\pi}v$