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

Options

A. v
B. `(pi//4)v`
C. `(4//pi)v`
D. Zero

Answer

C

Explanation

Relative velocityt, `v_(r)=|vec(v)_(1)-vec(v)_(2)|` where `v_(1)=v_(2)=v`
If angle between them be `theta`, then `v_(r)=sqrt(v^(2)+v^(2)-2v^(2) cos theta)=sqrt(2v^(2)(1-cos theta))=2v sin (theta/2)`
Hence, average relative velocity `vec(v)_(r)=(underset(0)overset(2pi)(int)2v" sin"theta/2d theta)/(underset(0)overset(2pi)(int)d theta)=(4v)/pi`