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A large number of particles are moving e...

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?

A

v

B

`(pi//4)v`

C

`(4//pi)v`

D

Zero

Text Solution

Verified by Experts

The correct Answer is:
C
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`
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