Consider a collection of a large number of particles each with speed `v` in a plane.The direction of velocity is randomly distributed in the collection.The magnitude of the average relative velocity of a particle with velocities of all other particles is

Options

A. `gtv`
B. `ltv`
C. `=v`
D. none of these

Answer

A

Explanation

Let `vecv_(1)` and `vecv_(2)` are the velocity of any two particles and `theta` is the angle between them.As each particles has same speed, so `v_(1)=v_(2)=v`
The relative velocity of particle `2` w.r.t `1` is given by `vecv_(21)=vecv_(2)-vecv_(1) or vec v_(21)=sqrt(v^(2)+v^(2)-2v v cos theta)`
`=2vsin (theta/2)`
As the velocities of the particles are randomly distributed, so `theta` varies from `0` to `2pi`.The magnitude of average velocity when averaged over all such pairs, Thus
`V_(21) = (int_(0)^(2pi) V_(21) d theta)/(int_(0)^(2pi) d theta) = (4V)/(pi) = 1.273 v rArr gt v`.