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 particle has same speed, so `v_(1)=v_(2)=v`. The relative of particle 2 w.r.t. 1 is given by `vecv_(21)=vecv_(2)-vecv_(1)` or `vev_(21)=sqrt(v^(2)+v^(2)-2v cos theta)=2v sin (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)=1273vrArrgtv`