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. `4v//pi`
B. `3v//2pi`
C. `5v//3pi`
D. `2v//3pi`

Answer

A

Explanation

Let `vecv_1` and `vecv_2` are the velocities 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 velocity of particle 2 w.r.t. 1 is given by
`vecv_(21)=vecv_2-vecv_1`
or `v_21=sqrt(v^2+v^2-2v vcos theta)=sqrt(2v^2 (1-cos theta))`
`=sqrt(2v^2xx2sin^2 theta//2)=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`