Consider a collection of a large number of particles each with speed v. The direction of velocity is randomly distributed in the collection. Show that the magnitude of the relative velocity between a pair of particles averaged over all the pairs in the collection is greater than v.

Options

A. `4v//pi`
B. greater than `4v // pi`
C. less than `4v //pi`
D. zero

Answer

A

Explanation

Figure shows the particles each moving with speed v but in different directions. Consider any two particles of their velocities.
The , velocity of B w.r.t , A, i.e,
`V_(AB) = V_(B) - V_(A)`
(in case of a pair of particle A and particle B)
i.e,`V_(BA) = sqrt(v^(2) +v^(2) - 2"vv" cos theta)`
` rArr V_(BA) = sqrt(2v^(2)(1-cos theta))`
`= 2v sin ((theta)/(2))`
So, relative vleocity of overall pairs
` v_("trel") =(int_(0)^(2pi) v_("rel") d theta)/(int_(0)^(2pi) d theta)`
`=(int_(0)^(2pi) 2v sin (theta//2) d theta)/(int_(0)^(2pi) d theta)`
`=(2v xx 2 [-cos.(theta)/(2)]_(0)^(2pi))/(2pi)`
`therefore " "v_("rel") = (4v)/(pi)`