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.

Explanation

Fig. 460 shows the particles each moving with same speed `v` but in different directions. Consider any two particles having angle `theta` between the directions of their velocities.

Thus `vec(v)_("rel") = vec(v)_(B) - vec(v)_(A)`
i.e., `v_("rel") = sqrt(v^(2) + v^(2) - 2 v cos theta)`
or `v_("rel") = sqrt(2v^(2) (1 - cos theta)) = 2 v sin (theta//2)`
So averaging `v_("rel")` over all pairs
`bar(v)_("rel") = (int_(0)^(2 pi)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 [-cos (theta//2)]_(0)^(2pi))/(2 pi)`
or `bar(v)_("rel") = (4v//pi) gtv` [as `4//pi gt 1`]