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

Step-by-step solution

Consider any two particles whose velocity vector particles making angle θ . Relative velocity v AB = v A - v B ⇒     v rel = v 2 + v 2 - 2 v 2  cos  θ ⇒     v rel = 2 v   sin θ 2 Average of v rel = ∫ 0 2 π v rel  d θ ∫ 0 2 π  d θ = 4 v π