question_answer

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

A) Zero                

B)        Greater than v    

C)        Less than v         

D)        v

Correct Answer: B

Solution :

[b] The magnitude of relative velocity between two particles, with their velocity vector at angle \[\theta \] is \[|\overset{\to }{\mathop{\Delta }}\,v|=\sqrt{{{v}^{2}}+{{v}^{2}}-2v.v\cos \theta }=2v\sin \left( \frac{\theta }{2} \right)\] Here \[\theta \] varies between \[0\] and \[2\pi \]. Therefore, average value of \[|\overset{\to }{\mathop{\Delta }}\,v|\] will be \[<|\overset{\to v}{\mathop{\Delta }}\,|>=\frac{\int\limits_{0}^{2\pi }{2v\,\sin \left( \frac{\theta }{2} \right)}}{2\pi }=\frac{4}{\pi }v>v\]