Obtain an expression for the velocity of centre of mass for n particles of system.

Explanation

Consider a system of n particles. Let `vec(r_(1)),vec(r_(2)),vec(r_3),...vec(r_(n))` be the position vectors of the particles of masses `m_(1),m_(2),m_(3),m_(n)` respectively w.r.t. the origin of a co-ordinate system.
If `vecR` is the position vector of centre of mass, then
`vec(r)_(cm)or vecR=(m_(1)vec(r_(1))+m_(2)vec(r_(2))+...m_(n)vec(r_(n)))/(m_(1)+m_(2)+...m_(n))`
`vec(R)=(m_(1)vec(r_(1))+m_(2)vec(r_(2))+...m_(n)vec(r_(n)))/(M)`
`therefore MvecR=m_(1)vec(r_(1))+m_(2)vec(r_(2))+...m_(n)vec(r_(n))" "...(1)`
Assuming that the mass of the system does not change with time differentiating (1) w.r.t. time,
`M(dvecR)/(dt)=m_(1)(dvec(r_(1)))/(dt)+m_(2)(dvec(r_(2)))/(dt)+...m_(n)(dvec(r_(n)))/(dt)`
but `(dvecR)/(dt)=vecV` velocity of centre of mass,
`(dvec(r_(1)))/(dt),(dvec(r_(2)))/(dt),....,(dvec(r_(n)))/(dt)` respectively are the velocities `vec(v_(1)),vec(v_(2)),...vec(v_(n))` of n particle.
`therefore MvecV=m_(1)vec(v_(1))+m_(2)vec(v_(2))+...m_(n)vec(v_(n))" "...(2)` is the velocity of centre of mass for given system. This formula can be written as also, `vecV=(m_(1)vec(v_(1))+m_(2)vec(v_(2))+...m_(n)vec(v_(n)))/(m_(1)+m_(2)+...m_(n))`