The velocities of two particles of masses `m_(1)` and `m_(2)` relative to an observer in an inertial frame are `v_(1)` and `v_(2)`. Determine the velocity of the center of mass relative to the observer and the velocity of each particle relative to the center of mass.

Explanation

From definition
`v_(com)=(dr_(com))/(dt)=(1)/(M)sum_(i)m_(i)(dr_(i))/(dt)=(sum_(i)m_(i)v_(i))/(M)`
The velocity of the center of mass relative to the observer ia
`v_(com)=(m_(1)v_(1)+m_(2)v_(2))/(m_(1)+m_(2))`
The veolcity of each particle relative to the center of mass (figure) using the relative motion equations for velocities is
`v_(1c)'=v_(1)-v_(com)=v_(1)-(m_(1)v_(1)+m_(2)v_(2))/(m_(1)+m_(2))`
`=(m_(2)(v_(1)-v_(2)))/(m_(1)+m_(2))=(m_(2)v_(12))/(m_(1)+m_(2))`
`v_(2c)'=v_(2)-v_(com)=(m_(1)(v_(1)-v_(2)))/(m_(1)+m_(2))=-(m_(1)v_(12))/(m_(1)+m_(2))`
where `v_(12)=v_(1)-v_(2)` is the relative velocity of the two particles.
Thus, in the C-frame, the two particles appear to be moving in opposite directions with velocities inversely proportional to their masses.
Also relative to the center of mass, the two particles move with equal but opposite momentum since
`p_(1)'=m_(1)v_(1c)'=(m_(1)m_(2)v_(12))/((m_(1)+m_(2)))=p_(2)'`
The expressions for two particle problems are much simpler when they are related to the C-frame of reference.