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I had two objects with masses m1 and m2 and velocities v1 and v2.

I found the velocity of the centre of mass:

vcm = (m1v1+m2v2)/(m1+m2)

And found the velocities of the two objects relative to the cm.

u1 = v1-vcm

u2 = v2 - vcm

And used conservation of momentum in the center of mass frame. And magically I got that:

m1u1 + m2u2 = 0

which I am guessing is the nice property of carrying out the math in the cm-frame. I just don't understand what leads to the above. Why should the equation:

mu1 + mu2 = m1v1 + m2v2 - m1vcm - m2vcm = 0

It is surely pretty obvious for you, so try to explain it for me :) And try to explain why it is obvious physically in terms of the center of mass being the point where only external forces work.