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Centre of mass frame collisions?

Hello TSR scholars :smile:. I do not understand why this statement I noted from a lecturer is true:

"The total momentum in the centre of mass frame is always 0 since there are no external forces"
[in reference to our topic of 1 dimensional, 2 particle collisions without external forces]

Here is what I understand thus far (screen shot below). Please note that "1or2" subscript refers to whether the first or second particle, and the prime denotes a velocity in the centre of mass frame:

Screen Shot 2015-12-30 at 19.55.36.png

The problem is that sub'ing the first two lines into the bottom two equations, do not give = 0 as per the RHS in either case.

I expect I'm missing something obvious somewhere! Thanks in advance for any help.
I think some more context may help :smile:
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Avatar for Sm0key
Sm0key
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I think I managed to work it out myself in the end.

I just pictured the CoM to be sat at the origin hence the Com has 0 velocity by definition thus at any point in time (pre or pst collision) it's p = 0.

And I managed to get those equations to = 0 too. :smile:

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