Trying to understand the conservation of relativistic momentum, I thought of this problem.(adsbygoogle = window.adsbygoogle || []).push({});

It's very simple, and possibly my mistake is embarrassing so I apologize in advance

Two particles of identical mass m and speeds v and -v in some frame S collide inelastically and they stop. The relativistic momentum in S is zero all the time.

Now, in system S' moving at -v, the initial momentum should be

p = m v' / √(1-v'^{2}/c^{2}) [1]

where v' is the velocity of the other particle in S'

v' = 2v / (1+v^{2}/c^{2}) [2]

if I substitute [2] into [1] I can get the momentum before impact as a function of v. After some manipulation I get

p = 2mv / (1 - v^{2}/c^{2})

But the momentum in S' after the impact is that of a particle of mass 2m moving with speed v. So its momentum should be

p = 2mv / √(1 - v^{2}/c^{2}))

so - where did I go wrong?

Thank you

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# A simple problem of conservation of momentum

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