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Sorry, I don't have seen your example yet, anyway, concerning the transformation from a ref. frame where both particle 1 and particle 2 have non zero velocity, I find, instead of equation (33) of "Introduction to Relativistic Collisions":neilparker62 said:...

For point 2 I would refer you to the worked example where the pre-collision mass is that of the two colliding protons and the post-collision mass is that of the two protons plus the 'generated' kaons.

##{E_3^0}^2={E_1^0}^2+{E_2^0}^2+2E_1^0 E_2^0+2T_1E_2^0+2T_1T_2+2T_2E_1^0-2c^2\vec{p_1}·\vec{p_2}## (*)

Then we change frame on K', co-moving with particle 2; ##\vec{v_1}'## is now the relativistic relative velocity between particle 1 and particle 2 and:

##T_2'=\vec{p_2}'=0##

so (*) becomes equal to (33).

I apologize if you have already described this in your last example.

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