Laws of conservation in special relativity

[vela]

in COM frame where \vec{p}=\vec{0}
I get
2E_0=\sqrt{E_0^2+p_1'^2c^2}+\sqrt{E_0^2+p_2'^2c^2}+...+\sqrt{E_0^2+p_n'^2c^2}
\vec{0}=\vec{p}'_1+\vec{p}'_2+...+\vec{p}'_n
and it does not have sense. Where is the mistake? It is obvious that the mistake is in the equation
2E_0=\sqrt{E_0^2+p_1'^2c^2}+\sqrt{E_0^2+p_2'^2c^2}+...+\sqrt{E_0^2+p_n'^2c^2}. Could you please help me to understand where is the error?

Before the collision, the two protons are not at rest, so the total energy is $2E'$, not $2E_0$. After the collision, the particles are at rest, so the energy of each particle is just $E_0$. The equation you wrote down has the initial protons at rest and the final particles all moving.

 

[Vitani1]

What you did in your first post (I think) is okay. Given that you are only required to compute the threshold energy to create an antiproton/proton pair (by threshold I mean you are not concerned with the kinetic energy of the pair after creating - you just need to calculate the energy required for them to exist) then your answer of 82 needs to be divided by 2 since you are dealing with pairs. At least this is what I think.