Electron-positron collisions and decay in Special Relativity

[Keano16]

Homework Statement

A particle physicist seeks to create a new fundamental particle with rest energy
200GeV by colliding electrons and positrons. What is the minimum positron energy
required when electrons and positrons traveling in opposite directions with equal speeds are
collided together?

The new particle is produced using this method with the minimum necessary
energy, and rapidly decays into two identical particles of rest energy 91.2GeV

Homework Equations

E²-p²c²=m²c⁴

The Attempt at a Solution

I am trying to do this question using 4-momentum. For a positron I have it as (E, p1, 0, 0) and for the electron I have it as once again (E, p2, 0, 0) (we are allowed to approximate the mass of a positron to that of an electron), the resultant being (2E, p1+p2, 0, 0).

I then equate (2E)²-(p1+p2)²=(200x10³)² (Working in MeV)

However, the problem I get is that as these particles are of essentially equal mass, and moving in opposite directions with the same speed, does p1+p2 in the 4-momentum effectively become 0?

If so, I get 4E²=(200x10³)²
giving E=100GeV

I think there's something I am missing, as in the next part it says two particles of rest mass 91.2GeV are produced, which exceeds this energy. Or does this energy translate to kinetic energy for the 200GeV particle?

Any help would be appreciated.

 

[Fightfish]

There's no need to work so hard; simple Conservation of energy will solve the first part. Just 200/2 = 100GeV does the trick.
For the second part, the "missing energy" is indeed the kinetic energy of the decay products.

 

[Keano16]

Ah right I see now. Thanks!