PX155 - I8 - particle accelerators

$E^{2} - p^{2} c^{2}$ is lorentz invariant, and conserved in collisions because $E$ and $p$ are conserved
eg: a particle of rest mass, $m_{0} = 1 G e V / c^{2}$ , is accelerated to a total energy, $E = 100 G e V$ , and collided with an identical but stationary particle. calculate the rest mass of the new particle created from these two particles
- conserving energy: $E_{f} = E_{0} + E$
- conserving momentum: $p_{f} = p$

M_{0}^{2} c^{4} = E_{f}^{2} - p_{f}^{2} c^{2} = (E_{0} + E)^{2} - p_{f}^{2} c^{2}

- we have $E_{0}^{2} = E^{2} - p^{2} c^{2}$ : $$M_{0}^{2}c^{4}= 2E_{0}^{2}+ 2E_{0}E

$ $ ∴ E_{0 f} = M_{0} c^{2} = \sqrt{2 \times 1 + 2 \times 100 \times 1} = 14.2 G e v / c^{2}