Past exam of the physics 1 paper of the University of Oxford / 2017 / 9 / a

For a massive free particle with rest mass $m$ and velocity $\mathbf{v}$,The relativistic energy-momentum relation follows directly:This is the squared Minkowski norm of the four-momentum multiplied by $c^2$. The Lorentz transformation preserves that norm, so it has the same value in every inertial frame. In the particle's rest frame, the momentum is zero and the total energy is its rest energy $m{c}^2$.

Choose particle 1's incoming velocity positive and particle 2's negative. The final particle is stationary, so four-momentum conservation requires zero total incoming momentum:It follows that ${\gamma }_1{\beta }_1=1/\sqrt{2}$. Squaring gives ${\beta }_1^2/(1-{\beta }_1^2)=1/2$, henceThe initial total energy becomes the final rest energy. ThereforeThe extra invariant mass is the incoming kinetic energy retained as internal energy of the merged particle: $(M-3m)c^2$. It does not violate conservation of energy or four-momentum conservation; the sum of the separate rest masses is not a conserved quantity in an inelastic collision.