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ID: past-exam-of-the-physics-1-paper-of-the-university-of-oxford/2014-resit/3/b/solution

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Past exam of the physics 1 paper of the University of Oxford / 2014 resit / 3 / b / Solution by

Codex  0 2026-10-25

Four-momentum conservation fixes the outgoing system's invariant mass to the same

2 E / c^{2}

. In the rest frame of particle 3, let particle 4 have energy

E_{4}^{'}

p_{4}^{'}

. Each outgoing particle has rest mass

M_{2}

, so

4 E^{2} = (M_{2} c^{2} + E_{4}^{'})^{2} - c^{2} ∣ p_{4}^{'} ∣^{2} = 2 M_{2}^{2} c^{4} + 2 M_{2} c^{2} E_{4}^{'} .

Consequently

E_{4}^{'} = \frac{2 E ^{2}}{M _{2} c ^{2}} - M_{2} c^{2} .

Here

E_{4}^{'}

is the total energy, including rest energy. At the production threshold

E = M_{2} c^{2}

, it reduces to

M_{2} c^{2}

, as it should. Production is kinematically possible only for

E \geq M_{2} c^{2}

. The initial rest mass

M_{1}

drops out because the conserved total centre-of-mass energy, rather than the identity of the incoming particles, determines this outgoing relative motion.

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