Past exam of the mathematics course of the University of Cambridge / 2002 / ib / Paper 4 / 9D (Special Relativity)

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- Solution 9D

Solution

Work in the parent's rest frame. Momentum conservation makes the final spatial momenta equal and opposite:

p_{m} = - p_{0}

with magnitude

p

. The massless energy is

E_{0} = p c

, while the massive energy–momentum relation is

E_{m}^{2} = m^{2} c^{4} + p^{2} c^{2}

. Conservation of energy gives

E_{m} + p c = M c^{2}

. Squaring

E_{m} = M c^{2} - p c

and cancelling the

p^{2} c^{2}

terms gives

p = \frac{M ^{2} - m ^{2}}{2 M} c, E_{0} = \frac{M ^{2} - m ^{2}}{2 M} c^{2}, E_{m} = \frac{M ^{2} + m ^{2}}{2 M} c^{2} .

The two directions are arbitrary but opposite in this frame. These are total energies, including the massive daughter's rest energy, as required for relativistic two-body decay.

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