An electron and a positron travelling at small speeds collide and...

The annihilation of an electron and a positron at small speeds (negligible kinetic energy) produces two gamma photons to conserve linear momentum. The initial momentum is approximately zero, so the two photons must travel in opposite directions with equal momenta, and therefore equal energies.

The total initial energy of the system is the sum of the rest mass energies of the electron and the positron:

$E_{\text{initial}} = m_e c^2 + m_e c^2 = 2m_e c^2$

The rest mass energy of an electron (or positron) is:

$m_e c^2 = 0.51 \text{ MeV}$

Total initial energy $= 2 \times 0.51 \text{ MeV} = 1.02 \text{ MeV}$

By conservation of energy, the sum of the energies of the two gamma photons must equal the total initial energy:

$2E_{\gamma} = 1.02 \text{ MeV}$

$E_{\gamma} = \dfrac{1.02 \text{ MeV}}{2} = 0.51 \text{ MeV}$

Thus, the energy of each resulting gamma photon is $0.51 \text{ MeV}$.