Problem 9.29

9.29 a).
We know that
$\Delta E = (\gamma_{1} - \gamma_{2}) m c^{2}$
For an electron,
m_e c² = .511 MeV
Solving for the change in energy, we have
$\Delta E = (\sqrt{\frac{1}{1 - .81}} - \sqrt{\frac{1}{1 - .25}}) (.511 MeV)$
$\Delta E = .582 MeV$

b)
Similarly, we have
$\Delta E = (\gamma_{1} - \gamma_{2}) m c^{2}$
$\Delta E = (\sqrt{\frac{1}{1 - (.99)^{2}}} - \sqrt{\frac{1}{1 - .81}}) (.511 MeV)$
$\Delta E = 2.45 MeV$