Solution to 2001 Problem 99

We apply conservation of (relativistic) energy and conservation of (relativistic) momentum. Let E denote the energy of the photon that was destroyed. The initial momentum is E/c and the initial energy is

\begin{align*}E_{\text{init}} = E + mc^2\end{align*}
We are told that the positron and the two electrons move off with the same speed at the end. So, they must each have a momentum equal to E/(3c). Therefore, the total energy at the end is
\begin{align*}3 \sqrt{m^2 c^4 + E^2/9}\end{align*}
Equating the initial and final energies and then squaring both sides of the equation, we find that
\begin{align*}E^2 + 2 E m c^2 + m^2 c^4 = 9 m ^2 c^4 + E^2 \Rightarrow E = \boxed{4 m c^2}\end{align*}
Therefore, answer (D) is correct.

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