Two-photon production from 0.8c electron-positron annihilation

[Krappy]

Homework Statement

An electron traveling at 0.8c annihilates with a positron that is at rest, producing two photons. One of the produced photons travels in the direction of the electron. Determine the energy of each of the photons.

Homework Equations

$$p_{photon} = \frac{E}{c}$$

$$P_{electron} = \frac{m_e v}{\sqrt{1-v^2/c^2}}$$

The Attempt at a Solution

$$p_i = p_f$$

$$\frac{m_e v}{\sqrt{1-v^2/c^2}} = 2p_{photon}$$

$$E = \frac{m_e 0.9 c^2}{2 \sqrt{1-0.81}} = 8.46 \times 10^{-14} J$$First of all, I already searched about this frequent topic, but didn't manage to solve the issues. My "issue" is regarding the final momentum. I assume that both photons will have the same momentum, but I don't think that's right.

Thank You ;)

 

[diazona]

No, it isn't right. I suppose you mean you assumed that both photons will have the same magnitude of momentum, though? That's only true if the total momentum is zero. You could analyze the problem in a reference frame where that is true.

 

[Krappy]

Yes, sorry, I meant the same magnitude.

Now I tried to solve this with using both Energy and Momentum conservation and it gave me something like $$10^19 eV$$. But I think there's an easier way to solve it. Could you please post your resolution?

Regards

 

[vela]

Do you know how to use four-vectors?

 

[Phrak]

Go to the center of mass frame. Solve the problem, and transform back.