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I have a problem that is making me crazy. Consider the following collision

[tex]A + B \rightarrow C[/tex]

which results in both particles (A and B) being destroyed and C being created.

I know the rest mass of all particles. Also, in the lab system, B is stationary and A is moving toward B. What is asked for is the momentum of particle A needed to allow for particle C to be created.

So I used the following squared four vectors:

[tex](E_A + m_B, p_A)^2 = (m_C, 0)^2[/tex]

with

[tex]E_A = \sqrt{(p_A^2 + m_A^2)}[/tex]

and solved for [tex]p_A[/tex]. What I got was different from what I got when I used energy conservation with

[tex]E_A + E_B = E_C \Leftrightarrow \sqrt{(p_A^2 + m_A^2)} + m_B = m_C[/tex]

Then I realized that I completely neglected the momentum on the right side in my first approach, so instead I did

[tex](E_A + m_B, p_A)^2 = (m_C, p_A)^2[/tex]

([tex]p_A[/tex] on the right side due to momentum conversion)

and got to the same result as with the second approach. Which is correct?

To make things even more confusing, I remembered an example our professor gave us:

[tex]\gamma + p \rightarrow p + \pi_- + \pi_+[/tex]

and he used the following squared four vectors:

[tex](E_\gamma + m_p, p_\gamma)^2 = (m_p + 2*m_\pi, 0)^2[/tex]

Note that he completely neglected the momentum on the right side. If neglecting the momentum was wrong in my first example, why is it correct here? I don't get it, please, someone enlighten me :)

Thanks in advance

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# Momentum conservation in collisions.

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