I'm looking at this wikipedia derivation (link) for the threshold energy of a reaction of the form $$ 1+2 \rightarrow a + b + c $$
but I get to a different result because wikipedia says
$$ E_1 = \frac{E_{CM}^2-(m_1+m_2)^2}{2m_2} $$ but I get
$$ E_1 = \frac{E_{CM}^2-(m_1^2+m_2^2)}{2m_2} $$
who's right and who's wrong?
I get the exact same results for beta and gamma shown in the page but the last step seems to give me a different result. These are my steps $$ \gamma(E_1+m_2-\beta p_1) = E_{CM} $$ $$ \gamma (E_1 - \frac{p_1^2}{E_1+m_2}+m_2) = E_{CM} $$ $$ \gamma \frac{\left[ E_1(E_1+m_2) - p_1^2 + m_2(E_1+m_2)\right]}{E_1+m_2} = E_{CM} $$ $$ \frac{\gamma}{E_1+m_2}\left[ E_1^2-p_1^2 +m_2^2+ 2E_1 m_2\right] = E_{CM} $$ $$ m_1^2+m_2^2 + 2m_2E_1 = E_{CM}^2 $$ $$ E_1 = \frac{E_{CM}^2-m_1^2-m_2^2}{2m_2} $$