A stationary particle of rest mass `m_0` disintegrates into three particles with rest masses `m_1`, `m_2`, and `m_3`. Find the maximum total energy that, for example, the particle `m_1` may posses.

Explanation

We have
`E_1+E_2+E_3~~m_0c^2`, `vecp_1+vecp_2+vecp_3=0`
Hence `(m_0c^2-E_1)^2-c^2vecp_1^2=(E_2+E_3)^2-(vecp_2+vecp_3)^2c^2`
The `L.H.S. =(m_02c^4-E_1)^2-c^2vecp_1=(m_0^2+m_1^2)c^4-2m_0c^2E_1`
The R.H.S. is an invariant. We can evaluate it in any frame. Choose the CM frame of teh particles 2 and 3.
In this frame `R.H.S.=(E_2^'+E_3^')^2=(m_2+m_3)^2c^4`
Thus `(m_0^2+m_1^2)c^4-2m_0c^2E_1=(m_2+m_3)^2c^4`
or `2m_0c^2E_1le{m_0^2+m_1^2-(m_2+m_3)^2}c^`, or `E_1le(m_0^2+m_1^2-(m_2+m_3)^2)/(2m_0)c^2`