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## Homework Statement

(a) Consider that decay when the pion is

__at rest.__Find the speed β of the muon and the energy [itex]E_{v}[/itex] of the neutrino. (Work in general algebraic terms, and set c=1. Plug in numerical values as late as possible! You may find it useful to define a quantity [itex]r=m_{\pi}/m_{\mu} [/itex]

## Homework Equations

1. [itex] E^2 = M^2 + P^2 [/itex]

2. [itex] β_{\mu} = P_{\mu}/E_{\mu} [/itex]

## The Attempt at a Solution

By conservation of momentum [itex] P_{\mu} = - P_{\nu} [/itex]

By conservation of energy [itex] E_{\mu} + E_{\nu} = m_{\pi} [/itex]

Using Equation 1 and the fact the mass of the neutrino is zero. I go through algebra to get

[itex] E_{\nu} = \frac{m_{\pi}}{2} - \frac{m_{\mu}^2}{2 m_{\pi}} [/itex]

I'm unsure how I can find the momentum and energy of the muon from there without going into long algebraic computations. Is there an easier way? Thank you.