Solving for Muon Speed & Neutrino Energy

[stefan10]

Homework Statement

(a) Consider that decay when the pion is at rest. Find the speed β of the muon and the energy $E_{v}$ 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 $r=m_{\pi}/m_{\mu}$

Homework Equations

1. $E^2 = M^2 + P^2$

2. $β_{\mu} = P_{\mu}/E_{\mu}$

The Attempt at a Solution

By conservation of momentum $P_{\mu} = - P_{\nu}$

By conservation of energy $E_{\mu} + E_{\nu} = m_{\pi}$

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

$E_{\nu} = \frac{m_{\pi}}{2} - \frac{m_{\mu}^2}{2 m_{\pi}}$

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.

 

[Simon Bridge]

Using Equation 1 and the fact the mass of the neutrino is zero.

You now the mass of the neutrino is not zero right? But it may be OK to neglect it.

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?

Not really. The hint is the shortcut.

Once you start you'll find the calculations are not all that bad.