# Classical mechanics exercise, pion decay!

## Homework Statement

If anyone could help me with this classical mechanics exercise I would be very grateful! The exercise is as follows:

The muon (μ) is a particle with mass mμ=207me, with me being the electron mass. The pion (∏) has a mass of m=273me. The pion can decay into a muon plus a massless neutrino, v, in the reaction  ∏ → μ + v. Find the kinetic energy of the muon when a pion decays at rest. Hint: Use conservation of both energy and momentum.

## Homework Equations

The momentum conservation looks like this:

Pbefore=Pafter

Where the momentum is given by:

P=mμγ(u)

The energy conservation is:

Ekin, before=Ekin, after

The kinetic energy is given by:

Ekin=mc^2(γ(u)-1)

quantities we know:

mμ=207 me

m=273 me

mv=0

u=0 (the speed of the pion)

## The Attempt at a Solution

I have tried to write the equations, please correct me if I'm wrong:

Ekin, before = m c^2 (γ(u)-1)

Ekin, after = Ekin, μ + Ekin, v = mμ c^2 (γ(u)-1) + mv c^2 (γ(u)-1)

Pbefore=P=m u γ(u)=0

Pafter=Pμ+Pv=mμ uμ γ(u) + mv uv γ(u) = mμ uμ γ(u) + $\frac{E}{c}$ (because a massless particle can still have momentum)

Now I'm a bit stuck. What is the kinetic energy of the neutrino with zero mass, zero? How do I find the energy, E in the momentum expression, is that the kinetic energy? It seems to me that there are too many unknowns, but I'm pretty sure I'm wrong here!

If anyone could help me, I would I will be very grateful!
mr. bean.

The pion is initially at rest so the total energy is equal to the rest mass of the pion, there is not kinetic energy before so get rid of the γ. Start there and then imagine that when the particle decays in order to make it so that the total momentum is still zero the muon and the neutrino must travel in opposite directions with equal momentum. From this you can solve for E.

vela
Staff Emeritus