Classical mechanics exercise, pion decay

• mr. bean

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!