# Conservation of linear momentum

## Homework Statement

A positive pion at rest decays to a positive muon and a neutrino. The kinetic energy of the muon has been measured to be T(muon) = 4.1 MeV. The mass of the muon is known from other experiments to be 105.7 MeV. Find the mass of the pion. Do this nonrelativistically, and then repeat your calculation relativistically.

## Homework Equations

Nonrelativistic: T = p^2 / 2m
Relativistic: T = E - mc^2; E^2 = (pc)^2 + (mc^2)^2
p(pion) = p(muon) + p(neutrino) = 0

## The Attempt at a Solution

Since I'm given T and m for the muon, I can find p(muon) from the above formulas, both nonrelativistically and relativistically. And by conservation of linear momentum, I know that p(neutrino) = -p(muon). But from here I'm stumped. I can't find out anything more about the neutrino, because there's no further data. And even if I could, I wouldn't know what to do with it. For example, are the masses of the muon and neutrino supposed to be simply added to find the mass of the pion? I feel like I'm not being given enough information to solve this.

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The mass of the neutrino ought to be zero, so its Kinetic Energy is pc .
Then you can add all 3 Energies together (including mass Energy), to get the initial Energy.

The mass of the neutrino ought to be zero, so its Kinetic Energy is pc .
Then you can add all 3 Energies together (including mass Energy), to get the initial Energy.
Is everybody just supposed to know that m(neutrino) = 0? Maybe so, I have very little formal physics training...

Anyway, so using the relativistic equations, we can get:

T(muon) = 4.1 = E(muon) - m(muon)c^2
= 4.1 = E(muon) - (105.7 / c^2)(c^2)
So E(muon) = 4.1 + 105.7 = 109.8 MeV

Now E^2 = (pc)^2 + (mc^2)^2
So 109.8^2 = (pc)^2 + 105.7^2
And pc(muon) = 29.72.

By conservation of linear momentum, p(muon) = p(neutrino).

So pc(neutrino) = 29.72.
And E(neutrino)^2 = (pc)^2 + (mc^2)^2
= 29.72^2 + (0)(c^2)^2
= 29.72^2
So E(neutrino) = 29.72 MeV

Now by conservation of energy, E before pion decay = E after = E(muon) + E(neutrino).
So E(pion) = 109.8 + 29.72 = 139.52 MeV

And since the pion is at rest, E(pion) = mc^2 = 139.52 MeV, which is the right answer.

What about the nonrelativistic case? Here we have T = p^2 / 2m.

So T(muon) = 4.1 = p^2 / (2)(105.7 / c^2)
So p(muon) = [(4.1)(2)(105.7 / c^2)]^.5
= 29.44 / c
Or pc(muon) = 29.44, which is very close to what we found in the relativistic case.

And is the idea that that's as far as we can go nonrelativistically? I don't see any nonrelativistic equations that allow you to have a nonzero momentum with a mass of zero. So it looks like a neutrino can't have kinetic energy in nonrelativistic theory. The neutrino is a relativistic animal. Is that right?