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Conservation of linear momentum

  1. Aug 24, 2011 #1
    1. The problem statement, all variables and given/known data
    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.


    2. Relevant 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


    3. 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.
     
  2. jcsd
  3. Aug 24, 2011 #2

    lightgrav

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    Homework Helper

    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.
     
  4. Aug 25, 2011 #3
    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?
     
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