- #1

Nate D

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## Homework Statement

In the rare decay ∏

^{+}→ e

^{+}+ v

_{e}, what is the momentum of the positron (e

^{+})? Assume the ∏

^{+}decays from rest. (m

_{∏}+ = 139.6 MeV/c^2, m

_{v}≈ 0, m

_{e}= 0.511 MeV/c^2)

## Homework Equations

Conservation of Energy: E

_{∏}= E

_{e}+ E

_{v}

Conservation of momentum: p

_{∏}= p

_{e}+ p

_{v}

0 = p

_{e}+ p

_{v}

p

_{e}= -p

_{v}

Invariant mass: E^2 = (pc)^2 + (mc^2)^2

## The Attempt at a Solution

My first step was to ensure momentum was conserved, stating that once the pion decayed, the positron and neutrino went off in opposite directions. This yields the equation p

_{e}= -p

_{v}.

Next I went about finding the rest energy of the pion, which following the equation, E

_{∏}= m

_{∏}c^2. Using this I found the rest energy of the pion to simply be 139.6 MeV.

After that I began to use the formula, E^2 = (pc)^2 + (mc^2)^2.

I thought that E in this case is the energy of the system, 139.6 MeV, m is the rest mass of the positron, .511MeV/c^2, and p is the momentum of the system that I am asked to solve for.

Solving that equation I found p to be 139.599 MeV/c.

This is where I am confused/unsure and could use some help if possible. Is that momentum

the momentum of the entire system or of the positron after the decay? Also how do I factor in the neutrino considering it is massless? I know the equation E = pc can be used to solve for the energy of a massless particle, however I am not sure if that needs to be used for this problem.

This is my first post here, and I am new to forums in general so I am sorry if I forgot to follow a certain protocol of these forums! Thank you for any help that you may be able to give me!