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zeeshahmad

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## Homework Statement

A pion in its rest frame decays into a muon and a neutrino. Find the velocity of the muon and its mean-lifetime in the pion rest frame. (I've done this part).

The muon decays into an electron and two neutrinos. If the two neutrinos happen to travel in the same direction in the rest frame of the muon, find the velocity of the resulting electron in both the rest frame of the muon (done this as well) and in the rest frame of the original pion. Hence show that the maximum velocity of the electron in the pion's rest-frame satisfies:

[itex]\frac{U}{c}=\frac{\mu^{2}-m^{2}}{\mu^{2}+m^{2}}[/itex]

where

[itex]\mu[/itex] is the pion's rest-mass,

M is the muon's rest-mass, and

m is the electron's rest-mass

neutrino's are massless

Hint: You will need to appreciate that the relative orientation of the muon velocity to the electron velocity is an issue.

## Homework Equations

relative mass:

ie m(v) = γ*m(0)

## The Attempt at a Solution

For the parts I've done:

I used the conversation of mass and momentum to get two equations for the first split (pion -> muon+neutrino), while the I assign (-p) for the momentum of the neutrino and so (p/c) for its mass.

So I get (eventually after some algebra..):

[itex]\frac{V}{c}=\frac{\mu^{2}-M^{2}}{\mu^{2}+M^{2}}[/itex]

which is the first bit done.

I also get the velocity of the electron in the muon's rest-frame:

[itex]\frac{V}{c}=\frac{M^{2}-m^{2}}{M^{2}+m^{2}}[/itex]

But when I attempt to use the velocity addition rule, to find this velocity in the rest frame of the pion, I don't quite get the answer..it's messy..

And I don't understand where the "maximum" velocity of electron comes in!

I also don't get the meaning of the hint.

Thankyou for taking time to read..

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