Max Velocity of Electron in Pion Rest Frame: Homework Solution

[zeeshahmad]

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:
\frac{U}{c}=\frac{\mu^{2}-m^{2}}{\mu^{2}+m^{2}}

where
\mu 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..):
\frac{V}{c}=\frac{\mu^{2}-M^{2}}{\mu^{2}+M^{2}}
which is the first bit done.
I also get the velocity of the electron in the muon's rest-frame:
\frac{V}{c}=\frac{M^{2}-m^{2}}{M^{2}+m^{2}}

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..

 

[BruceW]

So I get (eventually after some algebra..):
\frac{V}{c}=\frac{\mu^{2}-M^{2}}{\mu^{2}+M^{2}}
which is the first bit done.
I also get the velocity of the electron in the muon's rest-frame:
\frac{V}{c}=\frac{M^{2}-m^{2}}{M^{2}+m^{2}}

These are correct.

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..

Darn straight it is complicated. This is the part where you must use the hint. What answer are you getting? Also, what do you mean you don't quite get the answer? They don't provide you with the answer to this bit.

Showing the maximum velocity of the electron in the pion's rest frame is actually easier than the previous part of the problem. Again, you must make use of the hint, and think of the case in which the electron has most energy (relative to the inertial frame in which the pion is at rest).

Also, welcome to physicsforums :)

 

[zeeshahmad]

Thank you for welcoming me! .. and helping..

So I've managed to simplify the mess:
I get the velocity of the electron in the pion's rest frame, U to be:
\frac{U}{c}=\frac{\mu^{2}(M^{2}-m^{2})}{M^{2}(\mu^{2}+m^{2})}
Do you think this is correct?

I'm thinking on what you said about the energy..
..although the above expression equals the required one in the question when \mu=M

 

[zeeshahmad]

I'm really not sure whether there should be an angle involved, since the hint mentions "orientation"?

 

[BruceW]

No, that's not right. You need to take into consideration "relative orientation of the muon velocity to the electron velocity." They want us to find the velocity of the electron according to the (original) pion rest frame, right? So think through what happens according to this frame. The pion decays into a muon going in one direction (let's call that the z axis), and a neutrino in the opposite direction. Then that muon decays into an electron and a couple of neutrinos. Now is there any reason why the electron would come out from the decay going in the z direction as well? And from this answer, will an angle be involved?

Edit: This is in reply to post #3, not post 4

 

[zeeshahmad]

Ah yes it will beThanks, I'll get back to you after trying to maths it out..

 

[zeeshahmad]

Thanks man! You're a life saver!
I assumed an angle of alpha to the Z axis in pions frame.
So when adding the velocities (using the rule...) take the component of velocity in muon's frame parallel to the Z axis. So when I got an "unsimplifyable" expression, I put cos(alpha)=1 and it worked out exactly as expected.
Once again, thank you.

 

[BruceW]

glad I could help, dude!