Relativistic elementary particle problem

[dreamspy]

Homework Statement

We have the decay process:

pion --> muon + anti muon neutrion

How long does the muon travel before it decays? (the pion is stationary before the decay)

The answer is supposed to be

d = c * t (m(pi)^2 + m(mu)^2) / (2 m(pi) * m(mu) )

where t is the mean lifetime of the muon in it's center of mass system.

The Attempt at a Solution

Now I know about the conservation laws of energy and momentum.

Also from the form of the formula for d, I see that we should have d/t = v , where v is the speed of the muon. The speed is related to the momentum by p = mv. So I should try to find the momentum p of the muon. Now I say that the neutrino is massless and define the change in rest energy to be:

delta E = c^2 ( m(pi) - m(mu) )

This energy should then give the muon and the neutrino some momentum and kynetic energy. So I should use the conservation laws to find p and then the problem should be finished. But I haven't been able to show this. Anyone have any clue how to do this?

regards
Frímann

 

[fzero]

Now I know about the conservation laws of energy and momentum.

Also from the form of the formula for d, I see that we should have d/t = v , where v is the speed of the muon. The speed is related to the momentum by p = mv.

You should be using relativistic expressions for all quantities. p =mv is not true when the effects of special relativity are important.

So I should try to find the momentum p of the muon. Now I say that the neutrino is massless and define the change in rest energy to be:

delta E = c^2 ( m(pi) - m(mu) )

This is not a useful quantity. You should compute the energy of the pion, muon and neutrino and directly apply energy conservation.

This energy should then give the muon and the neutrino some momentum and kynetic energy. So I should use the conservation laws to find p and then the problem should be finished. But I haven't been able to show this. Anyone have any clue how to do this?

regards
Frímann

Indeed you want to apply conservation of energy and momentum to find the energy and momentum of the muon. By comparing this to the rest energy of the muon, you can determine the Lorentz transformation and then use this to compute the time dilation.

 

[dreamspy]

actually I was able to finish the problem using the foundation I posted in my first post. I simply say that the extra energy is divided to the kinetic energy of the muon and the neutrino:

delta E = T(muon) + T(Neutrino)

The momentum of the muon and the neutrino are equal but in oposite directions:

|p(muon)| = |p(neutrino)| = p

Then I simply use:

T(muon) = E(muon) + E0(muon)

where E(muon)^2 = (cp)^2 + (E0)^2

and E0 = m(muon) c^2

Now equating the two functions for delta E will then give you the correct answer at least according to the problem. Although p=mv is not always valid.

regards
Frímann

 

[fzero]

actually I was able to finish the problem using the foundation I posted in my first post. I simply say that the extra energy is divided to the kinetic energy of the muon and the neutrino:

delta E = T(muon) + T(Neutrino)

OK, I just thought there's less room for confusion in general if you just apply energy conservation directly as E(initial) = E(final)

The momentum of the muon and the neutrino are equal but in oposite directions:

|p(muon)| = |p(neutrino)| = p

Then I simply use:

T(muon) = E(muon) + E0(muon)

You either mean

E(muon) = T(muon) + E0(muon)

or

T(muon) = E(muon) - E0(muon)

Although p=mv is not always valid.

Right. If you can work entirely in terms of the momentum and energy you won't have a problem. If you need to compute a velocity you need to use the correct formula.

 

[paranormal]

Dear Frímann,

Thank you for your question. I am a scientist and I would be happy to provide a response to your content.

Firstly, I would like to clarify that the formula you have provided for d is the correct one for calculating the distance traveled by the muon before it decays. However, it is important to note that this formula is derived from special relativity and therefore, it only applies in the context of high energy particles.

As you have correctly mentioned, the speed of the muon can be calculated using the conservation laws of energy and momentum. In this case, we can use the fact that the total energy and momentum before and after the decay must be equal. Since the pion is stationary before the decay, its momentum is zero. This means that the total momentum after the decay must also be zero, as the muon and neutrino move in opposite directions. Using this information, we can write the following equations:

Total energy before decay = Total energy after decay

c^2 * m(pi) = c^2 * (m(mu) + E_k) + c * p(mu) + c * p(nu)

Total momentum before decay = Total momentum after decay

0 = p(mu) + p(nu)

Where E_k is the kinetic energy of the muon and p(mu) and p(nu) are the momenta of the muon and neutrino respectively.

We know that the mass of the neutrino is negligible, so we can ignore its momentum in our equations. This leaves us with two equations and two unknowns, p(mu) and E_k. Solving these equations will give us the momentum and kinetic energy of the muon, which we can then use to calculate its speed and subsequently the distance traveled before decaying.

I hope this helps. Let me know if you have any further questions.

Best regards,