- #1

ian2012

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## Homework Statement

A pi+, produced by a cosmic ray, has an energy of 280MeV in the laboratory frame S. It undergoes a decay: pi+ ---> muon+ & muon-neutrino. The muon produced continues to travel along the same direction as the pi+ and has a mean lifetime of 2.2mirco seconds in the muon's rest frame (S''). Calculate the expected distance it would have traveled in the laboratory frame S before decaying.

## Homework Equations

In the rest frame of the pion (S'):

[tex]E^{'}_{\mu}=\frac{(m^{2}_{\pi}+m^{2}_{\mu})c^{2}}{2m_{\pi}}[/tex]

[tex]P^{'}_{\mu}=\frac{(m^{2}_{\pi}-m^{2}_{\mu})c}{2m_{\pi}}[/tex]

and other relativistic relations and Lorentz Transforms

[tex]m_{\pi}=139.6\frac{MeV}{c^{2}}, m_{\mu}=105.7\frac{MeV}{c^{2}}[/tex]

## The Attempt at a Solution

I have attempted the problem, I don't know if i am correct.

The obvious thing to do is find the mean lifetime in the frame S' and the distance in S' in order to solve the equation:

[tex]x_{\mu}=\gamma(x^{'}_{\mu}+vt^{'}_{\mu})[/tex]

To find t' I have used the fact that time dilates in moving frames therefore t' = gamma x t'' (2.2 mircoseconds)

I then used the relativistic momentum equation to find the velocity of muon in the pion rest frame S': v' = 0.28177/gamma

Then i used LT to find the distance traveled in the pion frame = gamma x v x t'' = 0.28177 x t''.

Then I used the energy of the pion in the lab frame to find the velocity of the pion in the lab frame using the relativistic energy equation and the momentum equation to give = 1.739/gamma.

Subbing these into the above equation I got the expected distance = gamma x 4.45x10^-6 metres.

Is this correct? Is the gamma supposed to be there?