# Decay of a muon-probability of decay

1. Feb 11, 2012

### sawhai

1. The problem statement, all variables and given/known data

An energetic muon is created by the interaction of a cosmic ray 20 km away from the surface of the earth. How energetic does the muon have to be to be detected on earth before it decays with a 10% probability?
For a single muon, what is the probability that it will not have decayed after time Δt?

Show that, assuming the muon would be detected if it doesn’t decay, the length of time since creation of the muon Δt where 10% of the muons will be detected is given by Δt = [ln(10)= ln(2)]τ ' 3:3τ .

2. Relevant equations
For a decay half-life τ, the fractional proportion of a large population of N0 muons remaining after a time Δt is N=N0 = exp(- ln(2)Δt=τ ). How can relate this to a singe muon without having t and λ?

3. The attempt at a solution
I set up the decay equation but not sure what to do next.

2. Feb 12, 2012

### sawhai

Any one with an idea?

3. Feb 13, 2012

### JesseC

The half-life of particles moving relativistically is greater than that of particles in the rest frame due to the effects of time dilation.

You need to modify the decay equation to take account of time dilation. You might want to think about how the energy of the muon depends upon its speed.