Special Relativity and muon decay

[astr0]

The proper mean lifetime of a muon is 2.20 µs, which is denoted as τ. Consider a muon, created in Earth's upper atmosphere, speeding toward the surface 8.00 km below, at a speed of 0.980c. What is the likelihood that the muon will survive its trip to Earth's surface before decaying? The probability of a muon decaying is given by P = 1 - e^{-Δt/τ}, where Δt is the time interval as measured in the reference frame in question. Also, calculate the probability from the point of view of an observer moving with the muon.

I figured that this is dealing with time dilation, so I used the formula T=\frac{T_{0}}{\sqrt{1-(v^{2}/c^{2})}}

I know that v = 0.980c
And that T_{0} = 2.2x10^{-6} s

But doing this and solving for T, then plugging T into the probability equation does not give me the correct answer. What am I missing? Do I need to somehow account for the height?

 

[vela]

What does the T you calculated represent physically?

 

[astr0]

The time in the frame of reference that is outside of the muon.

 

[vela]

What time specifically? If you understand this, you should be able to solve the problem.

 

[astr0]

Then I'm not sure I understand it.

 

[vela]

Well, you have at least four different times in this problem:

1. Δt in the Earth's frame,
2. τ in the Earth's frame,
3. Δt in the muon's frame, and
4. τ in the muon's frame.

Which one is equal to T₀=2.2x10^-6 s and which one is equal to the time you calculated using the time-dilation formula? How can you calculate the others?