Muons reaching Earth's surface problem

[fruggz]

Question: Muons created in the upper atmosphere are sometimes able to reach the Earth's surface. Imagine that one such muon travels the 60km from the upper atmosphere to the ground (in the Earth's frame) in one muon half-life of 1.52us (in the muon's frame). How thick is that part of the Earth's atmosphere from the muon's creation point to the ground in the muon's frame?

Attempted solution: Assuming the muon travels at 0.99c, in on half life cycle in it's own frame it would travel 0.99c*1.52us. This yields d=451 meters. That seems a little too easy for my liking, is it correct? My physics class basically got a crash course in relativity, that is 10 chapters in two weeks and we did very few examples so this seems foreign to me.

 

[Orodruin]

Assuming the muon travels at 0.99c, in on half life cycle in it's own frame it would travel 0.99c*1.52us.

Be very careful with your wording. The muons travel exactly zero distance in their own frame by definition. In the muon frame, the Earth travels the quoted distance.

But you are right, you are being too simplistic here:

Assuming the muon travels at 0.99c

You cannot just assume a velocity, you have to compute it. (Although it will be pretty close to c, so the final value will not be much different.)

 

[Ibix]

Yes, that's a little too easy. Sorry...

You're guessing at the velocity of the muon. Can you work it out? Assume that the muon was created at t=0 and x=0.

Hint - it's a rare introductory problem to special relativity that doesn't require you to use the Lorentz transforms.

 

[Orodruin]

Hint - it's a rare introductory problem to special relativity that doesn't require you to use the Lorentz transforms.

Lorentz transforming here is a bit overkill. You can easily find the velocity without performing one. With the given information, the OP gives a pretty good approximation as long as the velocity is anywhere near c.

 

[fruggz]

I don't quite understand how I can calculate the exact velocity?

 

[Ibix]

I'd recommend using the Lorentz transforms. Can you find those? Do you know what quantities given in the question correspond to which variables in the Lorentz transform formulae?

Orodruin is proposing a short-cut based on the formula for time dilation. I'd steer clear of that until you're confident with Lorentz transforms - then you'll be able to see when Orodruin's approach will work and when it won't. Up to you, though.

 

[fruggz]

Well I had actually thought of using time dilation myself, I just couldn't figure out how to relate it to this problem. If I take the muons to be the home frame with the "earth traveling towards them" as Orodruin had pointed out that makes the 1.52us the proper time right? But there I don't know what to do because I don't know the dilated time without knowing the velocity.

 

[Ibix]

That's why I went for the Lorentz transforms. Look them up (depending which frame you call which, you may find the inverse transforms more helpful).

Alternatively, you can approximate as Orodruin suggested in #4.