Muon travelling faster than the speed of light

[ehrenfest]

A muon is observed to travel 800 meters before disintigrating. The lifetime of a muon is 2 * 10 ^ -6. So the observer concludes that the muon traveled at a speed of

4 * 10^8 m/s which is faster than c.

Why is the observer wrong?

I though it was impossible for anything to travel faster than light in any inertial reference. I don't see how you can get around the fact that the muon moved so far in that amount of time and at least its average velocity was faster than c.

I do not see where you can apply the Lorentz factor.

 

[StatusX]

The key is that that's the muon's lifetime in it's rest frame. Because of time dilation, it will observed to last longer than that when it's traveling close to the speed of light.

 

[neutrino]

The lifetime in which frame?

 

[Dick]

Besides which the muon lifetime is only a mean lifetime. For a muon to last twice the mean is not a bit unusual. Don't assume all muons last exactly 2x10^(-6) sec. But the answer they are probably after is the one StatusX pointed out.

 

[ehrenfest]

I see, StatusX. And you're right Dick this is a simplified problem.

So, if I want to calculate the actual speed of the muon, I want to set the rest frame time times the lorentz factor equal to the laboratory frame time. You can solve for v and get v = 2.078 * 10^8, if my calculations are valid.

 

[Dick]

I get more like 1.98*10^8, but that's close enough.

 

[ehrenfest]

I get more like 1.98*10^8, but that's close enough.

No. Let's figure this out.

I solved,

t_lab = 2*10^-6/(1- 800^2/(t^2*c^2))
for t to get

t = 3.848 * 10 ^ -6.

I then divided 800 by that.

 

[Dick]

Where's the sqrt in your gamma factor? Try t_lab=t_rest/sqrt(1-v^2/c^2) and solve for v.

 

[Dick]

Sorry. That's foolish. You don't know t_lab. As you said with some stuff fixed, t=t_rest/sqrt(1-d^2/(t^2*c^2)). But now I get 2.40*10^8 m/s.

 

[learningphysics]

I'm getting exactly v = 2.4 * 10^8 m/s

 

[Dick]

I'm getting exactly v = 2.4 * 10^8 m/s

That's a bingo!

 

[ehrenfest]

Where's the sqrt in your gamma factor? Try t_lab=t_rest/sqrt(1-v^2/c^2) and solve for v.

Yes. Adding the square root gets me 2.4 * 10 ^ 8. Thanks.