# Homework Help: Muon travelling faster than the speed of light

1. Aug 19, 2007

### 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.

2. Aug 19, 2007

### 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 travelling close to the speed of light.

3. Aug 19, 2007

### neutrino

4. Aug 19, 2007

### 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.

5. Aug 19, 2007

### 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.

6. Aug 19, 2007

### Dick

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

7. Aug 19, 2007

### ehrenfest

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.

8. Aug 19, 2007

### Dick

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

Last edited: Aug 19, 2007
9. Aug 19, 2007

### 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.

Last edited: Aug 19, 2007
10. Aug 19, 2007

### learningphysics

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

11. Aug 19, 2007

### Dick

That's a bingo!

12. Aug 19, 2007

### ehrenfest

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