Determining Speed of Muons: Exercise Problem

[liquidFuzz]

I try to determine the speed of muons. In an book I have there's an exercise regarding muons.

Given that they half of them decay in $2.2 * 10^{-6}$ s, how fast do they have to travel if half of them is to reach earth, ocean floor?

I tried to calculate the speed but I get to a point where I have two unknowns and one equation...

$t' = \gamma t$

How do I get further..? Assuming a speed close to c to get a t', or?

 

[bapowell]

One of the times is the life-time of the muon in the muon's rest frame: 2.2 * 10^-6

 

[liquidFuzz]

Perspicacious, thanks a million!

Any ideas of how to treat the velocity and time?

 

[jimbobian]

Well t' is the time from the muon's frame and you can use your equation to calculate the decay rate as observed by someone on Earth if you know the speed of the muon. You know the opposite, from the information you have you can calculate the required decay rate and use the equation to get the necessary velocity of the muon to achieve that.

 

[jimbobian]

Just to let you know, I have worked out the speed based on the information you gave and a few other rough pieces of information. The results is actually surprisingly close to the measured speed that I found in an MIT paper, so if you are still stuck then I can keep pointing you in the right direction.

 

[bapowell]

t is the proper time, and gives the muon's decay rate in the muon's rest frame. This is 2.2 * 10^-6 sec. t' is the dilated muon lifetime as measured by someone here on earth. It will be larger than t. Take v = x/t', where x is the distance that the muon must travel.

 

[liquidFuzz]

Thanks for the input. Thinking about it I'm pretty sure we did this calculation back in school, maybe I can find some old scribbling in one of my note books.

(I'm going on a weeks holiday to morrow, so I'm not sure I'll be able to tinker or post here. I'll update you asap though.)

 

[yuiop]

First you need to know the distance the muon travels (the height of the atmosphere as measured in the Earth rest frame). Call this x.

The speed of the muon in the Earth frame (relative to the speed of light) is then v/c = x/t'.

Using a bit of algebra:

$$\frac{v}{c} = \frac{x}{t'} = \frac{x}{\gamma t} = \frac{x \sqrt{1-v^2/c^2}}{t}$$

$$\frac{v^2}{c^2} = \frac{x^2 (1-v^2/c^2)}{t^2}$$

Solve for v/c:

$$\frac{v}{c} = \pm \frac{x}{\sqrt{x^2 + t^2}}$$

You already know t (the lifetime of the muon in the muon's rest frame) and you can easily find x from a Google search, so there are now no unknowns on the right and only one unknown on the left.