Calculating Muon's Time Dilation and Travel Distance in a Particle Accelerator

[spaghetti3451]

Homework Statement

The muon has been measured to have a mass of $0.106\ \text{GeV}$ and a rest frame lifetime of $2.19 \times 10^{-6}$ seconds. Imagine that such a muon is moving in the circular storage ring of a particle accelerator, $1$ kilometer in diameter, such that the muon's total energy is $1000\ \text{GeV}$. How long would it appear to live from the experimenter's point of view? How many radians would it travel around the ring?

Homework Equations

The Attempt at a Solution

Proper time runs in the muon's rest frame. Therefore, the experimenter observes a dilated lifetime of the muon.

Therefore, from the experimenter's point of view,

lifetime $= \gamma \tau = \frac{E}{mc^{2}} \tau = \big(\frac{1000}{.106}\big)(2.19 \times 10^{-6}) = 20.7 \times 10^{-3}$ seconds.Using the invariant interval in the muon's rest frame, and the experimenter frame,

$- (\delta \tau)^{2} = - (\delta t)^{2} + (\delta x)^{2}$

$\delta x = \sqrt{(\delta t)^{2}- (\delta \tau)^{2}}$

$\delta x = 0.0207$ m.

Therefore, number of radians $= \frac{0.0207}{500} = 4.14 \times 10^{-5}$.

Are my answers correct?

 

[PeroK]

Your answer for $t$ looks correct. The answer for $x$ cannot possibly be correct (you should be able to see it's far too small). What have you forgotten?

 

[spaghetti3451]

Hmm... missed out on a factor of $c$.

Here's the corrected version:

$- (c \delta \tau)^{2} = - (c \delta t)^{2} + (\delta x)^{2}$

$\delta x = c \sqrt{(\delta t)^{2}- (\delta \tau)^{2}}$

$\delta x = 6.21 \times 10^{6}$ m.

Therefore, number of radians $= \frac{6.21 \times 10^{6}}{500} = 12420$.Is it correct now?

 

[vela]

Homework Statement

The muon has been measured to have a mass of $0.106\ \text{GeV}$ and a rest frame lifetime of $2.19 \times 10^{-6}$ seconds. Imagine that such a muon is moving in the circular storage ring of a particle accelerator, $1$ kilometer in diameter, such that the muon's total energy is $1000\ \text{GeV}$. How long would it appear to live from the experimenter's point of view? How many radians would it travel around the ring?

Homework Equations

The Attempt at a Solution

Proper time runs in the muon's rest frame. Therefore, the experimenter observes a dilated lifetime of the muon.

Therefore, from the experimenter's point of view,

lifetime $= \gamma \tau = \frac{E}{mc^{2}} \tau = \big(\frac{1000}{.106}\big)(2.19 \times 10^{-6}) = 20.7 \times 10^{-3}$ seconds.Using the invariant interval in the muon's rest frame, and the experimenter frame,

$- (\delta \tau)^{2} = - (\delta t)^{2} + (\delta x)^{2}$

$\delta x = \sqrt{(\delta t)^{2}- (\delta \tau)^{2}}$

$\delta x = 0.0207$ m.

Therefore, number of radians $= \frac{0.0207}{500} = 4.14 \times 10^{-5}$.

Are my answers correct?

You know the muon is essentially traveling at the speed of light, so how far does it travel before decaying?

 

[PeroK]

Hmm... missed out on a factor of $c$.

Here's the corrected version:

$- (c \delta \tau)^{2} = - (c \delta t)^{2} + (\delta x)^{2}$

$\delta x = c \sqrt{(\delta t)^{2}- (\delta \tau)^{2}}$

$\delta x = 6.21 \times 10^{6}$ m.

Therefore, number of radians $= \frac{6.21 \times 10^{6}}{500} = 12420$.Is it correct now?

As pointed out above, when a particle has such a high gamma factor, it's speed is approximately $c$, so $x \approx ct$. You can see from your equation that $\sqrt{(\delta t)^{2}- (\delta \tau)^{2}} \approx \delta t$. Which amounts to the same thing.

 

[spaghetti3451]

Ah! Right! My answer's correct, but the formula for the invariant interval is redundant because the muon essentially travels at the speed of light.

 

[spaghetti3451]

Is that it?

 

[PeroK]

Is that it?

Yes.

 

[spaghetti3451]

I see that, in all the problems I solve, I can work out most of the steps by itself. But at some places, I get stuck and need help.

Also, I don't always see the subtle issues like the fact that the muon speed is the speed of light in the problem above and solve problems in a longwinded way.

How do I get myself rid of these difficulties?

 

[PeroK]

I see that, in all the problems I solve, I can work out most of the steps by itself. But at some places, I get stuck and need help.

Also, I don't always see the subtle issues like the fact that the muon speed is the speed of light in the problem above and solve problems in a longwinded way.

How do I get myself rid of these difficulties?

I wouldn't worry about not spotting shortcuts, but try to remember this one. It's quite common in these problems. However, you shouldn't have been happy with your answer of 2cm. Get into the habit of critically analysing your answers: whether they are algebraic or numeric. You can't always sanity check an answer, but often you can get a feel for when things are not right.

 

[spaghetti3451]

Is there a correlation between being able to solve problems from problem sets and being a high quality researcher whose papers get published in reputed journals?

I mean, do these skills ultimately help when you become a research physicist and have to solve open-ended problems?