Relativistic Distance/Time Problem

[chef99]

Homework Statement

A muon has a lifetime of 2.20 x10^-6 s when at rest, after which time it decays into other particles.

a) Ignore any effects of relativity discussed in this section. If the muon was moving at 0.99c, how far would it travel before decaying into other particles, according to Newtonian mechanics?

b) How long would the muon last, according to an observer in the earth’s frame of reference who viewed the muon moving at 0.99c?

c) How far would the muon actually travel, when viewed moving at 0.99c?

d) Compare the two distances traveled. Explain why this type of evidence is excellent support for the theory of relativity.


2. Homework Equations

d = vt

Δtm = Δts / 1-√v²/c²

The Attempt at a Solution

a)
d = (0.99c)(3.00 x10⁸m/s)

d = 2.97 x10^-8m/s

(2.97 x10^-8m/s)(2.2 x10^-6s)

= 653.4 mb) Δtm = Δts / 1-√v²/c²

Δtm = 2.2 x10⁶ / √1- 0.9801

Δtm = 1.56 x10^-5 s

The muon will last 1.56 x10^-5 s in the earth’s frame of reference, moving at 0.99cc) Using special relativity, the muon would last for d_SR= vτ, where τ is now the relativistic lifetime of the muon.

d_SR= vτ

d_SR= (0.99c)(3.00 x10⁸m/s)(1.56 x10^-5 s)

d_SR= 4633.2 md) These results are a good demonstration of relativity because they show that time dilation becomes significant as the velocity approaches the speed of light.
I'm just not sure if I did b) and c) right, in c) when it says "How long will it last", I assume that means how many seconds, but I could be wrong. It seemed logical in c) to use the same equation as a) but simply change the value of τ to the relativistic value, but as I haven't had to use this equation before I'm not sure if it is the right one to be using. I would appreciate any feedback

 

[mfb]

(b) and (c) are okay. There is a missing minus sign for the muon lifetime in (b) and the choice of $\tau$ for our reference frame in (c) is unfortunate (it is often used for proper time, time in the reference frame of the particle).

(d) is wrong.

These results are a good demonstration of relativity because they show that time dilation becomes significant as the velocity approaches the speed of light.

How would it show that? You just calculated something. What tells you that these calculations have anything to do with reality?
To support a theory you have to compare its predictions with experiments - and ideally show that a competing theory is inconsistent with the results.

 

[chef99]

(b) and (c) are okay. There is a missing minus sign for the muon lifetime in (b) and the choice of $\tau$ for our reference frame in (c) is unfortunate (it is often used for proper time, time in the reference frame of the particle).

(d) is wrong.How would it show that? You just calculated something. What tells you that these calculations have anything to do with reality?
To support a theory you have to compare its predictions with experiments - and ideally show that a competing theory is inconsistent with the results.

What would be the appropriate sign/symbol to use? As I said I wasn't given equations to use in any previous problems, so this was just the closest equation I could find that made the most sense. For that matter is that even the correct equation to use? Also does a) satisfy the "according to Newtonian mechanics" requirement?

For d), perhaps my explanation was too brief. The distance traveled in c) is far greater than in a), 4633.2 m and 653 m, which I think shows a definite effect of relativity, as a) does not account for near relativistic speeds having an impact on the amount of distance the muon covers. Is this not correct?

 

[mfb]

$t$ for the lab frame, $\tau$ for the view of the muon would be the standard notation.

For that matter is that even the correct equation to use? Also does a) satisfy the "according to Newtonian mechanics" requirement?

Yes, all that is fine.

For d), perhaps my explanation was too brief. The distance traveled in c) is far greater than in a), 4633.2 m and 653 m, which I think shows a definite effect of relativity, as a) does not account for near relativistic speeds having an impact on the amount of distance the muon covers. Is this not correct?

You just rephrased your previous statement. Yes, classical mechanics and special relativity make different predictions. And now? How do you figure out which one is right? How do you test these predictions?

 

[chef99]

$t$ for the lab frame, $\tau$ for the view of the muon would be the standard notation.Yes, all that is fine.You just rephrased your previous statement. Yes, classical mechanics and special relativity make different predictions. And now? How do you figure out which one is right? How do you test these predictions?

The other thing I can think of is that this supports the theory of special relativity because it predicts that viewed from the lab frame (earth), the muon will travel farther because it will have the velocity of itself plus that of the earth, despite it appearing to the observer that they are stationary because the Earth is in fact moving. Viewed from the muon (τ), the Earth will be moving along with the muon, meaning it will not travel as far because it appears to be moving slower compared to the earth. This is supported by the results I think, whereas Newtonian Mechanics would have predicted the same answer for both.
Is this a better explanation?

 

[mfb]

In the lab frame Earth is not moving.

You still didn't mention any experimental result. Which measurement can we make to distinguish between the two predictions? A measurement where special relativity predicts one thing and classical mechanics predicts something different?

How was the muon discovered? Would this discovery have been possible if classical mechanics would be true? Is it possible with special relativity?

 

[perot]

Calling question (d)

Explain why this type of evidence is excellent support for the theory of relativity.

a poor one amounts to praising it. It talks about "this evidence", but completely fails to mention what evidence it is talking about.

The student might not know how the muon was discovered. Or they might know, but fail to make the connection to this homework. This failure is then taken as evidence that the student has failed to understand the matter at hand.

Then again, it is possible it was mentioned in the original text and chef99 just failed to include it when copying it.

 

[chef99]

Calling question (d)

a poor one amounts to praising it. It talks about "this evidence", but completely fails to mention what evidence it is talking about.

The student might not know how the muon was discovered. Or they might know, but fail to make the connection to this homework. This failure is then taken as evidence that the student has failed to understand the matter at hand.

Then again, it is possible it was mentioned in the original text and chef99 just failed to include it when copying it.

I have now tripled checked the original text and it does not specify.