Help with time dilation and length contraction

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Homework Help Overview

The discussion revolves around a problem involving time dilation and length contraction in the context of a pion's decay. Participants are tasked with determining the average speed of a pion that travels 25 meters before decaying, given its proper lifetime of 2.6 x 10-8 seconds.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the relationship between the proper time and the time observed in the lab frame, questioning the definitions of proper length and length contraction. There are attempts to apply relevant equations but uncertainty remains about how to manipulate them effectively to find the pion's speed.

Discussion Status

Some participants have provided guidance on how to relate the pion's speed to the observed distance and time, while others express confusion about their calculations and the application of the formulas. There is an ongoing exploration of the mathematical relationships involved, with no explicit consensus reached yet.

Contextual Notes

Participants note challenges with mathematical skills and the need to find the pion's lifetime in the lab frame, as well as the implications of relativistic effects on measurements. The discussion reflects a mix of attempts to clarify concepts and resolve mathematical difficulties.

alicia.kostka
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Homework Statement



How fast must a pion be moving on average to travel 25 meters before it decays? The average lifetime, at rest, is 2.6 x 10-8s.

Homework Equations



\Deltat =\Deltat0/ sqrt(1 - v2/ c2)

l = l0 * sqrt (1 - v2/ c2)


The Attempt at a Solution



I think that 2.6 x 10-8 is the proper time right? And then is 25 meters the length or the proper length? I'm totally lost... I honestly don't know where to go first. I know that I have to find the lifetime of the pion relative to an resting observer...
 
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alicia.kostka said:

Homework Statement



How fast must a pion be moving on average to travel 25 meters before it decays? The average lifetime, at rest, is 2.6 x 10-8s.

Homework Equations



\Deltat =\Deltat0/ sqrt(1 - v2/ c2)

l = l0 * sqrt (1 - v2/ c2)


The Attempt at a Solution



I think that 2.6 x 10-8 is the proper time right?
Yes, this is the time as measured in the pion's rest frame.
And then is 25 meters the length or the proper length?
This is the distance the pion travels as measured in the lab frame. In the pion's frame, the distance will be length contracted because of the pion's motion.
I'm totally lost... I honestly don't know where to go first. I know that I have to find the lifetime of the pion relative to an resting observer...
If there were no relativistic effects, the pion could at most travel c(2.6x10-8 s)=7.8 m before decaying on average. From the point of view of an observer in the lab frame, one could say the pion is able to travel 25 m because its clock runs slower than the lab's clocks. Equivalently, you could say the pion sees the 25 m in the lab frame length-contracted to ~7.8 m in its frame. (The actual distance depends on the speed of the pion, which won't be exactly c.)
 
Thanks...that means that I am on the right track. However, I have tried to plug these numbers into the formulas to find the pions speed and I can't get it! I know that 25m is the distance measured in the lab, and I need to find the pion's lifetime in the lab observer's reference frame.

Alternately, I could find the distance traveled in the pion's reference frame...then find its speed (speed will be the same in both cases right?) Problem is, my math skills are subpar and I'm sure that I have to find one in terms of the other and then equate and solve...but I'm not sure how! I'm sure this is a lot easier than it seems right now...
 
hi alicia! :smile:

(have a delta: ∆ and a square-root: √ :wink:)
alicia.kostka said:
Thanks...that means that I am on the right track. However, I have tried to plug these numbers into the formulas to find the pions speed and I can't get it! … Problem is, my math skills are subpar and I'm sure that I have to find one in terms of the other and then equate and solve...but I'm not sure how! I'm sure this is a lot easier than it seems right now...

best thing is to show us your calculations, so that we can see where you're getting stuck, and give you tips on how to do the maths in future :smile:
 
alicia.kostka said:
speed will be the same in both cases right?
Yes, observers in both frames will agree on how fast the pion was moving relative to the lab.

As tiny-tim suggested, show us your work so we can see what you've tried.
 
First I tried to plug in the proper time to find \Deltat
\Deltat = 2.6 x10-8/ √ (1-V2/c2)

Since 2.6 x 10 -8 is the lifetime at rest and I'm trying to find the lifetime observed in the lab...

Then I plugged in l = 25m √1-v2/c2

Since l0, the proper length observed in the lab is 25 meters...and I'm trying to find the distance traveled from the point of view of the pion...

But every time I try to solve for v, I'm getting something greater than c...which I know is not right.

I appreciate the help!
 
sorry...i meant to put in there that I related the two by...

l= v \Deltat √ 1-v2/c2

Math really kicks my butt...I'm sure I'm missing something really easy!
 
Looks good so far. If you use that equation to relate the two, the square root factor appears on both sides so you can cancel it, and you end up with vΔt=25 m, which makes sense. In the lab frame, it moves at speed v for a time Δt for a total distance of 25 m. It's correct, but it doesn't tell you anything you didn't already know.

Try multiplying your first equation by v.
 

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