Help with time dilation and length contraction

In summary: That will give you Δt=v2Δt/c2. Now you can make a substitution into the equation you just wrote, which will eliminate Δt and leave you with only v on one side. Then you can solve for v.In summary, to find the average speed of a pion that travels 25 meters before decaying, we can use the formula Δt=v2Δt/c2 and relate it to the formula l= vΔt √1-v2/c2. By substituting the first equation into the second and solving for v, we can determine that the pion must be moving at a speed of 0.9931c on average.
  • #1
alicia.kostka
7
0

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



[tex]\Delta[/tex]t =[tex]\Delta[/tex]t0/ 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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  • #2
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



[tex]\Delta[/tex]t =[tex]\Delta[/tex]t0/ 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.)
 
  • #3
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...
 
  • #4
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:
 
  • #5
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.
 
  • #6
First I tried to plug in the proper time to find [tex]\Delta[/tex]t
[tex]\Delta[/tex]t = 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!
 
  • #7
sorry...i meant to put in there that I related the two by...

l= v [tex]\Delta[/tex]t √ 1-v2/c2

Math really kicks my butt...I'm sure I'm missing something really easy!
 
  • #8
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.
 

1. What is time dilation?

Time dilation is a concept in physics that states that time passes at different rates for objects moving at different speeds. As an object approaches the speed of light, time appears to slow down for that object from the perspective of an observer.

2. How does time dilation affect space travel?

For objects traveling at high speeds, time dilation can significantly impact the passage of time. This means that astronauts traveling at high speeds will experience time passing slower than those on Earth. As a result, space travelers may age at a slower rate compared to those on Earth.

3. What is length contraction?

Length contraction is a phenomenon that occurs when an object is moving at high speeds. From the perspective of an observer, the object appears to be shorter in the direction of motion. This is due to the fact that time is passing slower for the object, causing its length to appear shorter.

4. How does length contraction relate to time dilation?

Length contraction and time dilation are closely related and are both consequences of Einstein's theory of relativity. As an object approaches the speed of light, time appears to slow down and the object appears to contract in length. This is due to the fact that time and space are intertwined and change in relation to an object's speed.

5. Is time dilation and length contraction just a theory?

No, time dilation and length contraction have been proven through various experiments and observations, including atomic clocks on airplanes, particle accelerators, and GPS satellites. These concepts are fundamental to our understanding of the universe and have been extensively studied and verified by scientists.

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