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The proper mean lifetime of ##\pi## mesons(pion) is ##2.6x10^{-8}s##. Suppose a beam of such particles has a speed of .9c.

a) What would their mean life be as measured in the lab?

b) How far would they travel(on average)before they decay?

c)What would your answer be to part (b) if you neglected time dilation?

d) What is the interval in spacetime between the creation of a typical pion and its decay?

For part a) I just found the dilated time interval.

##Δt={\frac{Δt'}{{\sqrt{1-v^2/c^2}}}}## Where Δt' is the proper time given, and V is the speed at which the pion's reference frame, S', is moving wrt to the lab, S.

Simply plugging in I got ##Δt=6.65x10^{-8}s##

For part b) I'm assuming it means with respect to the lab, so would I just multiply ##Δt## by V?

(This is where the my confusion from the my thread in the physics section comes from: https://www.physicsforums.com/showthread.php?t=736893)

For this step should I be able to find the same distance if I applied the length contraction formula or used Δt and V?

Edit: After messing with some formulas I found that multiplying v by ##Δt## wouldn't give the same as length contraction.

I found that, V, the velocity relative to the lab, S, could be expressed as proper length/ proper time. (Right?)

So ##Δt • V## would give ##L_p • \gamma##

While Length Contraction, L, is given by: ##L={\frac{L_p}{\gamma}}##

I also found an example in my book that is essentially part b) and they used ##Δx=vΔt##. So then what is the difference between L and Δx?

a) What would their mean life be as measured in the lab?

b) How far would they travel(on average)before they decay?

c)What would your answer be to part (b) if you neglected time dilation?

d) What is the interval in spacetime between the creation of a typical pion and its decay?

For part a) I just found the dilated time interval.

##Δt={\frac{Δt'}{{\sqrt{1-v^2/c^2}}}}## Where Δt' is the proper time given, and V is the speed at which the pion's reference frame, S', is moving wrt to the lab, S.

Simply plugging in I got ##Δt=6.65x10^{-8}s##

For part b) I'm assuming it means with respect to the lab, so would I just multiply ##Δt## by V?

(This is where the my confusion from the my thread in the physics section comes from: https://www.physicsforums.com/showthread.php?t=736893)

For this step should I be able to find the same distance if I applied the length contraction formula or used Δt and V?

Edit: After messing with some formulas I found that multiplying v by ##Δt## wouldn't give the same as length contraction.

I found that, V, the velocity relative to the lab, S, could be expressed as proper length/ proper time. (Right?)

So ##Δt • V## would give ##L_p • \gamma##

While Length Contraction, L, is given by: ##L={\frac{L_p}{\gamma}}##

I also found an example in my book that is essentially part b) and they used ##Δx=vΔt##. So then what is the difference between L and Δx?

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