Special Relativity and pion

[Herjo345]

How fast must a pion be moving, on average, to travel 9.0 m before it decays? The average lifetime, at rest, is 2.60 ✕ 10-8 s. (Answer in terms of c)

I'm not exactly sure what equations I have to use but I believe it relates to time dilation.


I originally didn't realize it was a time dilation problem and calculated that the speed would be .8666c but since it relates to time dilation I am not sure how to go about solving it.

Thank you for your help.

 

[collinsmark]

How fast must a pion be moving, on average, to travel 9.0 m before it decays? The average lifetime, at rest, is 2.60 ✕ 10-8 s. (Answer in terms of c)

I'm not exactly sure what equations I have to use but I believe it relates to time dilation.I originally didn't realize it was a time dilation problem and calculated that the speed would be .8666c but since it relates to time dilation I am not sure how to go about solving it.

Thank you for your help.

You have a bit of algebra ahead of you. But let's just set up the equations first.

If $\tau$ is the time it takes the pion to decay in it's own frame of reference ($\tau$ = 2.60 × 10^-8 s), what is t, the amount of time it takes to decay in your stationary frame of reference?

(you can answer this intermediate result in terms of c and v; or in terms of $\gamma$, your pick for now ).

 

[Herjo345]

I could use the equation (ΔT/Δt)=√(1-(v/c)^2) but I'm still a little unsure on the reference frames. Is the first reference from from the pion and the second from a stationary observer?

 

[Herjo345]

I figured it out, thank you for your help.

 

[HalfLostAndSearching]

I would like to clarify some concepts in this question. Special relativity is a theory that explains the relationship between space and time, particularly in the presence of high speeds. It states that the laws of physics are the same for all observers in uniform motion, regardless of their relative velocities. Pion, on the other hand, is a subatomic particle that is unstable and decays into other particles after a certain amount of time.

To answer the question, we need to use the concept of time dilation in special relativity. Time dilation states that time appears to run slower for an observer in motion compared to an observer at rest. This means that the average lifetime of a pion, which is 2.60 ✕ 10-8 s when at rest, will appear longer for an observer moving at a high speed.

We can use the equation for time dilation, t' = t/√(1-v^2/c^2), where t' is the observed time, t is the rest time, v is the velocity, and c is the speed of light. We can rearrange this equation to solve for v, which is the velocity of the pion.

v = √(1- (t'/t)^2)c

Plugging in the values, we get:

v = √(1- (9.0/2.60 ✕ 10-8)^2) ✕ 3.00 ✕ 10^8 m/s

v = 0.999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999