special relativity- time expansion

[answerseeker]

i dont understand when to use substitution as used in the answer to this question:

how fast must a pion be moving, on average, to travel 10m before it decays? average lifetime is 2.6*10^-8.

i know the answer is D=V( to/ sqroot 1-v^2/c^2) but i dont understand why and how to know when to put the time expansion equation into the D=vt equation.
When would u ever do this again, and why isnt the pion's lifetime= t, b/c its at rest.

[Davorak]

t_0 = \gamma t = \frac{t}{\sqrt{1-\frac{v^2}{c^2}}}


Where t_0 is the time in the labortory frame and t is the time for the pion. So:

D=V t_0 = V t \gamma= V \frac{t}{\sqrt{1-\frac{v^2}{c^2}}}

[answerseeker]

so T is the rest time for the pion... what about 10m? that isnt from pion's point of view, but how do you know that?

[Davorak]

The equation I posted last time was for the distance in the laboratory frame. The question is asking for 10m in the laboratory frame or else you would not need to inclulde the gamma factor(\gamma).

In SR, from the pions point of view it is at rest and it is the rest of the world that is moving around the pion. It does not make sence to ask how far it moves in its own frame.

[answerseeker]

o, much clearer now, thnx