special relativity

[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.

[russ_watters]

When would u ever do this again, and why isnt the pion's lifetime= t, b/c its at rest. From the pion's point of view, its at rest, from yours, its not. The question is asking about its lifespan from yours.

[rayjohn01]

Good Quest Answer . Lets say you are person A -- and all you have or can see is light
Then to you light speed is fixed for ever and as far as you know you are at rest ( assuming only you are NOT subject to hidden forces ) . Now we introduce person B .
Assume first off that you cannot see B and he cannot see you -- then you both think you are at rest and just light is speeding around -- But now if you can see each other there are two view points yours and his and you will both judge the other moving by some speed wrt themselves and hence to light . You (A) will apply the correction
to B according to your view of his speed v wrt 'c' so for instance you will say his time slowed or lifetime increased ---- But B will say the same of you .
B ( the pion ) will see the Earth shrink in distance because you and the earth are seemingly traveling together at v .
In short you always apply the correction but from your viewpoint -- if you wish to put yourself in anothers shoes then you apply it to what THEY see .
Ray.

[answerseeker]

so how do you know that 10m is from YOUR point of view, not the pion's? since its lifespan is from It's point of view. how do you decipher which goes with who's point of view?

[HallsofIvy]

Because YOU were the one who measured it! pions do not carry meter sticks with them!

[JesseM]

so how do you know that 10m is from YOUR point of view, not the pion's? since its lifespan is from It's point of view. how do you decipher which goes with who's point of view? Every object is at rest "from its own perspective" (ie in its own reference frame). So if something is said to be moving at some nonzero velocity v, you know that must be from the perspective of someone else.