# Special relativity and substitution

In summary: In this particular case, the person measuring the speed is person A, and the pion is at rest in its own reference frame. To person A, it seems as if the pion is moving at a speed of 10 meters per second, so person A applies the appropriate correction to the pion's time according to its own perspective (ie assumes that the pion is moving at a speed that is slower in person A's reference frame than it actually is).
i don't 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 don't 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 isn't the pion's lifetime= t, b/c its at rest.

When would u ever do this again, and why isn't 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.

Good Quest Answer . Let's 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.

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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?

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

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.

## 1. What is special relativity?

Special relativity is a theory developed by Albert Einstein that explains the relationship between space and time. It states that the laws of physics are the same for all observers in uniform motion, regardless of their relative velocity.

## 2. How does special relativity differ from classical mechanics?

Special relativity differs from classical mechanics in that it takes into account the fact that the speed of light is constant and the laws of physics are the same for all observers in uniform motion. It also introduces the concept of time dilation and length contraction, which are not present in classical mechanics.

## 3. What is the principle of relativity?

The principle of relativity states that the laws of physics are the same for all inertial observers (observers in uniform motion). This means that there is no preferred frame of reference and all physical laws must be the same for all observers.

## 4. What is the equation for time dilation in special relativity?

The equation for time dilation in special relativity is t' = t / √(1 - v^2/c^2), where t' is the time observed by an observer in relative motion, t is the time measured by an observer at rest, v is the relative velocity between the two observers, and c is the speed of light.

## 5. What is the principle of substitution in special relativity?

The principle of substitution in special relativity states that we can use the equations of special relativity to calculate physical quantities in a moving frame of reference by substituting the relative velocity and time measured in that frame into the equations. This allows us to predict how physical quantities will change for observers in different frames of reference.

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