Special relativity- time expansion

In summary, the question is asking for the speed at which a pion must travel in order to decay after traveling 10m, given its average lifetime. The equation D=V(to/sqrt(1-v^2/c^2)) can be used, but it is important to consider the context and frame of reference in which the distance and time are being measured. In this case, the equation is being applied in the laboratory frame, where the pion is not at rest. From the pion's perspective, it is at rest and it does not make sense to ask how far it has traveled in its own frame.
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.

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

Where $t_0$ is the time in the labortory frame and $t$ is the time for the pion. So:
$$D=V t_0 = V t \gamma= V \frac{t}{\sqrt{1-\frac{v^2}{c^2}}}$$

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

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 sense to ask how far it moves in its own frame.

o, much clearer now, thnx

1. What is time dilation in special relativity?

Time dilation in special relativity refers to the phenomenon where time appears to run slower for an observer in motion relative to another observer who is at rest. This is due to the fact that time and space are not absolute, and are affected by the speed at which an object is moving.

2. How does time dilation affect the perception of time?

Time dilation can cause the perception of time to differ for observers in relative motion. For example, an astronaut traveling at high speeds in a spaceship would experience time passing slower compared to a person on Earth. This means that the astronaut's biological clock would run slower, and they would age less than someone on Earth who is not moving at high speeds.

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

The equation for time dilation in special relativity is t' = t / √(1 - v²/c²), where t' is the perceived time for the moving observer, t is the time for the stationary observer, v is the relative velocity between the two observers, and c is the speed of light.

4. How does gravity affect time dilation in special relativity?

In special relativity, gravity is not considered. However, in general relativity, which is a more comprehensive theory of gravity, it is believed that gravity can also cause time dilation. This is because gravity is not a force, but rather a curvature of spacetime, and this curvature can affect the passage of time.

5. Is time dilation only observed in extreme cases?

No, time dilation can be observed in everyday situations as well. For example, the clocks on GPS satellites experience time dilation due to their high speeds and distance from Earth's gravitational pull. However, the effects are very small and can only be measured with extremely precise instruments.

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