An observer on Earth sees you travel 1000 m at half the speed of light. What distance do you think that you travelled?
To answer this question, we need to consider the effects of special relativity, namely time dilation and length contraction.
According to special relativity, as an object moves closer to the speed of light, time dilates, meaning that time passes more slowly for the moving object compared to a stationary observer. Additionally, length contraction occurs, which means that the length of the moving object appears shorter from the perspective of the stationary observer.
In this scenario, an observer on Earth sees you travel 1000 meters. Let's assume the observer measures this distance with their own reference frame. We can calculate the distance you perceive from your reference frame using the concept of length contraction.
To determine the distance you traveled, we need to know your velocity relative to Earth. You are traveling at half the speed of light, which is approximately 1.5 x 10^8 meters per second.
Using the formula for length contraction:
Length_contracted = Length_proper / γ
where γ (gamma) is the Lorentz factor given by:
γ = 1 / sqrt(1 - (v^2 / c^2))
v is your velocity (1.5 x 10^8 m/s) and c is the speed of light (3 x 10^8 m/s).
Plugging in the values, we have:
γ = 1 / sqrt(1 - ((1.5 x 10^8)^2 / (3 x 10^8)^2))
γ = 1 / sqrt(1 - 0.25)
γ = 1 / sqrt(0.75)
γ ≈ 1.1547
Now, we can calculate the distance you traveled as observed from your reference frame:
Length_proper = Length_contracted * γ
Length_proper = 1000 meters * 1.1547
Length_proper ≈ 1154.7 meters
Therefore, from your perspective, you traveled approximately 1154.7 meters.