An observer on Earth sees you travel 1000 m at half the speed of light. What distance do you think that you travelled?
To determine the distance you think you traveled, we can use the concept of time dilation from special relativity. Time dilation states that when an object is moving relative to an observer, time passes more slowly for the moving object compared to the stationary one.
Given that you traveled at half the speed of light, which is approximately 3x10^8 meters per second, we can calculate the time dilation factor. According to the Lorentz factor formula, γ (gamma) is equal to 1 divided by the square root of (1 - (v^2 / c^2)), where v is the velocity and c is the speed of light.
Since you traveled at half the speed of light, v = 0.5c. Plugging this value into the formula, we have:
γ = 1 / √(1 - (0.5c)^2 / c^2)
γ = 1 / √(1 - 0.25)
γ = 1 / √(0.75)
γ ≈ 1.155
This means that time on your clock is passing roughly 1.155 times slower than it would for the observer on Earth.
Now, we need to consider the distance you travel according to your own reference frame. From your perspective, your distance traveled is simply the distance contracted by the same factor γ. So, if the observer on Earth measures the distance as 1000 meters, you would perceive it as:
Distance = 1000 meters / γ
Distance ≈ 865 meters
Therefore, you would perceive that you traveled approximately 865 meters, even though the observer on Earth measures it as 1000 meters.