To calculate the perceived length of an object moving at a significant fraction of the speed of light (0.5c) relative to an observer on Earth, we need to take into account the effects of special relativity, specifically length contraction.
According to the theory of special relativity, objects that are moving at high velocities relative to an observer will appear shorter in the direction of their motion. The formula for length contraction is given by:
L' = L * sqrt(1 - (v^2/c^2))
Where: L' is the perceived length of the object L is the rest length of the object (1 meter in this case) v is the velocity of the object relative to the observer (0.5c) c is the speed of light in a vacuum (approximately 299,792,458 meters per second)
Plugging in the values, we can calculate the perceived length:
L' = 1 meter * sqrt(1 - (0.5c)^2/c^2) L' = 1 meter * sqrt(1 - 0.25) L' = 1 meter * sqrt(0.75) L' ≈ 0.866 meters
Therefore, if a 1-meter thick stick is moving with a velocity of 0.5c relative to Earth, it would be perceived to have a length of approximately 0.866 meters due to length contraction.