To calculate the perceived length of an object moving at relativistic speeds, we need to apply the Lorentz contraction formula. According to special relativity, objects appear shorter in the direction of their motion when observed from a reference frame in which they are moving.
The formula for Lorentz contraction is as follows:
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. v is the relative velocity between the observer and the object. c is the speed of light in a vacuum.
In this case, we have: L = 1 meter (rest length of the stick) v = 0.5c (relative velocity of the stick)
Now, let's calculate the perceived length:
L' = 1 meter * sqrt(1 - (0.5c)^2/c^2) = 1 meter * sqrt(1 - 0.25) = 1 meter * sqrt(0.75) ≈ 0.866 meter
Therefore, the perceived length of the 1-meter thick stick, moving with a velocity of 0.5c relative to Earth, would be approximately 0.866 meters when observed from Earth.