The observer on the spaceship would measure a length of:
L' = 1 meter /
(1 - v²/c²) meters
What would an observer on the spaceship measure for a meterstick on the space station?
Objects in motion appear shorter in the direction of their motion compared to an observer at rest. The amount of contraction depends on the object's velocity relative to the observer.
The observer on the space station is at rest relative to the meterstick on the space station. Therefore, she measures its true length of 1 meter.
The observer on the spaceship is moving relative to the meterstick on the space station. Therefore, she will measure a shorter length for the meterstick due to length contraction.
Calculating the exact contracted length requires knowing the relative velocity between the spaceship and the space station. However, we can express the answer in terms of this unknown velocity (v) using the Lorentz transformation:
Contracted length (L') = L rest /
(1 - v²/c²)
where:
L' is the contracted length measured by the spaceship observer
L_rest is the rest length of the meterstick (1 meter)
v is the relative velocity between the spaceship and the space station
c is the speed of light (approximately 3 x 10^8 m/s)
Therefore, the observer on the spaceship would measure a length of:
L' = 1 meter /
(1 - v²/c²) meters