Carrying on with our example of the spaceship traveling to a distant planet, let's think about what it means for measuring distance. The one thing that might puzzle you is this: everything is relative, so a person on the Earth sees the clock on the spaceship running slow. Similarly, the person on the Earth is moving at 0.95c relative to the observer on the spaceship, so the observer on the ship sees their own clock behaving perfectly and the clock on the Earth moving slow. So, if the clock on the spaceship is measuring time properly according to an observer moving with the clock, how can we account for the fact that the observer on the ship seems to cover a distance of 9.5 light years in 3.122 years, which would imply that they're traveling at a speed of 3.04c?

That absolutely can not be true. For one thing, one of the implications of relativity is that nothing can travel faster than c, the speed of light in vacuum. c is the ultimate speed limit in the universe. For another, two observers will always agree on their relative velocities. If the person on the Earth sees the spaceship moving at 0.95c, the observer on the spaceship agrees that the Earth is moving at 0.95c with respect to the spaceship (and because the other planet is not moving relative to the Earth), everyone's in agreement that the relative velocity between the spaceship and planet is 0.95c.

So, distance is velocity multiplied by time and we know the velocity and time measured by the observer on the spacecraft is 0.95c and 3.122 years. This implies that they measure a distance for the trip of 2.97 light-years, much smaller than the 9.5 light-year distance measured by the observer on the Earth.

This is in fact exactly what happens; a person who is moving measures a contracted length. In this case, the person on the Earth measures the proper length, because they are not moving relative to the far-off planet. The observer on the spaceship, however, is moving relative to the Earth-planet reference frame, so they measure a shorter distance for the distance from the Earth to the planet. The length measured by the moving observer is related to the proper length by the equation:

length contraction: L = Lo

In this case we can solve for the length measured by the observer on the spaceship:

L = 9.5 (1 - 0.952 )1/2 = 2.97 light-years.

This agrees with what we calculated above, as it should.

One important thing to note about length contraction: the contraction is only measured along the direction parallel to the motion of the observer. No contraction is seen in directions perpendicular to the motion.