If a spacecraft whose length is observed by its pilot to be 30 meters passes an observer at a relative velocity of .8 * c, where c is the speed of light, then provided the spacecraft's length is oriented in its direction of motion what will be the length of the craft as measured by the observer?
At relative velocity v any observer will measure the dimensions of objects in the other frame of reference to be decreased in the direction of motion. This follows easily from the fact that time is dilated, as seen in the preceding problem, but the details of this reasoning are not included here.
In the present situation the pilot is presumed to be at rest with respect to the spacecraft. The length as measured by the pilot is called the 'rest length' of the spacecraft. The observer in this situation is outside the spacecraft observing its passage.
Assuming that the spacecraft is oriented with its length in the direction of motion, its length as detected by the observer will therefore be less than that observed by the pilot who is presumably at rest with respect to the rocket.
The 'length contraction factor' will be `sqrt ( 1 - v^2 / c^2 ). In this case v = .8 * `c, so the observed length will be
It is important to understand that this observation is not an optical illusion. This is what will be observed by any measuring instrument, including an array of high-speed and very accurate electronic sensors.
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