I just have a quick question about time dilation/proper time because my physics book makes it a little confusing. Let's say we have an observer on Earth, and then an observer on a space ship. The space ship leaves Earth, flies to the Moon, and then returns to Earth. Who is the person measuring the proper time and why? I know that a clock "runs slower" when it is in motion because it is in frame S' which is the rest frame of the clock, but doesn't the observer on Earth also have a clock that is in it's rest frame?

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EDIT AFTER FIRST COMMENT: Time dilation isn't the appropriate effect to consider in this particular problem -- length contraction is. In Nick's frame, a length contracted ship passes by at speed $v$. In Molly's frame, a point-object (heh) Nick passes by an uncontracted ship at speed $v$. Clearly, this should happen quicker in Nick's frame because of the length contraction. Thinking in terms of time dilation simply doesn't help here. Think from the point of view of each observer and it will be quickly obvious which effect to use.