ANSWERS: 2
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The experiment will take an extremely long time so it depends on the battery life of the watches. If they both die at the same time then yes. If they die at separate times (more likely) then they will display different times. . If both watches keep time accurately throughout the experiment though, they should have the same time at the end.
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It depends on the speed at which you are going to the nearest star. If I travel at near the speed of light my clock slows down and my stopwatch does also. I get there at a time sooner than that measured by the other planet.
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