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but you've got to do a little more work.
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It's a lot easier here,
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because I get to pump into the equation two facts at one shot,
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that the time between the two ticks is 1 second
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and it's at rest with respect to me,
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which is why the Δx vanished between the two ticks.
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If I want to use your equations,
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Δx' won't be 0
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because the clock has moved,
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and you've got to do... you can do it.
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I have, I think, shown in my notes how to do that but it's unnecessary.
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This is the main result.