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That's the time for a round trip.
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You look at my clock and what do you think is happening?
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Remember, you are moving to the right, relative to me,
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so according to you my clock is moving to the left,
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and it looks like this.
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Light beam is going on a zigzag path, according to you.
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You agree that's very clear.
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For me, the pulse is going up and down,
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for you it's going on the hypotenuse.
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You can imagine how with the hypotenuse u squared
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and c squared are going to come from drawing the sides of the triangle.
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That's why the time would come out longer.