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I don't want to go into the rocket problem in any detail.
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It's good for you to know how these things are done.
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Here's a rocket whose mass at this instant is M,
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whose velocity is v right now.
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What rockets do is they emit gasses,
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and the gasses have a certain exhaust velocity.
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That velocity is called v0.
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In magnitude, it's pointing away from the rocket,
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and it has a fixed value relative to the rocket,
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not relative to the ground.
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If you are riding with the rocket
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and you look at the fumes coming out of the back,