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they will be leaving you at that speed v0.
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A short time later-- What happens a short time later,
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the rocket has a mass M - δ
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because it's lost some of its own body mass
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in the form of exhaust fumes.
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The exhaust fumes, I'm just showing them as a blob here,
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and the rocket's velocity now is not v, but v + Δv.
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And what's the velocity of the fumes?
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Here's where you've got to be careful.
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If your velocity was v at that instant,
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the velocity of the rocket fume,
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with respect to the ground is v - v0;