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I was wondering why the mass of the fuel is regarded as -dm and the mass of the rocket as m+dm? I understand that the total mass has to be unchanged (ie m). Why can't I say that the rocket's mass is m-dm and the mass of the fuel being ejected is dm? When I do that and continue with the suggested answer, I get v but in the opposite direction. Would appreciate any explanation

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Here, $\mathscr{d}m$ is inherently negative because rockets mass decreases with time.

$$ \mathscr{d}m = -|dm| $$

And the rocket's mass is,

$$ m + \mathscr{d}m = m - |\mathscr{d}m| $$

For reference, please look Young's University Physics with Modern Physics, Ch 8, Sec 8.6

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