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I recently tried to derive the rocket equation using conservation of momentum, and did not get very far; was wondering what I am missing. Here's my attempt:

Let's say that over some small time $dt$, the rocket ejects $dm$ mass at speed $c$ going the opposite way, and speeds up by $dv$ as a result. So, $$p_i = mv$$ $$p_f = (m-dm)(v+dv)-cdm$$ The $cdm$ term is subtracted since it is going the opposite way. Expanding $p_f$, simplifying the equation $p_i=p_f$, and ignoring the $dmdv$ term, we get $$(c+v)dm = mdv \Rightarrow \frac{dv}{c+v}=\frac{dm}{m}$$ This is clearly wrong because after integrating it says that $v$ has a linear relation to the mass. What forbidden moves have I done?

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You have forgotten that the velocity of the exhaust relative to the "observer" depends on the velocity of the rocket relative to the observer, as we have a constant velocity of the exhaust relative to the rocket. $c$ is not a constant; therefore, you need to take into account that $c=c(v)=-v_e-v$, where $v_e$ is the velocity of the exhaust relative to the rocket.$^*$


$^*$Since in your $p_f$ equation you made the $c\text dm$ term negative, technically $c$ is the velocity of the observer relative to the exhaust. If you want the velocities to all be relative to the observer in the $p_f$ equation that term should be positive. Also keep in mind that just because the exhaust is traveling backwards relative to the rocket does not necessarily mean that the exhaust is traveling backwards relative to the observer. Your mistakes here most likely come from not being careful enough in really thinking about these relative velocities.

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