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Consider the motion of a rocket of mass $m$ in space with gas expelled at relative velocity $u$. I found two different version of writing the momentum of the system on Klepper Kolenkow and Morin book. Both agree about momentum of the system at the istant $t$.

$$P(t)=mv$$

While the difference is on the momentum at istant $t+dt$. On Morin I found:

$$P(t+dt)=(m-dm)(v+dv)+dm(v-u)$$

While on Klepper Kolenkow it is claimed

$$P(t+dt)=(m-dm)(v+dv)+dm(v+dv-u)$$

The fact is: the mass $dm$ of gas is travelling at relative velocity $u$ with respect to the rocket. But is the rocket to be considered moving at velocity $v$ or $v+dv$ when writing the velocity of the gas?

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The two equations are the same to first order, which is all that is important. If I were writing down the equation for the total momentum P(t+dt) myself, I would probably jot down the first equation (that of Morin) since I would be thinking of the instantaneous velocity of the rocket at time t rather than at time t+dt. But, again, the distinction is not important. The only difference between the equations is that when expanded out the second equation gives an additional term of (dm)(dv), which is a 2nd-order infinitesimal and therefore can be neglected with respect to all the first order infinitesimal terms.

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