# How to write the velocity of gas in rocket motion

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?