I'm having quite some issues deriving the ideal rocket equation, and I suspect it is due to lack of identifying scalar terms and vector terms. Here is how I derive it :
Let $v$ be the velocity of the rocket and $u$ be the velocity of the exhaust, relative to us, the observer.
$$P_i = mv$$
$$P_f = (m-dm)(v+dv)-udm \space\space\space$$ (Here I've put the negative sign before $u$ as it is in the opposite direction)
$$\Delta P = P_f-P_i = mdv-vdm-udm\space$$(ignoring the term $dmdv$)
$$F_{ext}\space dt = mdv-(u+v)dm$$
This is where my confusion begins. What exactly is relative velocity? Is it $(u+v)$ or is it $-(u+v)$ ?
If I consider the rocket is moving as $v\hat{i}$ and the gas is moving at $-u\hat{i}$, then their relative speed is $u+v$. From the rocket's perspective, the relative velocity of the gas is $v_{rel} =(u+v)(-\hat{i})$.
Then we can say, $u=(v+v_{rel}) = v\hat{i}+(v+u)(-\hat{i})=-u\hat{i}\space,$ which is exactly what we wanted. So, we can write (along $\hat{i}$, taking that to be positive) :
$$F_{ext}dt=mdv + v_{rel}dm$$
We can also do this in another way, where we take the direction of the gas to be positive. In that case, our expressions would have an extra negative sign on both sides, but after that, the final expression would be exactly the same.
My confusion is regarding relative velocity. If the speed of the rocket is $v\hat{i}$ and the exhaust is $-u\hat{i}$, shouldn't we say $u=v+v_{rel}$ ?
Many sources write $u=v-v_{rel}$ instead. For example, the comments below this [question]:Rocket equation derivation mistake by my professor
Is it because, they are considering $v_{rel}$ a scalar here, and using the extra negative sign to denote the opposite direction, but since I'm considering $v_{rel}$ a vector here, and so, I don't have to plug in an extra negative sign here ?
Can I say :
$$\vec{u} = \vec{v}+\vec{v_{rel}} = v\hat{i} + (v+u)(-\hat{i})$$
Then this will automatically give me the negative sign in front of $u$.
I suspect what the others do is consider the scalar version of the above equation. Can anyone tell me if I'm correct, is my expression for relative velocity correct, or should there be a negative sign as most sources, including Wikipedia claim there is.
