I Doubt about variable mass systems

[kuruman]

OK, thanks. Were you able to see the posts in the thread I referred you to in post #26?

The OP in that thread is guilty of setting up the reader for confusion.
First the OP says "The rocket now has velocity v+dv and mass M+dM, where the change in mass dM is a negative quantity.". That's fine.
Then the OP says "If I'm allowed to change M+dM with M-dM and -dM with dM then Eq. 9-38 becomes
M * v = dM * U + (M - dM) * (v + dv)"
Equation 9.38 is
M * v = -dM * U + (M + dM) * (v + dv) ... (9-38)

Well, having defined dM as a negative quantity, one is not allowed to change dM to -dM but to -|dM| to avoid conflict with the definition plus confusion about the variables. Then Eq. 9-38 should be
M * v = |dM| * U + (M - |dM|) * (v + dv) ... (9-38a)
OP writes the equation
M * v = dM * U + (M - dM) * (v + dv) ... (9.38b)

Do you see the difference between equations (9.38a) and (9.38b)? They are the same but only if dM in (9.39b) is defined as a positive quantity which is in direct contradiction to its original definition. Is that what confused you?

The equation $$m~\frac{dv}{dt}=(u-v)\frac{dm}{dt}$$ gets around the issue whether $dm$ is positive or negative. The mass transfer to or from the system of interest 1 (e.g the rocket) and the secondary system 2 (e.g. the fuel) is such that $dm_1+dm_2=0.$ Study the derivation in the insight. Which one is positive and which one is negative is irrelevant. In the equation $m$ and $dm$ are $m_1$ and $dm_1$ with the subscripts dropped as explained in the derivation.