Applying Newton's 2nd to a rocket

[Dustinsfl]

A rockt with mass \(M\) is sitting motionless in space. The rocket is powered by steam with velocity \(V_s\) and mass per time \(C = \frac{dm}{dt}\). Write an equation for the rockets acceleration in terms of \(M, {} V_s\), and \(C\).

So \(\mathbf{F}_s = \mathbf{F}_r\) by Newton's 3rd. By Newton's 2nd,
\begin{align}
\mathbf{F}_s &= \mathbf{F}_r\\
\frac{d(mV_s)}{dt} &= \frac{d(mV_r)}{dt}\\
m\frac{dV_s}{dt} + V_s\frac{dm}{dt} &= m\frac{dV_r}{dt} + V_r\frac{dm}{dt}\\
V_s\cdot C &= ma_r
\end{align}
Here we assumed the steam isn't accelerating and \(\frac{dm}{dt} = 0\) on the RHS. I understand assuming the steam acceleration is zero, but why is the \(\frac{dm}{dt}\) on the RHS zero and not \(C\)?

 

[Aaron Bex]

The \(\frac{dm}{dt}\) term on the RHS is zero because the mass of the rocket does not change over time. The steam is the only thing that is changing in mass, so its rate of change, \(C\), is the only factor that needs to be accounted for on the right side of the equation.