How to Apply Newton’s Second Law to Variable Mass Systems

[kuruman]

Introduction
The applicability of Newton’s second law in the oft-quoted “general form”  $$\begin{align}\frac{d\mathbf{P}}{dt}=\mathbf{F}_{\text{ext}}\end{align}$$ was an issue in a recent thread (see post #4) in cases of systems with variable mass.  The following example illustrates the kind of confusion that could arise from the (mis)application of Equation (1):
A rocket is hovering in place above ground near the Earth’s surface. Assume that the combustion gases are expelled at constant rate $\beta=dm/dt$ with velocity $w$ relative to the rocket.  What condition must hold for the rocket to hover in place?
A novice might start with Equation (1) and go down the garden path only to reach a quick impasse as shown below.
Attempted solution
We start with the general form of Newton’s second law, Equation (1) $$\frac{dP}{dt}=M\frac{dV}{dt}+V\frac{dM}{dt}=-Mg$$ If...

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