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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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