The Rocket Equation and that Minus Sign: An Avoidable Pitfall in the Language of Physics

Abstract

Derivations of the rocket equation that use a momentum-conservation argument are notorious for often encountering a problem that would produce an incorrect sign in their result, if it were not “fixed” in various ad hoc ways that can always be shown to be mathematically invalid. Here, we show that the problem of the unwanted minus sign typically (perhaps always) results from a misunderstanding of the language of calculus. When one adheres to a simple and fundamental rule of calculus notation, the derivation of the rocket equation becomes straightforward, and no unwanted minus sign appears.