Document Type


Publication Date


Publication Title

American Journal of Physics






868- 874

Publisher Name

American Association of Physics Teachers


We analyze a category of problems that is of interest in many physical situations, including those encountered in introductory physics classes: systems with two well-delineated parts that exchange energy, eventually reaching a shared equilibrium with a loss of mechanical or electrical energy. Such systems can be constrained by a constant of the system (e.g., mass, charge, momentum, or angular momentum) that uniquely determines the mechanical or electrical energy of the equilibrium state, regardless of the dissipation mechanism. A representative example would be a perfectly inelastic collision between two objects in one dimension, for which momentum conservation requires that some of the initial kinetic energy is dissipated by conversion to thermal or other forms as the two objects reach a common final velocity. We discuss how this feature manifests in a suite of four well-known and disparate problems that all share a common mathematical formalism. These examples, in which the energy dissipated during the process can be difficult to solve directly from dissipation rates, can be approached by students in a first-year physics class by considering conservation laws and can therefore be useful for teaching about energy transformations and conserved quantities. We then illustrate how to extend this method by applying it to a final example.


This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in American Journal of Physics, 87(11), 868- 874 (2019) and may be found at

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Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 License.

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