Pierre de Fermat, among other things, enunciated a principle of least time for the behavior of nature, regarding light being refracted at the boundary between two transparent media. The principle was a physical hypothesis that led Fermat, via some ingenious mathematics, to a prediction of the observed law of refraction. As it turned out, the prediction matched the known results, thus supporting the proposed principle.
In this piece, we invoke the same least-time idea but in the well-known context of a lifeguard running across sand at a beach. We show with a little calculus how the law of refraction, Snell’s law, emerges. Then, we consider what a least-time path should look like if the sand were to vary in its softness between the lifeguard station and the water.
Finally, we see that the same mathematical ideas can be applied to the brachistochrone problem that Johann Bernoulli proposed and we give a relatively easy solution.
In this piece, we invoke the same least-time idea but in the well-known context of a lifeguard running across sand at a beach. We show with a little calculus how the law of refraction, Snell’s law, emerges. Then, we consider what a least-time path should look like if the sand were to vary in its softness between the lifeguard station and the water.
Finally, we see that the same mathematical ideas can be applied to the brachistochrone problem that Johann Bernoulli proposed and we give a relatively easy solution.
| blog_16_beach_and_brachistochrone.pdf |
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