The Calculus of Rainbows

The geometry of reflection and refraction explains the apparent position of a rainbow relative to the sun, and calculus shows why light is concentrated in the rainbow. Exercises include the derivation of Snell’s Law and the Law of Reflection, an explanation for the different colors in the rainbow, and an exploration of secondary and tertiary rainbows. This is intended as a one-hour lecture for honors calculus, with exercises which students can work through on their own. Observation of Rainbows What do you need to make a rainbow? Of course, you need at least two things, rain and sun. Closer examination reveals more details–but not more ingredients! Not all rainbows are alike. Suppose you see a rainbow at sunset (or sunrise). rainbow setting sun observer angle to