# Euclidean Geometry in Physics

I've started tutoring my 13 year old niece in math. She learning geometry this year (it's a year-round school). Obviously, it'll just be basic Euclidean geometry -- though I might try to get to a little non-Euclidean stuff at the end.

Anyway, she says she's interested in physics (how she can interested in something she's never studied IDK, but I guess it's good she's interested in science). I was trying to think up ways to teach her some physics principles through geometry. Unfortunately, without calculus, the only physics I can think of that uses geometry is the center of mass of an object.

Are there any other physics principles/ problems I could go over if all she knows is some basic algebra and geometry?

• PS, her school does Algebra II before Geometry. So she understands elementary algebra fairly well (though obviously not any linear algebra, group theory, etc). – user156332 Jun 10 '14 at 16:49

A few thoughts:

• Resolving vectors into components is all about trigonometric ratios, which in turn are all about similar triangles.
• Inverse-square laws (gravity, electrical force) can be interpreted geometrically in terms of the surface area of a sphere.
• Planetary orbits and ellipses.

I am sure I will think of more and will add to this list as they occur to me.

I am a fan of collisions. Get some simple euclidean shapes (billiard balls, or dice on ice) and show her how to calculate the angles that they'll move in after hitting each other.

• This is a great idea, there is almost no calculus in collision problems. It's all algebra. That said, developing basic trigonometry is an essential task to solve problems nicely. – James S. Cook Jun 10 '14 at 18:04

I remember that we did some geometrical optics in high school. Maybe it is a little to advanced, but maybe you'd like to judge by yourself. So what I concretly remember is:

• Snells law
• Lens optics, especially the "thin lense formula"

Snells law uses very basic trigonometry (actually only the definition of sin), but you could avoid this by just using ratios.

• I forgot all about geometric optics, I think I have a book on that around here somewhere. Thanks for the suggestion. – user156332 Jun 10 '14 at 17:19

People already gave some good ideas, but I'll pitch in what could be a more systematic approach

Without calculus is somewhat difficult to study physics after Newton. Fortunately a lot was known before him. An account of this can be seen in Rene Dugas' "A History of Mechanics". In this book he explains how people did mechanics in the past, with lots of diagrams and doing explicitly the geometrical arguments people gave before Newton (and after as well, but that's not the point).

If I recall correctly almost all proofs before Newton required just euclidean geometry and some simple algebra, plus a bit of trigonometry every now and then. But you should be able not only to state but also to prove almost every assertion one learns in high school level mechanics, such as uniformly accelerated motion, static equilibrium and forces and so on.

It would take you some time to "adapt" the book for lecturing her, but it should enable you to go somewhat far.

• Thanks for the suggestion. I'll look into it. – user156332 Jun 10 '14 at 19:14