Newton's equation
Last week we dug into Newton's equation of motions as a concrete example of differential equations---equations with derivatives---the topic we had been talking about the last few sessions. Newton's equations generally describe how objects move through space subject to a force, and we can model the behavior of an object or objects by defining what forces act on them.
We talked about 3 cases in particular:
- when there are no forces (the boring case where objects keep going where they were originally going)
- when there is a constant force (like gravity's effect on an object on earth, which pulls object in a certain direction)
- when multiple objects attract each other through a law like gravity.
We touched on the fact that in science class we learn formulas for determining how far things fall on earth after a certain time, but that those formulas come ultimately from using calculus to solve the different equation, as might be done in AP classes and certainly at university.
But we were more interested in the case where calculus cannot help us, for example in case (3), the so-called N-body problem. So we spent some time developing our intuitions about derivatives using an example of how far you travel when driving at different speeds at different times. Since Caleb had some calculus in school, he had a different way of looking at the problem than Alex and Luka, who could solve the problem using graphing and geometry.
In the end, we used the ideas from geometry to develop numerical algorithms for the computer. My hope was to convey how simple the algorithm is, because it was literally encoding the geometrical solution that Alex and Luka came up with. With the algorithm, we could explore case (3) above to simulate how planets might orbit each other in space and watch them in action with animated results. I think this, if nothing else, caught their interest!