- #1

Will Flannery

- 122

- 36

This difficulty arises in the first high school physics class, and reappears in practically every university physics class. In high school physics the formula for the acceleration of a falling object is derived, A(t) = G∙M/(r(t)∙r(t)), and this differential equation is unsolvable*. So, the substitution is made A(t) = -9.8 and the student can calculate simple trajectories. The interesting systems, orbits and rocket trajectories, are off the table.

The same thing happens in university classes in classical mechanics, electric circuit analysis, rigid body motion, heat transfer, stress and strain in materials, fluid dynamics, and electrodynamics. Process models are derived that are solvable only for the the simplest or trivial systems.

The fix is to teach a high school class in scientific programming that uses Euler's method to compute approximate solutions to differential equations, and to apply it to problems in several areas of physics. The course is independent of the math curriculum beyond high school algebra and geometry.

Euler's method is very simple and can be taught to high school science students with no prior exposure to calculus in a one hour lecture, and used to calculate by hand the trajectory of a falling object.

It works like this: given the starting position r(0) and velocity v(0), the approximate trajectory is advanced in time, from time t to time t + ∆t as follows:

r(t+∆t) = r(t) + v(t)∙∆t

v(t+∆t) = v(t) + A(t)∙∆t

If ∆t is small, say 20 milliseconds (as in Honeywell inertial navigation units), the estimate is accurate. This is Euler's method.

The two computational statements can be translated 1 to 1 to MATLAB (or the language of your choice, preferably one with great graphing capabilities).

You can do the same thing for orbits and rocket trajectories, and problems in electric circuit analysis, and 2-d rigid body motion.

I've written a paper

http://www.berkeleyscience.com/The%20Coming%20Revolution%20in%20Physics%20Ed.pdf

that describes a high school course in scientific programming and includes complete solutions for representative problems in central force motion, electric circuit analysis, and 2-D rigid body motion. Each problem/solution consists of:

#1 physical laws - simple DEs

#2 process model - the process models have 1 or 2 line derivations

#3 computational equations - by rote using Euler's method

#4 actual code - 1 loop containing 1 statement for each computational equation

#5 graphs of results - MATLAB and Python have great built-in graphing functions

An example: the Apollo trajectory.

Course synopsis:

Euler's method extended to partial differential equations is the finite difference method (FDM), and it too is trivially easy. The FDM can be used to compute solutions for problems in heat transfer, the wave equation, stress and strain in materials, fluid dynamics and electrodynamics. Deriving the process models becomes more difficult as more geometrical reasoning is involved and tensors appear in the analyses of stress and strain in materials and fluid dynamics. Selected topics from this material could be covered in a 2nd semester university course.

* what Newton did: see http://farside.ph.utexas.edu/teaching/336k/lectures/node32.html