What is this all about?

Over the last decade there has been a consistent and evolving discussion about what physics and engineering students learn about applying their knowledge to 'real life' situations. As we have talked about in class the real world is often difficult to describe in ways which can be described by the esthetically pleasing analytic solutions we typically explore in introductory classes. This challenge is exacerbated by a natural inclination in the physics community to see analytic solutions as the ultimate goal of physics problem solving. Particularly with problems that evolve in time in complex ways such analytic solutions may not be reasonably achievable.

On the other hand there are a wide range of numerical techniques that have been developed to address these circumstances. You have experienced some of these in calculus class where you discussed approximating integrals with rectangles and shells. I trust you realized that in such problems that if the integral of the function you create does not exist you can still manually calculate the volume or area of the rectangle and get a reasonable estimate of the result. These numerical tools for addressing very complex problems have their own challenges in that manually calculating the area of many rectangles is painful repetitive work. While it undoubtedly suffers from some artistic simplification this is what is at the root of the scene in the movie 'Hidden Figures' where they realize that they can solve an analytically intractable problem with an iterative numberical approach. Those of you who have seen the movie undoubtedly noticed that in those days a 'calculator' was a human being who did the work that you expect from your digital device called a calculator.

In the days before meaningful desktop computers numerical analysis was literally done by hand with paper and pencil and tracked in carefully kept notebooks. The process works but it is slow and very resource intensive. As the digital world came into being one very important impact was to permit us to use technology to execute the repetitive calculations that are part of all numerical approximation processes. As we discussed in a previous class this is true even for apparently simple things like the cosine of an angle. Years ago we had large books on our desks with the values of trig functions for angles down to the arcsec. Now your calculator actually calculates the value of the trig function from the Taylor series. If you had to use the Taylor series to calculate the value of a trig function you needed you would go nuts!

Possible actual examples range from telephoto len, and now phone camera lens design, to the heating of turbine blades in jet engines. You explore some of these numerical processes in GE102 (if you've had it) using the tools embedded in Excel. The same course at OSU and many other universities uses various programming packages like Matlab, Mathematica, and Maple instead of Excel. Python is one of these programming languages.

The precipitating event for me around these questions of developing student skills with numerical solutions to 'real' or 'authentic' problems happened last spring. I asked the ENGR 212 (Dynamics) students if they could build a model in Excel that would explain why toast typically lands jelly side down when it slips off the edge of the counter. I also asked them to explore why, if you PUSH the toast as it goes over the edge it typically DOESN'T flip over. [feel free to experiement:)]. I set a time limit of 6 hrs and sent them off. These were and are all capable students but they found the process very challenging at least in part because the answers they got from the physics made no sense. In many cases their models predicted that the toast would spin hundreds of times on the way to the floor which they knew was crazy. They all documented their thoughtful and reasonable work and I was left wondering why they ended up with such unrealistic solutions.

Part of the problem was that Excel is hard to do serious numerical analysis in and that was the only tool they had. Partly it had to do with the way they had been taught to approach problem solving in college (I include me in that list of teachers). As I sat down to construct some teaching materials to try to address the challenge I decided the best way to figure the skills I wished you had as students was to do the problem in detail myself and pay attention to where what I was doing did or didn't match what I had been doing in class. It was an eye opener for sure!

So here's the plan: I'm going to walk through what I did on the toast problem exploring how I have been learning to code in python and how I approach a complex problem. Just a little bit each week. After getting familiar with the tool we are going to use [Jupyter Notebooks] you will pick a numerical problem to start working on for the rest of the term. There will be criteria and approvals needed for the project and each of you will need to work independently. I will model how to slowly make the simplified model more complex and you will take next steps with your problem. Every couple of weeks you will have a progress report to turn in which will flow naturally from your Jupyter notebook so it shouldn't feel like an added task.

Questions?