Graphical Solutions _and_ Symbolic Algebra in Physics

Graphical solutions are terrific tools to build understanding in motion, momentum, energy, and nearly every area of physics. Even better than that, they allow a large number of students to successfully solve problems that really struggle with a strictly algebraic approach. Kelly O'shea and others have some great resources for you to peruse; the approach can be eye-opening for those of us brought up in the traditional system.

One thing that I struggled with a few years ago when first exposed to graphical problem-solving was all of the numbers. With my honors physics classes, I put a lot of emphasis on building their skills in exclusively symbolic solutions to problems - students are expected to do every solution with "no numbers until the end." That is, they have to first derive an expression giving the desired quantity in terms of nothing but given quantities; only then can they use any numbers. This approach has several big advantages:

• It's a pain to carry around units in the algebra (and necessary, because it's unthinkable to have "naked numbers"); symbolic algebra means that we don't have to
• We can re-evaluate for different values of the parameters easily
• We can check how our solution depends on the variables - should the Atwood's acceleration increase or decrease when this mass increases?
• We can learn how our solution depends on the variables - maybe we didn't know that the mass would cancel out of the expression for the minimum speed in the Gravitron!
• We can check how our solution behaves in special cases - should the acceleration go to g if that mass goes to zero?
• We can learn how the solution behaves in special cases - hey, the position function of a falling ball with drag becomes linear in the large-t limit!
• We can still check the units of the answer, without having to carry them through the algebra - if the units don't work, then it's not even worth plugging in the numbers
• This is a skill that ultimately is a part of how "big kids" do science - it's an great skill to have going into college science and math courses (the course that I took this summer had, out of about sixty HW and exam exercises, exactly three problems that involved numbers!)
• It helps to refine and develop student algebra skills (the more abstract end of the concrete-to-abstract progression of their skill development)

These advantages don't have to be lost in graphical problem-solving - students can still label the graph (naming every labeled quantity, whether it's known or not) and can still build slope and area relationships from the graph (involving those variables that they labeled). Solving these symbolically gets them the best of both worlds: graphical analysis and its lower barrier to entry and improved understanding, and symbolic algebra, with the skills listed above.

What does it look like? Here's a bit of student work, from a recent assessment on CVPM (constant velocity particle model) and the a whiteboard with the first (!) CAPM (constant acceleration particle model) problem that three students did this year.

VPython, Energy, and Stability

The content in AP Physics C about stability and its relationship to energy is a pretty thin introduction to a fairly deep idea. Classifying equilibria by looking at the derivative of potential energy can be a quick add-on, or we can make it a little deeper with the help of VPython.

When it first comes up, we go through the stable/unstable/neutral equilibrium discussion and the corresponding shapes of potential energy graphs, but I really wanted a way for students to apply that. I've thought of a few, only three of which I have tried out so far:

• Have VPython plot a potential energy vs. tip angle curve for a box that's standing on its end - compare for different heights/widths of box
• Have VPython graph total potential energy vs. stretch amount for a mass on a vertical spring
• Have VPython graph potential energy for a mass of a spring that doesn't necessarily stay vertical (on y vs. x axes, with U value signified by color)
• Investigate the Lenard-Jones (interatomic) potential, both for meaning and location of equilibrium points and to fit an approximate quadratic to it near the equilibrium point, justifying the use of the ball-and-spring model of matter.
• Later in the year, during rotation, have them create a physical pendulum program with a box on a pivot not through its center - modify this to show the graph of potential energy as a function of angle

For the first and fifth, I have some results using VPython to show. (The fourth is great, too, especially if you're using Matter and Interactions!) For the box tipping, the graph is neat, not least of which because of the discontinuity in the derivative at 0. Calculus wouldn't find this minimum on without intervention from the student, and it's easy to forget to check those endpoints! This gives a nice graphical reminder. My one concern is that the geometry is fairly harrowing - determining the height of the CM as a function of angle isn't super-easy, and requires a high-quality diagram. 

Here's a screen shot:

When you vary the parameters, that central well can become deeper (more stable) or shallower (less stable), and vanishes as the box gets narrower (and becomes a pencil).

For the final one, I think that the program, being a modification of a previous one, isn't too difficult to do, though it does bring in the complication of having a "meta loop," where the whole simulation runs several times, with different placements of the axis. 

Here's a screen shot that links to a video of the program running:

The graph here is incredibly rich. Not only do we see two different kinds of equilibrium (three, if you let the pivot be at the middle for the first run through), but we see how the system is forced away from or towards those equilibrium more or less violently as the pivot moves. This helps to bring together their common-sense understandings of stability and the physical principle that we're trying to investigate.