“This is a boss equation.”

We’ve been pushing graphical methods to solve problems (see: Vector Addition Diagrams, IFF Charts, LOL diagrams, etc) like nobody’s business. So even just a few units into the year, these physics kids may be graphical problem-solving wizards, but a lot of them still hunger for equations in an insatiable sort of way. Even though we  (my classes) don’t have any equations for kinematics, the topic still offers a nice opportunity that satisfies both the equation-lovers and the thinking/understanding-lovers (not that those two groups are mutually exclusive, but I think if you’re reading this post, you know what I mean).

Around projectile motion time (so after kinematics/forces and momentum), when a student or group gets a bit ahead of the others, I ask them if they want to know the next challenge (of course they do—which the Honors Physics kids will freely admit and which the regular Physics! kids will deny loudly and with frequency alongside their persistent attempts to get me to divulge the challenge).

Come up with an equation for ∆x (displacement) that doesn’t have ∆t in it. This equation should work for any case of constant acceleration (CAPM) motion. Make sure you start with a graph that has an initial velocity (not starting from rest) so that you end up with a general equation that works for any CAPM case.

It’s a little funny for them to be deriving this equation since they actually don’t have an equation for ∆x that does have ∆t in it. They draw and use a graph to solve every problem. Using the area on the velocity-vs-time graph to find an equation for ∆x has always resulted in something that involved ∆t, though, so the challenge is usually readily apparent to the student (especially since they’ve used these graphs for at least a couple of months at that point).

Transfer of ownership

I used to (even last year, in Honors Physics) derive this equation for them. It can be a really helpful one to have, so giving it to the kids seemed to be the nice thing to do. I would walk them through the algebra, telling them that they didn’t need to write down all the steps. I wanted them to see how I got it, but I wouldn’t ask them to do it again for me on their own. In the end, they all had the neat little package of v_{f}^{2} = v_{i}^2 + 2 a \Delta x, and they stored it neatly on their equation sheets. They had to be reminded when it was okay to use the equation (and to change the symbols to indicate the y-direction when using it with projectile motion), but when they remembered that it existed, it made a few problems much easier to do. It was decidedly my equation. I would be surprised if more than 1 in 10 students who used it could describe where they got the equation other than saying “from Ms. O’Shea” or “from the equation sheet”.

Last year, I decided to do a completely graphical treatment of kinematics in my regular Physics! classes. I did the same for my Honors Physics classes until the projectile motion unit, then I derived the equations for them, figuring they had gotten the value out of the graphical solutions and were ready for an “easier” way. Even though I could see the value of solving problems graphically, I was still clinging to my own biases about solving with equations (the way I’d learned to do it).

Anyway. In one of last year’s regular sections, while working through some projectile problems, one student pointed out that every time they solved a problem that needed simultaneous equations (one from the slope of the velocity graph, one from the area under the velocity graph), he ended up with equations that always looked a certain way. I encouraged him to try and flesh out the pattern more by keeping everything as a symbol instead of plugging in numbers, then I got out of the way. Others in the class picked up on what he was doing, and soon a few of them had taken over the front board with their work. They ended up with an equation that only worked for objects that started from rest. I prompted them to start with a more general graph, and they came up with the more general version of the equation. They started using their equation anytime they knew it would work, though some were timid about doing so since it “seemed like cheating” to skip the steps of using the graph!

Multiple Manifestations

This year, I’ve completely drunk the graphical solution kool-aid. I’ve derived no kinematics equations (not even in Honors Physics). Instead, I’ve issued the challenge I described above when the time seemed right. Honestly, it’s been even more fun than watching last year’s class work out the equation. That same situation keeps happening for student after student. As soon as they notice how helpful the equation is, they want to know it. My only rule is that they are only allowed to use it if they have derived it themselves.

One student knew the equation already because she had taken a memorize-facts-about-physics class at another school. She had no idea why the equation worked, just that she had been told it did. It took her three tries to work her way cleanly through the algebra (all of which she did outside of class); she entered the room beaming when she had triumphantly discovered the equation all on her own.

One kid found me during brunch on a Saturday and asked me to time him as he sat down to derive the equation for the first time (< 10 minutes).

Some, on seeing classmates use it, remark aloud, “Yep, it’s about time I actually figure that out, huh?”

Since I don’t prescribe the steps of the derivation, there are a few different forms that the equation takes. For many, it is \Delta x = \frac{v_f^{2}- v_i^{2}}{2a}. At least one student thinks of it as setting the ∆t’s equal to each other in the slope and area equations, and he always states it as \frac{2\Delta x}{v_{f}+v_{i}} = \frac{\Delta v}{a}. Everyone has her own way of thinking about the relationship. No one needs an equation sheet or to be reminded when it is okay to use it.

After 30 or 40 minutes of making (and fixing and making and fixing) algebra mistakes, a student wrestled the relationship down onto the page. He sat back, looked at his work, and thought it over. Then he announced, “This is a boss equation.” Even better, though, it is his boss equation.

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