Dealing with the fear of being a boring teacher.

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teaching

# When Models Fail: Gravity

This post is written to imply the tone and hand gestures of Shannon McLaughlin, a badass teacher in the Des Moines metro, who was probably Green Beret, MI6, and a union roofer all at the same time.

Today a student and I tried to energy-pie-chart out the journey of Apollo 11 from launchpad to the Moon. It quickly devolved into a confusing and awesome conversation about what gravity does when it steals energy from you.

We often call this “potential” energy in science class, and lately I’ve been questioning the pedagogy surrounding the whole thing. Don’t get me wrong, obviously potentials are important, but most students have the following misconceptions:

• Potential energy is stored in an object as some kind of glowing liquid thought up by Stan Lee.
• Potential energy will ALWAYS become kinetic energy.
• Gravitational potential energy is a function of the time that gravity has to act on you (as in, a projectile on the way down has more potential energy because it has been in ‘gravity’ longer than when it was on the way up)

At the root of all of this is the pedagogical decision to introduce students to the standard gravitational potential energy early on:

$PE=mgh$

This equation makes sense for the regime in which we use it, and plenty of physics worksheets, exams, and detritus contain all sorts of qualifying bull shit like “…near Earth…”, which makes us all feel great about our physics degrees, but most students don’t see why we waste the ink.

That’s a problem.

A huge effing problem.

If you really believe in Modeling, then you have to make a huge freaking deal about the limitations of models, almost to the point of caring more about limitations than actual implementations of the models in the first place.

PE=mgh fails for a lot of reasons. I made the following illustration to help my students through this, and I’m wondering what you all think I should add. This is an intentionally flawed infographic designed to get students to reject the linear model of gravitational potential energy.

Getting to the Moon. The Earth is green, the moon is grey (small, click to embiggen), the distances and radii are all to scale. Redness indicates influence of the Earth (as m/r). Blueness indicates predominant influence of the moon (as m/r) Scale = 300.78 km/px. The graphic is intentionally flawed, because I like breaking models more than using them

Here’s a raw version with 1/r potentials mapped. The moon, Earth, and distances are to scale. Again, red=earth influence; blue=moon influence) I built this pixel-by-pixel in Processing (Processing.org). Email me if you want the code.

Here’s the deal: If you use the traditional model of potential energy, then there must be an appreciable amount of potential energy from the moon when you’re on the launch pad (first pie chart).

That’s insane.

I mean, what the hell, then, does the wedge from Andromeda look like? Jupiter? The Sun?

That’s a conversation I want to have before dropping the following:

$F_g=\frac{Gm_1m_2}{r^2}$

$U=-\frac{Gm_1m_2}{r}$

I hope you grock why I’m writing this. It has nothing to do with knowing physics, and everything to do with how students think.

Help me make this even more mentally uncomfortable.

Oh, and this is SO AWESOME (evidently xkcd is my whole curriculum)

7 thoughts on “When Models Fail: Gravity
• Leif Segen says:

I went snooping in the xkcd forums because the gravity wells poster contradicts itself for Earth’s numbers. Turns out there is a more accurate representation of the solar system.

• [...] other part of why this numerical model served an interesting purpose in my class was as inspired by Shawn Cornally’s post about misconceptions surrounding gravitational potential and our friend mgh. I had also just [...]

• Micah S says:

After reading your first paragraph, I was jumping at the bit to recommend the XKCD infographic to you. Glad you already found it :)

Oh, and the line by Leif in the comment above about “exposing lies from the previous class” is key; I myself try to be upfront and explicit about more sophisticated models superseding simpler models so that the student won’t think it was all lies — if you give them the idea the previous class was all lies, what’s to stop them from thinking that you’re not just teaching lies as well?

Which is why showing how [F = mg] relates to [F = Gm1m2/r^2] will make the transition from [PE = mgh] to [U = -Gm1m2/r] much more palatable and comfortable. Simpler models that are easier to work with -> sophisticated models that work in more situations.

(Wait until they hit special relativity and learn that F doesn’t really equal ma, which may kill their faith in physics, depending on whether you’ve prepared them for it or not.)

• Leif Segen says:

So far I’m interpreting that your pie charts* are showing that the PEg = mgh model is ridonculous because it predicts and unreasonably large amount of gravitational energy in the Apollo-11-Moon system. And I’m hearing that, if any front loading is going to be done, it has got to be meaningful and true(!).

*I first heard about energy pie charts recently. I’ve been all over the energy bar charts for years. I think they’re better for comparing only two or three states of a system.

I remember feeling like every physics class I took was exposing the lies from the previous class. Of course, models and analogies are key in learning and teaching. So the more upfront we can be about that, the better?

Regarding how students think, they can wonder why the negative sign and why there isn’t a squared term for potential energy. One can be addressed with bar charts fairly nicely, I think. The other may take some calculus and a conceptual leap or two.

• Leif Segen says:

Okay, I first thought that the colors indicate some sort of positive and negative potential à la this PhET sim. But, Doh!, gravitation is solely attractive. Now, I believe the colors are about which side of potential’s local maxim you’re on. I feel like it would be awesome to add equipotential lines to this some how. Oh! … and to do some 3D rotation. (See 40 seconds in to this Simpsons clip – even though it introduces another set of misconceptions/limitations of that type of model.)

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