I’ve been putting some thought into the first “big idea” I talked about for Theoretical Mechanics in my last post (differential equations). Once again, Danny Caballero has really helped me with this, you should really check out his stuff. I also got some great help today from a math colleague of mine, as I wanted to know how he motivates differential equations.

As noted in my last post, this, to me, is really about equations of motion (which happen to always be differential equations). I want to spend some time in the first day (before we really get into our flipped class groove) getting the students to reflect on the notion, utility, and subtlety of equations of motion. Danny’s material pointed out the difficulty some students have moving beyond x as an answer to x(t) as a function. I want to brainstorm some ways to really get at that in the first class.

I’ve been thinking about how so many introductory physics problems revolve around equations of motion but only scratch their surface. Here’s an example:

Determine the forces on a block on an inclined plane. Then determine the speed of the block at the bottom if it is released from rest.

The first part, “determine the forces,” is really writing down the equation of motion for the block. The second part, is determining the solution of those equations **at one point**. I emphasize that, because my math colleague has me thinking about trying to motivate students to search for functional answers, not single answers. The function, which is the solution to the differential equations (the equations of motion), is everything you need to know about the whole history of the block. Asking that function about the speed at the bottom is just one of an infinite number of questions that the function can answer.

In a differential equations class, most homework is centered around finding a function that satisfies a given set of differential equations. That, to me, is a lot like what we’re doing in Theoretical Mechanics. We’re interested in finding various trajectories of things, not just how fast it might be going at a particular point.

What I’m thinking about doing is asking the students to redo “simple” general physics problems (or maybe just one), but to have them change the focus a little to the underlying functions involved. Here are a few examples I’ve been kicking around

- Mass on an inclined plane. Determine the function that the x-variable obeys. Use it to determine the speed at some point.
- Mass on a parabola-shaped wire. Identify why this problem is more difficult than the mass on an inclined plane. See how far you can get to determine the equation(s) of motion.
- (this is an early problem in the text I used to use) Consider air resistance for a projectile problem.
- Something like Kelly O’Shea’s goal-less problems. Have them consider a somewhat simple problem, but don’t tell them what I’ll want to know later (in the hopes that they’ll focus on function solutions, instead of single answer solutions).

Ok, as usual, I’d love some feedback. I have to say, I really like using this blog as a syllabus brainstorm device. The feedback I’ve been getting all month has really been helpful, so thanks!

P.S. Here’s a typical problem in the text I used to use:

A pendulum of length b and mass bob m is oscillating at small angles when the length of the pendulum string is shortened at a velocity of s. Find the equations of motion.

I like how it puts the focus on the equations of motion. But I’d much prefer my students turned this in:

I did this with the following Mathematica:

Manipulate[ Graphics[{PointSize[0.05], Red, Point[{x[t], y[t]} /. sol], Black, Line[{{0, 0}, {x[t], y[t]} /. sol}]}, PlotRange -> 1], {t, 0, 9}]

Andy, I really like the idea of getting them to redo the canonical intro problems from the equations of motion perspective as a jump off point to highlight how your course is both the same and different than the intro mechanics from before. I suspect that the jump to the textbook problem you used to use was probably too big of a jump such that many of them would lose the ability to make connections with stuff they had previously done.

I assume you plan to continue with the heavy use of Mathematica in this course?

Yeah, that last problem was just for fun, after I was looking around for a problem that ended with “find the equations of motion.” That’ll come a few chapters in.

Yep, Mathematica will still play a very heavy role in this course.

Chris Goedde had a great offline suggestion to have students describe the trajectory of a particle moving at constant speed along a tilted line that doesn’t pass through the origin. They need to do it in both cartesian and polar coordinates. I think it really addresses the notion of students seeing x, y, r, etc as functions. It also makes sure they understand the unit vectors.

However, I would not be shocking to see this team finally put something special together.

His passes don’t have zip.