Freefalling into Extension 2 Mechanics

This post refers to the NSW (Australia) Mathematics Extension 2 course - the highest level mathematics taught in our high school system, but should hopefully have relevance for anyone teaching introductory mechanics.

My class is just beginning to explore the harder mechanics content in our senior mathematics course, exploring the equations of resisted motion, so it was with great delight I brought the exciting work of Felix Baumgartner into my classroom:

Here are some reflections on introducing this topic:

Did you lay a strong foundation right at the start of learning calculus? Coming from a physics/engineering background, I taught all the calculus concepts right from the start as a study of rates of change, using this as a motivation for the mathematics. My classes started with motion using a motion detector even before we looked at the concept of the derivative. When we studied the properties of the derivative, the meaning of increasing, decreasing and stationary points, it was all done in the context of motion (lots of roller coasters!). When we studied the second derivative, we asked the question "Why?" - and looked at how much of the physical world works on the second derivative rather than the first. So by the time we moved on to specific topics of applying calculus to the physical world, my students already had wide exposure to the link between the mathematics and the physics.

Did you explain why we have equations of motion that connect acceleration to displacement? One of the hardest parts of the earlier work in our course on motion is understanding and working with these equations:

$$a = v \frac{dv}{dx},  a = \frac{1}{2} \frac{ d(v^2)}{dx}$$

By showing that acceleration is caused by forces, and that in turn forces are often dependent on position rather than time gives the motivation for these more complex equations. It's an excellent opportunity to tap into students' current science knowledge on gravitational and electrical fields - and again reinforces Newton's Laws of Motion (this time the second law). For students with a little more physics knowledge, it's very interesting to link the second form of this equation to the equation for kinetic energy $$KE = \frac{1}{2}mv^2$$ using some integration.

You can't emphasise Newton's First Law often enough.  It's amazing how many studies show that while university physics students can describe and use Newton's First Law of Motion, deep down they remain firmly wedded to the Aristotelian world view. I make it a point to emphasis the First Law of Motion each and every time I start a problem - and I keep an eye out for anyone hesitating or wavering. If necessary, I repeat my stories about ice-skaters and about the Voyager spacecraft continuing on their journey even though they have run out of fuel.

More diagrams, more diagrams! It is sad to see how few diagrams are presented in most mathematics textbooks when presenting the theory of dynamics. While there are diagrams in most worked examples, they aren't explicit in the construction of the diagram, leaving it for students (and teachers) to try to interpret why the diagram was done that way. I spent most of the introductory lessons on this topic just drawing pictures.

"That's why it's called RESISTANCE sir!" My favourite quote from one of my students. We were drawing diagrams and I was trying to find a clear way to show that the resisting force always opposes the current velocity direction. I was saying the word "opposes" a few times when one of my student yelled this out - we won't forget that in a hurry!

"Bait-and-switch" constants.  Your students are probably used to the little bait-and-switch games we play with constants of integration. The same games are played with the constants used for resistance forces:

$$R = kv,  R = mkv$$

Sadly many of our standard text books just switch on the fly between the two forms without explanation, adding in the mass whenever it's needed. It's very confusing for students (and teachers) when this is done so arbitrarily. And then it hit me: this is a totally legitimate game - we're just using a different constant to make our life easier. It sure would be nice to write:

$$R = k_1v,  R = mk_2v$$

I shouldn't complain since I happily went along with the game when integrating!  This is yet another small but cumulative thing that makes this topic challenging. I think it's important to be explicit about this little game.
Update: As explained in Robin's comment below, this trick is a bit too clever - verging on not being legitimate - because it gives the false impression that the resistive force depends on the mass - it doesn't. The trick only works for a specific case of the mass in this problem.

Students have difficulty seeing that the physics has nothing to do with the coordinate system. And it's not their fault - our teaching and our text books rarely show how arbitrary coordinate systems are - we just happily keep changing them to our convenience, potentially confusing our students. I think it's critical to draw lots of vector diagrams without any coordinate systems, and then make a clear and obvious choice with the class that we can choose any coordinate system that works for us.  We had a deep-learning moment in my class last week when I unwittingly applied a different orientation of the axes than was in our text book - a great opportunity to highlight this issue.

Terminal velocity is a really fun concept. Students are absolutely fascinated with it - the physical understanding is interesting, and the mathematical development is revelatory. We had some great discussion on different terminal velocities for different situations and these led directly into a more rigorous discussion of Felix's jump.

Don't think that girls aren't interested in watching extreme sport events. My class of fifteen girls was were absolutely riveted watching Felix make his jump. They insisted on watching the full length 10 minute video, totally transfixed for the duration. I recommend reading the Wikipedia page on Felix's "Red Bull Stratos" jump with your students prior to watching the video - it provides an excellent opportunity to discuss the language of motion, examine the different stages of the jump and provides meaningful context for this thrilling event.

Hold your breath and enjoy the whole jump:

The next post in this series looks at the wonderful Veritasium You Tube resources available for teaching mechanics.