I was grading some formative assessments I gave yesterday and a student drew their momentum bar chart differently. I LOVE that this shows the total momentum of the system as the total area of the bar! This would only work well with equal masses though (which happened to be the case here), as the heights of the bard are the same (on the vertical axis, velocity is on the horizontal axis).
Today I used a cart of mass 3m smashing and sticking to a cart of mass m to introduce IF charts. I really love how we can look at how the individual momentum changes (the momentum of cart 3m decreases while cart m increases) as well as look at initial final momentum (hence IF, initial total should equal final total so long as the external net force is zero).
I just love how this whiteboard shows how multiple representations support problem solving and allow for more complete understandings. This is problem 4 from practice 3 here. I got in the practice this year of making students always find the area of the F parallel vs. x graph for work done, because I like the calculus connection and the more completeness of the definition of work as integrating F dot dx (I don’t use dot product here, but it’s a quick extension if they see it again). Also, energy bar charts really revolutionized how well my students understand energy transfer, particularly when the energy of the system is changing due to work being done.
My colleague Ben gave me this idea. He showed me how you can write a second energy bar chart for objects that are not in your system and show how the energy that is transferred out of the main system is transferred into the secondary system. No matter the system definition, energy is transferred and stored, but what changes is simply how we are tracking that. This is definitely something I didn’t understand after my first year teaching energy this way (essentially from a 1st law of thermodynamics perspective), and I’m sure it will become more nuanced as I get more used to it. One thing I am thinking about emphasizing more next year is the link between free body diagrams and energy transfer, in that forces in or against the direction of motion cause energy to transfer, though sometimes that is within the system and sometimes adding to or taking from the energy stored in the system, depending on the definition. Lots to think about here.
Today I gave my version of Kelly’s Energy is Pain introduction to energy. For me this comes after 5 days of empirically showing how work produces changes in energy, though they don’t know those terms yet. I was careful to do two things in this process; first, I wanted to emphasize (and will keep doing so) that work is always done, though often it is within the system and we simply track the energy storage, not the changes due to work. The above picture illustrates this, as I had first (in blue) considered the air outside of the system, thus energy is transferred out of the system via work done by the air and then stored as thermal energy outside of the system, and then in green went back to what it looks like when we consider the air in the system, so that the work simply transfers the energy to thermal energy stored in the system. Same transfer, same storage, but the question is, are we tracking the storage or the transfer?
I slightly modified the typical money example for this purpose. I picked a student and defined our system as including both myself and the student. I said I had two dollar bills initially. Then I gave the student some money without stating how much, and then we both stated that we had a dollar each. Nothing left the system, but money was still transferred. Then I started over defining myself as the only part of the system. I stated openly that I am transferring a dollar to the student, and had him put it away. In that case we tracked my initial storage, how much left, and the final storage. In both cases the transfer and the storage were the same, but how we tracked them were different.
Today we worked on problem solving with the Central Force Particle Model (CFPM). I really, really like force vector addition diagrams (FADs) here, as after a week of qualitative analysis kids understand that if the motion is uniform, then the net force should be towards the center, so the FAD helps them get the direction of the force(s) causing the centripetal force correct. Also, I have really been enjoying doing mini-board meetings for problem solving whiteboarding, as this group is doing.