Modeling Instruction in Physics: Summary for Unit 7

Unit 7 is where we explored energy and work. What was unusual, for me at least, was the order in which we did our exploration. I've always started by studying work, largely because that's the way textbooks are set up. Starting with energy puts the focus on energy, which is the larger concept.

Instructional Goals:

1. View energy interactions in terms of transfer and storage
1. Develop concept of relationship among kinetic, potential and internal energy as modes of energy storage.
2. Empasize various tools to represent energy storage
3. Apply conservation of energy to mechanical systems
2. Variable force of spring model
1. Interpret graphical representation - area under curve on F vs x graph is defined as elastic energy stored in spring.
2. Develop mathematical representations - Force and Elastic Energy formulas
3. Develop concept of working as energy transfer mechanism
1. Introduce conservation of energy - work equals the change in energy
2. Working is the transfer of energy into or out of a system by means of an external force. W is computed by finding the area under an F vs x graph, where F is the force transferring energy.
3. Energy bar graphs and system schema represent the relationship between energy transfer and storage.

Model Development

In previous units, we started by making observations of objects or situations. (Even for forces, we were discussing particular situations and observing them.) Because energy is not directly observable, we started instead by considering what we "know" about energy. After a long list (with some contradictions in it), Bryan started guiding our thinking a little more specifically. He asked us if there was energy in a ping-pong ball he threw, and how we knew. We knew that if he threw it at someone, they'd feel it. Working from that idea, he asked what would happen if he threw it faster. We said there'd be more energy, because we'd feel it more. Then he asked what would happen if he changed the ball to a racquet ball or a bowling ball. We answered we'd feel that a lot more. So we said that moving energy depends on speed and mass. He used similar lines of questioning for holding a ball up in the air and for aiming a stretched spring at someone. In the end, we had ideas about what kinetic, gravitational and elastic energy depended on. (Of course, he didn't give us those names until after we'd talked the ideas in less scientific terms. Concept before vocabulary.)

This is where we began our first investigation - What is the relationship between the force required to stretch the spring and the length of the spring? (Force vs. displacement) This was also a reversal for me. Usually we talk about kinetic and gravitational energy before talking about springs at all. And again, I find myself liking this order better. If we can figure out force and energy for springs, then we can use the idea of transferring that energy into motion. It flows the way we expect the order of events to go. 

We got a nicely linear graph for our force vs. displacement, and as a class had a nice discussion about what the coefficient could mean. After checking the stretchiness of each others' springs, we decided it probably indicated how hard it was to stretch the spring. (The higher the number, the harder it was to stretch it.)

Then we considered two of the springs at once, one with a spring constant of 12 and one with a spring constant of 25. We attached them to carts, and wondered how we could get the springs to have the same amount of energy, as shown by the speed of the carts. We tried stretching them so the force was the same (12-spring stretched twice as far as the 25-spring). That didn't work. We tried stretching the springs equally (equal displacement), and that didn't work either. It was interesting, though, that which car was faster switched...

In trying to figure out what else we could make the same so that the elastic energy in each spring would be the same, Bryan asked an important question: "What else on the graph can we make the same?" We tried the force (the vertical value), and the displacement (horizontal value). The slopes were fixed by the springs, so all that was left was the area under the curve. After a quick bit of algebra to create an equation for the area in terms of just the displacement. We tested it, and viola! It worked. So now we had our equation for elastic energy.

Our next lab had us exploring the relationship between the energy in the spring and how fast the cart moved. After that, we explored how high on a ramp the cart ended. In both cases, we developed equations for the graphs, and using units, determined what the coefficients meant. The important part of these two labs is the understanding that the elastic energy in the spring is entirely transferred into the cart. 

Model Deployment

At this point, to develop the ideas of the conservation of energy, of a system and of work as the transference of energy, we did a worksheet that asked us to list what the system was (given), and the types of energy at the two points listed. We considered energy staying within the system, energy entering the system and energy leaving the system. After discussing our answers to this qualitative worksheet, we continued on to quantitative problems, sometimes using the equation W = F d, which was developed from the area under a constant force vs. displacement graph. 

Our practicum this time was to determine how much mass to hang on a spring so that the mass would move down and "kiss" an egg (tap it without breaking it). We had to determine our spring constant, and the elastic energy in the spring when the mass would be touching the top of the egg. We knew that all of the initial gravitational energy would have to be transformed into elastic energy so that the mass would be stopped at the top of the egg. We knew how high the mass would be starting, so we could use the elastic energy and the initial height to calculate how much mass would give us the right initial gravitational energy. My group convinced ourselves that our value wasn't right. We tried remeasuring and recalculating, but our second answer wasn't any better. So we went with the first one, and, well, our results speak for themselves: