I’ve been thinking a lot about the Energy Transfer Model (ETM). The Modeling Instruction curriculum seems to start this model by jumping right into the concept of Energy Transfer without much empirical model building, contrary to many of the earlier models. I really like the way Kelly starts energy, showing students how previous models don’t work to predict the desired outcome. Still, I was unsatisfied in that I felt like I would just be telling students what energy is and how it transfers without letting them get a feel for it for themselves. So I set out to design my own version of the beginning of ETM. I used this version of ETM in my college physics class after starting ETM the standard way in regular physics.
Day 1: Area of Force vs. Position graphs
Day 1 started just as Kelly’s post details above, though she has modified it since posting to use Pasco’s spring cart launcher instead of regular springs. The idea is simple. How can I make the final velocity of these carts the same if they are launched by two different springs? We spent 10 minutes playing with the carts, and I showed them at maximum compression, both at about 8 cm, the carts launch at different speeds. Predictably, the spring with the highest spring constant launches fastest. So how can we make them go the same speed using their Force vs Position graphs?
We (my colleague Ben, with whom I teach the regular class, and I) tested the springs and their constants fell very close to those stated in the documentation, so we used that to make expected F vs. x graphs rather than take real data. It worked just fine.
In all classes I did this (three different sections, one regular and two college), the first guess was to make the force equal for each spring. So we did that. My regular class just looked at the graph, saw that if we wanted a force just over 4 N we could use about 5 cm for the red spring and 3 cm for the blue one. For the college classes I asked them to choose an arbitrary compression for the red spring, then find the blue compression to give the same force.
Either way, it failed miserably.
Turns out that if two different springs are compressed to the same force value, they do in fact have the same average force, and thus the same average acceleration. However, the weaker spring has to be compressed further to get that same force value, and thus the same acceleration happens over a larger distance. The weaker spring actually gives a faster speed when the force each exerts is the same!
They get this. I asked them what would happen if you had two cars that had the same acceleration, but one accelerates for 10 meters and one for 20 meters. The 20 meter one ends up at a faster speed. Yep, that happens here too. The red spring car goes faster because it has the same acceleration on average as the blue spring but for a longer distance.
So anyway, what now? I had to guide them to check area. I did not do as awesome of a job as I would like using the area under velocity vs. time graphs to find displacement, and as a result area of graphs is not a formost thought for them. However, all classes jumped on the idea once I led them there (by referring back to kinematics graphs and the parts of those graphs that do in fact have physical meaning). Most students needed help with the idea that they should pick an arbitrary compression of the weak spring. Once there, however, we worked through the math and found the compression of the blue spring such that its area equaled that of the red spring with our arbitrary compression.
The launch was perfect. In all 3 sections.
Kids really like it when things work, and boy, does this work. It took about one 45 minute class period to get this done, but they definitely got the idea that the area under the F vs x graph meant something. I emphasized, over and over, that area under graphs, if it has a physical meaning, means a change in something. We don’t know, however, what that something is yet.
This is where the classes diverged. The regular class went into a lecture day on types of energy and energy pie charts. But that’s not what I want to write about.
To continue empirically, I wanted them to see that the area under the F vs. x graph (a change in something, as I kept calling it) was meaningful in other situations as well. So next we looked at ramps.
Day 2: Ramps and the Area of F vs. x graphs
On day 2 I told them we were going to again look at the Area of F vs. x graphs, but this time in a different situation. We started with a cart at rest at point A, arbitrary but constant. We wanted to end with the cart at rest at point B up the ramp, also arbitrary and constant. I had them pull carts from A to B in any way they wanted and to find the area under the F vs. x graph. Here’s a sample trial.
I learned some things. First of all, most of them didn’t end the cart at rest at point B at first. But we did, however, use that to establish that the faster the cart was going at B, the larger the area seemed to be. We will go back and quantify this later (part 3 or 4 of this series, I believe). So we went back and got some data for starting and ending at the same points each time, starting and ending at rest, but getting from A to B in different ways. Here’s some sample data.
In discussion it became evident that outliers appeared in one of two general cases; when the cart was difficult to actually stop at point B, and when the cart moved backward at some point. On the whole, it was pretty easy to convince them that the area was the same no matter how you got from A to B as long as the cart didn’t move backward and the cart was at rest again at B. Pretty awesome.
That same day I asked them what measurement would always correlate with the area. Horizontal distance up the ramp? Angle? Height? We were able to quickly show that though distance correlated with area, it didn’t work well if we kept the same distance and changed the angle (we got different areas then). Thus distance is not a universal predictor of the area. How about angle? Similar problem; for one angle you could get infinite areas. How about height? We spent the last minutes of this period showing that if we had an equal change in height, even for two different ramps (same cart of course), that the area was approximately the same. Cool.
Day 3 and 4: Finding the Correlation with Height and the Entrance of Energy
Day 3 was short classes, only 30 minutes because of a pep fest, and I think data collection and whiteboarding could probably be done in one class period. However, the conversation we had about types of energy at the end of day 4 fit really well and it was nice to have that there. But I’m getting ahead of myself.
Day 3, 30 minutes, was spent collecting area vs. change in height data. Some students changed the height just by pulling the cart further up the ramp, and some by changing the angle of the ramp, or a combination of the two. Part of the awesomeness of this lab is that it doesn’t matter; no matter how they change the height, if they collect data consistently and correctly, the results turn out well. (Students won’t, by the way, take data consistently and correctly; I had at least 2 groups in each class with non-sensical data. They don’t set the endpoints of the integral in Loggerpro correctly, or they don’t change the endpoints (thus making the change in height the same for all trials), or they measure change in distance rather than height, or they do one of I’m sure many other things that yield poor results. It’s a learning experience though, and the conversations that come from ‘bad’ data are often just as useful as those that come from ‘good.’)
In any case, the graphs were decently linear. Through a board meeting (circle sharing) groups quickly noticed that the intercept was zero, and that that made sense as if we don’t have any change in height, we shouldn’t have gone anywhere, so the area of F vs x would also be zero. They then noticed that some groups (conveniently with carts of different masses, *cough cough*) had different slopes. At some point someone notices that the slope appears to be approximately 10 times the mass. Hmmm, isn’t g really close to 10? Then we look at units. The slope must have units of Newtons, as y axis has units N*m and the x has units of meters. If the slope was mass times g, then the units would be in Newtons. Hmm. Note: In all this, I try to at ask questions with a couple of words max and let the conversation take its course.
This was convincing enough for my students that the slope should be mg. It was, pretty close, for the groups that had decent data. I then asked them to write a general equation to model our data. Most were able to get here;
where A is the area under the Force vs Position graph, in N*m.
I pointed out that even though this was a different situation than day 1, the area still gave us something meaningful. But seemingly unrelated to speed! We’re getting there. Let’s rearrange the above equation a bit.
which leads to
Here is where I finally defined that the quantity mgh is called Gravitational Interaction (or Potential) Energy. I took a side trip for a bit on energy as pain, as described very well (better than I could) in Kelly’s aforementioned post on building the ETM.
Thus what we have found is that the initial gravitational interaction energy plus a the Area under F vs x (which recall we had emphasized as a change something) gave us the final gravitational interaction energy. So I guess the area is a change in Energy, huh?
Starting with Day 5 we are going to look at how the area correlates with speed, and use that to figure out Kinetic Energy. We will then use that to transition in to Energy Bar Charts and the rest of the energy unit. More on that in later posts (I think 1731 words is enough for now, huh?)
Concluding thoughts, for now.
I really like that this method strongly emphasizes that the energy is changing due to the Work done (though we haven’t used that word yet), and I plan to use it to strengthen both their methods of using graphs and multiple representations to solve problems as well as to help with the idea of Work itself, which when taught traditionally has really only served to confuse my students. I don’t like, however, that for now I am ignoring changes in thermal energy, which the typical intro to ETM in Modeling Instruction emphasizes from the get go. I used to teach energy where we would ignore friction for weeks, then finally add it in and start all over, and didn’t like that. I think, however, that the idea that the F vs x graph influences the transfer of energy will transfer (hehe) to friction as well. We’ll see, and I’ll keep you updated.