# A New Found Appreciation

On my seemingly mountainous, yet steadily disintegrating, pile of Really Important Physics Things That I Never Learned in Undergrad (TM) was the importance of choosing a system when analyzing multiple objects in an interaction. Physics is, among other things, a study of how things interact, but when your analysis is only focused on one object, then deliberately choosing a system seems unnecessary. A rope pulling a crate is certainly “multiple objects interacting”, however in problems such as this, nobody actually cares about the rope. For much of the traditional kinematics/dynamics sequence, the choice of a system is largely invisible and often a single object. The car. The crate. The ball.

But this becomes a problem when anything involving a conservation law comes into play. Whether or not some quantity is conserved is completely dependent upon the chosen system. Upon reflection, I imagine the unspoken, implicit choice of “system” becomes something murky like “all the objects mentioned in the problem” or even “the entire universe.” I also suspect that not explicitly defining a system causes issues with students applying conservation laws [citation to be added later, possibly?]. If students internalize “energy conservation” as “energy doesn’t change”, then I can see that causing issues with incorporating work/change in energy into their problem solving process. Same goes with the impulse-momentum theorem and momentum.

This new appreciation for the explicit choice of a system started with a twitter conversation earlier this year in which I expressed my dislike of the phrase “closed system.” Yet until recently, I wasn’t sure how to help my students develop the same appreciation and understanding of what it means to define a system in relation to conservation laws. Until now. I think.

# Introducing Momentum

My students were working through the paradigm lab for the momentum unit from the AMTA Modeling Curriculum. Two frictionless carts, one of which is spring-loaded, “explode” away from one another. Students are tasked with placing them on the track at such a location that the carts reach the edge of the track at the same time. Through varying the mass of one of the carts, students discover the inherent ratio nature of momentum and interactions and come up with the following equation (momentum is defined as mass*velocity after they come up with the top one):

Afterwards, they worked on using momentum bar charts to represent the experiment. I doctored the numbers a little to keep the focus on the concept of thinking proportionally while not being overshadowed by seemingly random decimals and fractions. I asked them to determine the initial (as defined before the carts exploded) total momentum and the final total momentum (as defined as after the carts stopped touching), to which their surprise was zero. Peculiar!

To be perfectly honest, I wasn’t planning what came next until the night before I did it, which is when I remembered the twitter conversation from earlier this year. As it turns out, asking students to sum up the momenta of the carts came in handy later on.

# Taking a Second Look

My general strategy for students discovering foundational laws and principles is to provide them both with an experiment and a way to view the results in such a way that whatever it is I’m wanting them to discover screams at them with a megaphone “HEY! LISTEN! HEY!” until they can’t stand it anymore. In this case, I’m wanting them to discover the Law of Conservation of Momentum while simultaneously approaching the Impulse-Momentum Theorem. Students were prompted at the beginning of the class to organize their notebooks into two columns with three rows each.

I started with analyzing the carts separately because that’s what they’re used to. We’d not formally discussed how to analyze a collection of objects, much less that such a thing was “allowed.” I allowed students to work primarily in their groups without assistance from me, and they required little help as all of the diagrams and analysis was nothing new to them. Once I was certain that all the groups had gotten everything, I put my analysis on the whiteboard.

The diagrams with all the circles on the left are system schemas, which were introduced months before when students were first learning how to draw free-body diagrams. Emphasis was placed on determining whether the momentum of each “system” (i.e., cart) was equal or not equal to zero, along with the final change in momentum from before, to during, to after.

Next up was for students to re-do the analysis, but treating both carts as a single system.

# What’s in the booooxxxx?

What I was most worried about was the level of abstraction required for students to truly understand what it meant to analyze both carts as a single system. We hadn’t discussed center-of-mass, nor had we discussed what it would mean for the center of mass of an object or collection of objects to be located outside of those objects. While center-of-mass would be the most accurate way to articulate the effects due to conservation of momentum, it would require more new concepts and abstraction, which I was sure to be too much for my students to assimilate at once. Instead, I placed a box over both carts.

I emphasized that this sort of analysis is something that we’d done many times before. The carts themselves are made of individual atoms all interacting with each other, yet we never worried about it because it was unnecessary. Treating the carts as a “single” object was really no different, and the visual of the box really seemed to drive the point home.

During the times in which students were moving through each of the before, during, and after steps, I would raise the box, set the carts accordingly, and put it back down. I emphasized that we were only concerned with what happened to the box from the outside. Just like we didn’t care what individual atoms were doing before, we don’t really care about what happens inside the box. Without much help from me, students produced the following analysis in their notebooks. Again, emphasis is placed on determining the total change in momentum from before, during, to after the interaction.

At this point, students are asked to make a claim, and support it with evidence and reasoning. Most groups’ claims were of the following form:

If the forces on a system are balanced, then it’s momentum doesn’t change. If the forces are unbalanced, then it’s momentum does change.

Sounds a lot like a conservation law to me, which is typically stated like this:

In a closed system, the momentum of that system doesn’t change.

I’m with Frank; ditch the phrasing of “open” and “closed” system all together.

# Closing Thoughts

There’s no reason to wait until momentum to introduce the idea of analyzing systems of single and multiple objects. Next year, I certainly won’t wait so long. I imagine the same type of demonstration can be performed and analyzed to get the point across. This will be particularly useful once we get into energy, where the entire concepts of work and power are defined by what’s happening by or on the system and, of course, the link between the system definition and the law of conservation of energy.

This has also helped me realize something about the kind of physics teacher that I am. I am continually fascinated by all the connections I’m discovering that I never found on the first, second, or third time around with all of this content. Now, more than ever, I see physics as an intricately constructed puzzle-that’s-also-a-tower, built from the ground up from a few simple principles and definitions. I want to help my students construct this tower for themselves and see all the beauty in the details that I do. More practically, I think that students being aware of these details helps them achieve a more deep understanding of physics.

# Setting the Stage

I introduce the impulse-momentum theorem immediately after a lab introducing momentum. At this point, students have a general idea on what momentum is, including that it’s defined as the product of mass and velocity. The introduction lab helped them see the ratio-centric nature of momentum and interactions. All that said, the general understanding of momentum is still fairly basic and developing.

I start with a simple question that essentially has students state the impulse-momentum theorem conceptually.

I immediately get responses such as change the mass! and change the velocity!. It only takes a little digging to get them to expand upon change the velocity to include that an unbalanced force must be present on the object. Bam, impulse-momentum theorem:

To change the momentum of an object, the forces on that object must be unbalanced.

Then, we move to deriving it. I’ve been including more derivations lately, but I’ve been careful to only do it when I feel that the students can do it themselves (with a little poking in the right direction by myself). My only direction here was this:

So, we said above that a net force is required to change something’s momentum. But the only equation we have that includes net force says nothing about momentum. That doesn’t sit well with me. Let’s see if we can get momentum in there. Be sure to start from  $F_{net} = ma$

The “ideal” derivation I’d imagined looks something like this:

As it turns out, only a couple of students actually did it along the lines of the way I did it. Which is actually fine! There’s no single “right” way to go through stuff like this.What’s important is that most kids were able to figure out how to get a in there somewhere. What most students did was this:

And I was quite satisfied! My response fort when students asked the oft repeated physics class refrain Mr. Register, is this right?, was Almost! Watch me go through it and you’ll see… When I went through my version of the derivation, they seem to see the difference. What’s important is that they’ve gone through a pretty important physics derivation (mostly) on their own.

Ok, let’s move that delta-t over to the left side… and we have the impulse-momentum theorem!

# Enter the Mythbusters

Huzzah! We have an equation! It’s got a fancy sounding name! Let’s start plugging in numbers!

Nope.

I’m trying to put more focus on understanding equations conceptually, hence the full-stop on jumping straight to plug-n-chug. To accomplish that along with really grabbing my students’ attention, I use this clip from a Mythbusters episode.

I stop it before they actually crash the cars together and poll the room, which instantly explodes in conversation and wild theorizing. I have a few students share their thoughts, throw out a few Hmm…‘s and What do you mean by that?‘s, a few inquisitive looks here and there. A head tilt or two. At this point, students are biting at the bit to know the answer.

But finishing the video would, in my opinion, flop the entire lesson. While about half my students guessed the right answer, none of them had a fully well thought out explanation supported by even (in my opinion) a reasonable amount of evidence to support their claims. Essentially, they were guessing. Which is a fine starting point, but a poor ending one (see The Most Important Thing (TM) on this post for more of my thoughts on this). I’m also trying to develop my students’ scientific argument writing skills a la’ the Claim-Evidence-Reasoning framework.

I want my students to be able to explain, in detail, exactly why the answer is what it is.

# Analysis

The analysis I have my students do is aided greatly by momentum bar-charts, which they were briefly exposed to during the lab and notes introducing momentum. I also provide some data.

I chose to use tons and not pounds or kilograms because of the numbers involved. Using either of those gives momenta in the thousands, especially if the mi/hr is converted to m/s. Using tons and mi/hr keeps the numbers low. Students work through all the bars, and I go through with them to make sure they have a solid set of data from which they can later draw their conclusions. I give them the additional piece of information:

The time for every collision is the same.

The goal is for students to see that the change in momentum for the cars hitting each other is the same as the 50 mi/hr car hitting a wall. If the change in momentum is the same and the collision time, delta-t, is the same, then the net force on each car must be the same as the 50 mi/hr car hitting a wall.

One common thing I noticed is that some groups wanted to add the impulses from each of the 50 mi/hr cars hitting each other together to get 150. Once I pointed out that the impulse for each car was +75 and -75, which added to zero, they’d go back to the drawing board. Others ignored me completely and stuck to their guns. As it turns out, the total momentum being equal to zero is important, but that’s for another day.

Another thing I noticed and also found fascinating is that several of my students across different classes mentioned something their Driver’s Ed classes say. Apparently, they’re told that two cars getting hit head on at 50 is like hitting a wall at 100. I responded by asking the students if they thought their driver’s ed teachers were physicists or knew anything about physics.

Yeah, I guess?.

So, it’s true because somebody you perceive to be an authority told you so?

Yes. Uhh, no. Maybe?.

That was usually enough to get them to keep digging with the data they had.

Up until now, students were doing things in their notebooks. This was enough to fill a 50 minute class period. Day 2 consisted of them putting their arguments on whiteboards and debating with each other in small groups.

# Whiteboarding

I had students form pairs and each put their arguments in Claim-Evidence-Reasoning format. Then, I shuffled all the groups around the room so that each pair was presenting to another pair that doesn’t normally sit at their table. Each group was to present their argument and (heavily) encouraged to modify their own boards if they changed their mind as a result of seeing another group’s presentation. Or add more evidence to support their own arguments if both groups agreed.

This here was the highlight of my week.

Students explaining and defending their ideas to each other is so beyond more effective than me “sage on the stage”-ing it. And way more fun for them. Many groups got (playfully) passionate about their arguments. They were already biting at the bit for the answer at the end of the previous day. They were downright starving for it today.

Once I was satisfied that enough discussion had happened, I played the rest of the video. I then I went through the explanation myself.