One of the things I love about the modeling curriculum is its emphasis on multiple ways of representing physical quantities. I try to emphasize to my students the value in these multiple representations, and how deep understanding understanding comes from seeing the connections between them.

A prime example is momentum, which in a great reworking of the Modeling Curriculum by Kelly O’Shea follows Newton’s Second Law (Unbalanced Force Particle Model-UBFPM).

Consider the following fairly standard 2D momentum problem:

A 20kg rock, initially at rest, is blasted into 3 fragments. The first 10 kg fragment travels 45° North of East at 10 m/s. The second 5 kg fragment travles east at 20 m/s. Find the velocity of the third fragment.

I try to emphasize 3 different ways of solving this problem to my students.

1. Graphically. In many ways this is the method that gives the most physical insight into the problem. By solving this problem graphically, students can actually see which way the third fragment is traveling. In fact, just after drawing a picture of the momentum vectors for the first two objects, most students can make a pretty good guess about which way the third object will go.

The final momenta of the two known fragments.

Since the rock starts at rest, the total momentum must be zero. Here’s the illustration.

A vector construction that uses the fact that the total momentum is zero to find the momentum of the third fragment.

So by doing the construction and measuring the third vector, you can find that the momentum of the 3rd piece is $180 \frac{\textrm{kg\:m}}{\textrm{s}}$ at an angle of 22 degrees below the $y$-axis. Since we know the mass of the of the object is 5kg, we can calculate the velocity as $36\frac{\textrm{m}}{\textrm{s}}$, in the same direction as the momentum.

2. Components. Much more abstract, but also more powerful, are components. Since we’ve waited to study components until kids have a good graphical understanding, I think my students are much better able to understand the meaning of components, thought they still occasionally make mathamtical faux-pas like adding x and y components together. I do think that using an improved component notation that resembles ordered pairs (and is very similar to vpython) is a key part of how my students are able to comprehend components so easily. Unit vectors, or $\hat{i},\hat{j},\hat{k}$ notation are just too arcane for most students. Here’s how we do it:

$\begin{array}{rcl} \vec{p}_{1f}+\vec{p}_{2f}+\vec{p}_{3f}&=&0\\ \vec{p}_{3f}&=&-\left(\vec{p}_{1f}+\vec{p}_{2f}\right)\\ \vec{p}_{3f}&=&\left(\langle 100,0 \rangle \frac{\textrm{kg\:m}}{\textrm{s}}+\langle 70,70 \rangle \frac{\textrm{kg\:m}}{\textrm{s}}\right)\\ \vec{p}_{3f}&=&\langle -170,-70 \rangle\frac{\textrm{kg\:m}}{\textrm{s}} \end{array}$

3. Initial-Final (IF) Charts. The third way of representing momentum is novel to modeling physics, and as best I can remember was invented. The idea is to represent components of momentum before and after a collision using velocity-mass bar charts, where the area of the bars represents the momentum in the system. Here’s an example for our problem above:

IF charts representing the known x- and y-momentum before and the explosion.

These charts quite clearly show the initial momentum to be zero, and then the momenta of the two fragments to in the final chart. They can see that there are $170 \frac{\textrm{kg\:m}}{\textrm{s}}$ of momentum in the x-direction and $70 \frac{\textrm{kg\:m}}{\textrm{s}}$ of momentum in the y-direction. Since momentum is conserved in this case, it becomes pretty clear to my students that the chart must be completed by adding enough momentum to each chart to make the total momentum in each direction zero. And since they know the mass of the third fragment is 5 kg, they can figure out the x and y components of velocity, like so.

The complete IF chart for the momenta of the three fragments. It should clear that in both final charts, the negative area is equal to the positive area since total momentum is zero.

Really this is just a tool for representing the components of the momentum in a more visual way, but it does gain a few advantages. First, it clearly emphasizes conservation. In fact, many of my students were able to discover momentum conservation by being asked to draw these charts for a series of collisions and asking them to compare the initial and final situations.

The key idea is that all three of these representations agree with one another and provide unique insight into the problem. My students often ask which one is best, and I simply reply “they all are.” I want my students not to get into the habit of relying on a single tool for understanding, or thinking there is only one way to approach a problem.

These IF charts are also a great lead-in to energy bar charts, which are one of the primary tools used in the modeling curriculum to understand energy conservation. For example for the problem above, where the chemical potential energy of the explosion completely transformed into the kinetic energy of the fragments, you might draw an energy bar chart like so:

I’m also beginning to think that momentum is the idea first conservation law to introduce. First, you only have one type of momentum, and it can only change through one mechanism (impulse). In energy, you have many different types (chemical, gravitational, kinetic, thermal), and many different mechanisms of energy transfer (work, heat, radiation). I used to think the vector nature of momentum made it harder, but if your kids have developed a good graphical understanding of vectors, and get reminded enough why 2+2 isn’t always 4 in vector land, they seem to do ok.

Finally, I’d like to make a huge plug for Andy Rundquist’s post about having a unit of momentum. Right about we get to momentum, kids start to get hit with three units that look the same to a novice, but are radically different:

The newton: $1\:\textrm{N}=1 \:\frac{\textrm{kg\:m}}{\textrm{s}^2}$

the kilogram meter per second: $1 \:\frac{\textrm{kg\:m}}{\textrm{s}}$

the Joule: $1\:\textrm{J}=1 \:\frac{\textrm{kg\:m}^2}{\textrm{s}^2}$

Andy says, and I agree, that talking about kilogrammeterperseconds being conserved is awkward and inhibits student understanding. He introduced the unit pom, and I thought to give it a name: “parcel of momentum.” Suddenly, it becomes very easy to talk about pool balls exchanging poms via impulses when they collide with one another and it presents a tidy little picture of momentum conservation that students can really grasp. I only wish I had started with that, helping my students to see that the “total amount of poms never changes in a collision” before they got all hooked on the awkward SI unit. So how to we go about getting NIST on board?

1. February 12, 2011 12:16 pm

This is great (and not just because of the plug 🙂

When I think back to times when it seems students misunderstand something, I often realize that I was mixing and matching the various representations in my head and therefore in my explanations. Unfortunately I don’t do a good just helping them see the different representations like you talk about here. It’s a common problem for me: I’ve got something in my gut where all the representations live and interact. I can use whatever works best to convince myself of something. When I talk with students I don’t always (or even often, unfortunately) help them develop those models for themselves so that we can have a good conversation about it.

Have you witnessed students talking past each other because they’re each using different representations? How do you deal with that?

Thanks again for the plug: go POMs!

• February 13, 2011 12:02 am

I’m not sure I have seen students talking past each other using different representations—perhaps this is because I’ve been requiring them in most group work to use multiple representations. The thing I still a lot of is students not really seeing the connection between graphical vector drawings and algebraic work, so I still get a lot of adding of x and y components, etc.

2. February 12, 2011 9:24 pm

I’m all about the poms! Great idea. Like those graphs too. They wouldn’t have been made in Omnigraph sketcher now were they? Love that program too. Thanks.

• February 13, 2011 12:02 am

Yep. OGS is the best. Did you know you can make motion maps and free body diagrams with it? Though I prefer the sketching tools in pages for making FBDs.

3. February 13, 2011 12:37 pm

According to wikipedia ( http://en.wikipedia.org/wiki/SI_derived_unit ), the supplementary unit for momentum is the newton-second. Personally, I prefer not having so many names for units, since I have a poor memory. It is simpler if there are only a few units and I just have to remember the relationships, and not arbitrary names as well.

• February 13, 2011 12:44 pm

While you’re right, I find the Newton-second a bit opaque as well. Think of energy conservation—it’s very easy to talk of joules of heat being transferred from a hot to a cool liquid, or joules of kinetic energy being dissipated in the disfigurement of steel during a car cash. Somehow, talking about particles in a gas exchanging Newton-seconds of momentum during collisions doesn’t do it for me, while poms does make a bit more sense. Of course, with something like momentum, which is so easily defined at the elementary level as the product of mass and velocity, students can always get to the fundamental unit pretty easily.