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Momentum is King!

March 6, 2011
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About a week ago, I saw a great tweet by Andy Rundquist describing Newton’s laws in terms of momentum.

Andy’s three laws of momentum were:

  1. Particles swap momentum.
  2. Interactions determine swap rate.
  3. If you don’t swap, you don’t change.

There’s something incredibly pleasing about seeing Newton’s laws in this form. If I want to get all zen about it, I would say the emphasis on swapping and interactions remind us how how connected everything is, while Newton’s laws as they are regularly phrased tend to focus on an isolated particle.

I’ve already written in some detail about how I think the language used to describe Newton’s laws plays a huge role in how easy they are for students to comprehend.

As a reminder, in my class, we state Newton’s 3 laws as:

  1. An object’s velocity will remain constant unless it is acted upon by an net force.
  2. \vec{a}=\frac{\vec{F}_{net}}{m} The acceleration of an object is proportional to the net force and inversely proportional to the inertia of the object.
  3. If object A exerts a force on object B, then object B will exert a force on object A. These two forces will be of the same time, equal in size, and opposite in direction.

As an exercise, I asked my students to figure out which of Newton’s Laws corresponded to each of Andy’s laws, and the kids dispatched this task in short order. We also talked about why swap was such a good word in these laws, rather than something else like “give each other” or ” trade.” Swap implies a 1 for 1 exchange. If particle gains X POMs of momentum from particle B, particle B must lose X POMs of momentum.

Then we sort of got on a tangent, and began to talk about the a person on the earth.

A student said this person has a constant velocity. “Really?” I asked, and drew this picture of the person on the earth now and 12 hours later.

So what is the change in momentum of this person. Some students quickly jump to the conclusion that it’s zero, since the speed doesn’t change. The picture helps them to quickly see this mistake.

How fast is the person moving anyway? Here’s a great chance for us to do some simple estimation. Let’s let the earth be 25000 miles in circumference, and suppose that there are about 2000 m in every mi. So, we have that the person must travel about 6.25\times 10^7 m in 24 hours (approximately 100,000 seconds), which gives a speed of about 625 m/s (compare this to 723 m/s for the exact answer).

So what’s the change in momentum between these two times? Let’s assume a mass of 50kg, and that motion to the left is negative.
\begin{array}{rcl}  \vec{\Delta p}&=&\vec{p_f}-\vec{p_i}\\  &=& 50\;\textrm{kg}\cdot (600\;\frac{\textrm{m}}{\textrm{s}})-50\;\textrm{kg}\cdot (-600\;\frac{\textrm{m}}{\textrm{s}})\\  \;&=&60,000 \;\textrm{kg}\frac{\textrm{m}}{\textrm{s}} \\  \;&=&60,000 \;\textrm{POM}  \end{array}
So then I ask, what would happen if a person in this room suddenly received 60,000 POM of momentum? “They would die”, is the rather obvious response from my students. Pretty quickly, my student sees that the reason we don’t get hurt, even though our momentum changes by this much is that the transfer takes place over 12 hours. So

\begin{array}{rcl}  \vec{F}_{net}&=&\frac{\vec{\Delta p}}{\Delta t}\\  &=&\frac{60,000 \;\textrm{kg}\frac{\textrm{m}}{\textrm{s}}}{50,000 \; \textrm{s}}\\  &=&1.2 \textrm{N}  \end{array}

So the force required to cause this change in momentum turns out to average 1.2 N to the right. Soon my students compare this to the gravitational force, 500 N, and realize it’s tiny. And then we think how this force comes to be, and my students recognize it must be that the normal force is just slightly smaller than the gravitational force, so over any small enough time interval, your velocity is approximately constant, but over large enough time intervals, the change is significant. And we get all this before we’ve done anything with centripetal motion. Momentum is king!

Finally, as a special treat, Andy put together this great screencast that explains how conservation of energy really is just a result of momentum conservation (There’s a tiny bit of calculus here).

So I wonder, could I retool the introductory course to start with momentum, and use that as the central principle from which we explain a whole range of phenomena? Possibly, but I think there would be significant hurdles to overcome, particularly in the abstract nature of these ideas. But this seems to be idea for an second year AP course.

17 Comments leave one →
  1. March 6, 2011 11:12 pm

    Just a note on the screencast. This week I’m doing oral assessments in that class. One of the standards is to have students state Newton’s laws from a momentum perspective and another is to derive the work – energy relation. A few of the students have asked if they have to present that material with this “momentum is king” perspective. I told them that if they don’t, they have to discuss why they don’t like that perspective. Others have said “there is no perspective, just laws”. I’m curious to see what happens when either of those standards get randomly chosen this week.

  2. Pete McNamara permalink
    March 7, 2011 1:21 am

    I’m tempted to make momentum “king” next year by teaching momentum after velocity and before acceleration, making the progression v, p, a, F.

    In my class I teach the tale of the meter, which was originally defined to be 1/40,000,000 the circumference of the earth (and is a wild story), so I would go back to that tale, but I like the derivation.

  3. March 7, 2011 11:32 am

    Very nice post John! Love the rich description of what happened in your class. Would love to see some video of this type of classroom interaction. Let me know when something like this is going to happen. Would love to observe.

    also, been reading about nonlinguistic representation of ideas or concepts. The idea is that students learn more deeply when they are asked to add or associate a nonlinguistic model with the linguistic explanation of a concept. Brain-based research show this to be true. Here is a website for this idea:

    http://www.netc.org/focus/strategies/nonl.php

    Also, see chapter in Robert Marzano’s, AArt and Science of Teachimg.

    Bob

    • March 7, 2011 9:23 pm

      Bob,
      The nonlinguistic representation idea is very interesting. I’m going to have to give this some deeper thought on how I might have students construct some visual representations of some of the funamental laws we use. The modeling curriculum is already very heavy on visual representations—most of the kinematic equations are derived from velocity graphs, and there are diagrams that are used for analyzing collisions. But I haven’t ever really thought about having kids come up with their own visual representations of Newton’s laws. Thanks for the inspiration!

  4. March 7, 2011 9:17 pm

    Like Robert Ryshke the most valuable thing to me in this post is the type of interaction you modeled for your students here, of exploring an idea that is not in your textbook and using math as a tool to build a new framework for the new idea.

    I am more hesitant than you about the “make momentum king” idea. Does this perspective sync with ideas in advanced physics about the relative usefulness of energy vs momentum? I would talk to a physics prof before implementing a change like this.

    The laws in my class are something like:
    1. (The traditional version, with a heavy emphasis on *net* force.)
    2. A net force causes acceleration; an acceleration means there was a net force. (Also, F=ma.)
    3. Forces come in pairs that act on different objects, and are equal in size and opposite in direction.
    I’ve found that force as a concept is easy enough to understand, especially by high school. I’m moving toward “Forces are interactions;” still using “push or pull” in grade 8 though.

    My “answer this question by the end of the course” for this topic is, “If atoms are 99.9999999999999% empty space, why can’t I just push my hand through this table-top?”

    • March 7, 2011 9:27 pm

      Micah, I agree that force is a much easier concept for students to understand, and we start by setting up a very careful template for describing forces. We begin by noting that only objects can exert forces (no forces of vacuums), and that forces must be described using the form the [type of force] of the [object exerting force] on the [object experiencing force]. This pays huge dividends when it comes to understanding N3. If students fully label “gravity” as the gravitational force of the earth on the ball, then it is a trivial switch to realize the N3 pair must be the gravitational force of the ball on the earth. I also work really hard to get my students to see that at rest really is nothing more than a special case of constant velocity.

      • Pete McNamara permalink
        March 8, 2011 10:25 pm

        I am not sure that Force actually is an easier concept to understand, “really”. It relies on a much longer chain of ideas than momentum does. Students come into physics with a strong idea that force is proportional to velocity, and many have a lot of trouble distinguishing velocity and acceleration, intuitively. Isn’t momentum a smaller step from what they know than force?

        • March 9, 2011 8:05 am

          I think the idea that force is a push or a pull is easier to understand than momentum is a conserved quantity. I think all students need to confront the idea that a force isn’t required to keep an object going, as this is a major misconception for almost everyone. I really do like the momentum approach, I’m just not sure how workable it would be for high school freshmen.

        • Pete McNamara permalink
          March 9, 2011 11:02 am

          You may be right… this stuff is hard enough for college freshmen🙂. When I talk to my kids (I teach mostly HS seniors) they say a lot of things like “the ball keeps going until the force of the push runs out” or comparable things that suggest they have something almost like momentum in their head already, they just call it “force”.

          Acceleration is a really hard concept, and a lot of kids have trouble really distinguishing it from velocity. I wonder if turning velocity into momentum (which is more like “stuff” the object has) then looking at how quickly that changes would help. v->p->F->a?

        • March 9, 2011 9:47 pm

          Pete,
          You’ve given me something to think about over the summer. Currently, I teach mechanics in this order 1-d constant velocity -> balanced forces (N1 & N3) -> 1-d constant acceleration ->unbalanced forces (N2) -> conservation of momentum (2D) -> projectile motion. Could all this be replaced with a treatment that develops momentum from the outset and talks about constant momentum vs changing momentum with particles and then expands to systems? It’s worth thinking about for sure.

    • March 7, 2011 9:41 pm

      Micah, I’ve been thinking a lot about the quantum mechanical issues involved in this approach. Classically I haven’t run into any problems as the Lagrangian formulation just turns into more accounting (and, of course, makes things really easy to calculate). Still, I think momentum as king in the classical world is quite defensible (I’ve been fighting off my junior physics majors for a couple of weeks now).

      On the quantum side, the Schroedinger equation is two things: energy balance and particles are waves. The first part really isn’t that big of a deal since this approach is simply saying that while energy conservation is true, it’s not as interesting. The second part is not something I’ve been able to wrap my brain about yet. The notion of momentum representing the wavelength is odd and not present in the classical world. In the quantum world the wave is everything. Hmm, here I go again, I guess . . .

  5. Chris Goedde permalink
    March 8, 2011 11:14 am

    The calculus-based college textbook “Matter and Interactions” by Chabay and Sherwood takes a momentum first approach, if you want an example of what such a curriculum might look like. They introduce momentum in chapter 1, force and the “momentum principle” (Delta p = F * Delta t) in the second chapter, and acceleration doesn’t appear until chapter 4, and then it’s solely as an approximation.

    • March 9, 2011 8:03 am

      Chris, I love Matter and Interactions. IMO, it’s the best calc-based college text out there. I previously <a href="wrote a review about it. And one of the many reasons why I love it is that it starts with the momentum principle and uses it to explain so much.

  6. Paul Camp permalink
    May 18, 2011 5:36 pm

    “So I wonder, could I retool the introductory course to start with momentum, and use that as the central principle from which we explain a whole range of phenomena?”

    Already been done for you (Thomas Moore, Six Ideas that Shaped Physics).

    • May 22, 2011 10:50 pm

      Paul,
      Yes—I’ve got to get a copy of that text and take a look. I’m still not quite sure that approach could work for 9th graders, but I don’t want to disrespect the king. I do think Andy is on to something with his reformulation of Newton’s Laws.

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