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Long Chain of Reasoning exercise 1: Momentum conservation

January 29, 2011

Conservation laws are beautiful. They are among the primary “big ideas’ I want my kids to leave my class understanding. And I want them to understand them at a level beyond m2\vec{v}_{1i}+m2\vec{v}_{2i}=m2\vec{v}_{1f}+m2\vec{v}_{2f}. But the challenge in using conservation laws is knowing when to apply them. This is where I think my thoughts about following long chains of reasoning might come into play.

Here is a long chain of reasoning about a collision between two cars (A and B) on a level road.

  1. Car A and B experience a contact force between them.
  2. the change in momentum of A should be equal to the net force on A times the time of the collision. \Delta\vec{p}_A=\vec{F}_{net}\Delta t
  3. the change in momentum of B should be equal to the net force on B times the time of the collision. \Delta\vec{p}_B=\vec{F}_{net}\Delta t
  4. By N3, the force of A on B is opposite the force of B on A. \vec{F}_{\textrm{A\;on\;B}}=-\vec{F}_{\textrm{B\;on\;A}}
  5. Since the other forces are balanced (gravitational force and normal force) the net force on each object is this contact force.
  6. So the net forces are equal and opposite.
  7. Thus the change in momentum will be opposite. \Delta \vec{p}_A=-\Delta \vec{p}_B
  8. Thus the total change in momentum will be zero \Delta \vec{p}_B+\Delta \vec{p}_A=0
  9. We conclude that the momentum is conserved (does not change) during the collision.

My thought is to start my students by having them construct the above chain of reasoning. Then, present them with a new situation: the collision takes place on the side of a hill, and ask them to use the chain of reasoning to see if momentum will be conserved in that case. Since I’ve been on leave for 2 weeks, I haven’t gotten a chance to test this out in the classroom, but I’ll let you know how it goes.

5 Comments leave one →
  1. January 29, 2011 12:42 am

    I am assuming you are leading them through this (abstract) chain after they have (concrete) experimental evidence for momentum conservation, right?

    • January 29, 2011 12:47 am

      Yes, they’ve done the paradigm lab on collisions and even looked at the N3 videos you made while I’ve been on leave. This would be my way of capping off the unit and seeing how they can pull everything together.

      • February 2, 2011 6:49 am

        Here’s the photo (click for larger version) of the chains the students constructed

        Image and video hosting by TinyPic.

        After students made these chains, I asked them to use them to analyze 3 situations to see if momentum is conserved:

        • I drop a rock from my hand. (system: rock only)
        • A coasting rocket slams into an asteroid, and as it makes contact, its rocket engine begins to fire. (system: rocket + asteroid)
        • An asteroid is broken apart into 3 fragments by an explosion (system: asteroid)

        While I don’t think this was a huge success, I do think it was a helpful lesson for my students, and if we try to make chains of reasoning like this more often (and more importantly, I find way to assess this), I think they’ll get the hang of it.

  2. Alexandra Maniaci permalink
    January 30, 2011 7:29 pm

    My name is Alexandra Maniaci, I am a student in EDM 310 at USA and will be commenting on your blog. My Twitter name is @alexmaniaci and my blog URL is http://maniacialexandraedm310.blogspot.com. I think this exercise you set up will be enjoyable and educational for your students. The second situation in your assignment, with the cars being on the side of a hill instead of a level surface, is very thought-provoking and a great way to put a twist on a problem.

  3. February 17, 2011 9:37 am

    I let my students use these paper chains on the momentum test, and I think it helped them. I saw a couple of kids following their way through each of the links as they examined the question. One thing I think I might do to modify the chains would be to color code the links on how/why they are true—steps that always true by mathematical reasoning are green or something.

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