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.

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

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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.