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## 07 November 2015

### Mail Time: Should I teach the elastic collision equation with velocities?

A correspondent writes in, in reference to an AP Physics 1 class:
[Edited for space]:

I have been teaching elastic collisions problems using an elastic equation : Vf + Vi = Vf + Vi to solve problems that are missing two of the velocities. We discuss that as long KE and momentum are conserved then we can take KE and momentum equations and divide them to get the elastic collision equation with just velocities.

I recently was tutoring a former student who is now in a local college about solving these types of problems and she told me that her professor said that the equation does not work and that its not physics! She was told that I was completely wrong!!! I immediately went to a Giancoli textbook. Giancoli does derive this equation following same reasoning that I derive for my students.

BUT... I hate to think that I have been teaching this wrong!  I was hoping you might be able to offer some clarity.    I went through [the professor's] problems and compared my solutions to the professor's solutions and I do get the answers he gets, just in a lot fewer steps.  Any suggestions? Should I not teach elastic collisions this way?

1. You are teaching absolutely correctly.  I don't know what her professor is on about.  Remember, "professor" means neither "good teacher" nor "better than you at introductory physics."  It's so easy for a high school physics teacher to be intimidated by folks with PhDs, or by education "experts."  As long as you are carefully self-evaluating -- and you obviously are, based on paragraph 3 above -- then do things your way.  I can't emphasize enough that even my well-tested methods and ideas are not for everyone.  The best physics teachers, like the best chefs, are creators, not imitators.

2. On this specific issue of elastic collisions: You might consider why it's necessary to teach quantitative solutions to elastic collision problems at all.  Yes, you need to be able to check whether a collision is elastic by comparing KE before and after the collision.  But even with the simplified relative speed equation that you reference, solving for speed in elastic collisions is more calculation that we need for AP 1, or even for my taste in any intro course.  That's not to say you're wrong to teach it, as I did for years... I just don't think it does enough to be worth the time it takes to teach and solve the problems.

1. Greg, I agree with your answer. I have used that same Giancoli derivation and formula for many years. Students tend to think it is pretty slick but, all in all, it is plug and chug. Several years ago I gave my AP students some Conceptual Physics worksheets that just asked questions. They were pretty much stumped so I now teach the ideas conceptually first. A few quick calculations and that's it. The lab we do is entirely qualitative and observational.

2. Greg, I agree with your answer. I have used that same Giancoli derivation and formula for many years. Students tend to think it is pretty slick but, all in all, it is plug and chug. Several years ago I gave my AP students some Conceptual Physics worksheets that just asked questions. They were pretty much stumped so I now teach the ideas conceptually first. A few quick calculations and that's it. The lab we do is entirely qualitative and observational.

3. I too agree with the general principle that it's not necessary to teach rigorous calculations, with one caveat: I have a few students who lean on mathematics to help reinforce their conceptual understanding. For those students, who are typically very strong mathematically, I encourage them to make up reasonable values for unknown quantities (or just to manipulate symbols) and then draw conclusions that way. The goal, of course, is to wean them off their algebraic crutch, but that can take some time.

4. Actually this relationship is quite conceptually deep. The most basic interpretation is that the relative velocities are equal and opposite before and after the collision. The more sophisticated interpretation is that this is the consequence of the momenta of the two objects having equal and opposite momenta in the center of mass frame of reference.
However, the students professor was quite right about warning the student about using this equation because it is only valid in 1d elastic collisions, and it is likely that students will not use it correctly.

Personally I have become quite taken by teaching collisions through using the momentum vector and the dot product. It is actually quite shocking how it reduces the algebra and how generalizable it is to special relativity.

5. Late to the party (only just found this blog - it's very helpful), but here is why I'd be leary about that equation (or any equation derived from "first princiles").

Many more years ago than I'd like to admit, I was tutoring a very conscientious young lady who was earning B+ on her physics exams, but was worried that she didn't really understand what was going on. We started with banked track problems. I began drawing an inclined plane, and sketching an FBD, and she exclaimed, "oh, that's what a banked track is!" Her professor had derived a banked track equation, and she had gotten the problems right by successfully plugging into that equation, without having a clue about what was going on. (A similar situation might be what the college professor was responding to. If the student understands where the equation comes from, and when it applies, ok. However, some may use it as a shortcut, without understanding when it applies...)

I get the same thing from my students. For circular orbits : "Can we use v^2=Gm/r on the test?" I tell them, no. (At least not without deriving it.) "You need to show me that you understand the problem from 1st prunciples: Fnet = ma, and a=v^2/r works for any uniform circular motion. Don't memorize a separate equation for orbits, roller-coaster loops, banked tracks... know what is going on."

I hadn't even seen the particular equation above. (Or forgot it if I did.) I am intrigued about the relative velocity angle, and might show it to my kids.