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14 January 2012

Collisions -- how much detail?

In a typical college physics textbook, the end-of-chapter problems about collisions begin with the simple and move on to the unreasonably complex.  I'm frequently asked by AP teachers:  How far do I go before complexity becomes "unreasonable?"

In judging the depth necessary for this or any topic, first recognize the motivation of the textbook authors.  They're not making a considered, pedagogically sound choice about what material is important, or even about the best way to present said material.  No, they quite reasonably want the largest audience possible.  The publisher is far more likely to hear "I didn't choose your book because I like to derive the formulas for inelastic collisions in two dimensions with a coefficient of restitution e, which isn't covered" than "I rejected your book because it had too much information."  Thus, we get 103 page* tomes that touch on every possible aspect of "introductory" physics.

* And 102dollar

Don't ever use the textbook as a sole guide to what's important.  Of course that begs the question:  how do you figure out what's important when teaching an AP or college-prep high school course?  The simple answer is to look at AP exams since about 1996* for guidance.  Summarize to yourself the kinds of questions that are asked in each format (multiple choice or free response).  Then, certainly if you're teaching AP, be sure to cover the types of questions that showed up; and ignore anything else, even if it's in the textbook.

*1996 approximately marks the transition on AP Physics B exams between the "Shut up and Calculate" era to the "Justify your answer" era.  Sorta like when major league baseball lowered the pitcher's mound in 1969.

But what about those who DON'T teach AP, or who aren't particularly fond of the College Board's curriculum or their funny little ways?  What if we were to start from first principles, and decide philosophically how much detail SHOULD be included in an advanced high school physics course?  Fair question.

I want to cover the basics both conceptually and calculationally.  But we don't want to perform any calculations that are so complicated that the mathematical methods outweigh the physics approach.  In an algebra based course, I assume fluency in basic algebra, and in using sin, cos, tan.  A multi-variable system that cannot reasonably be solved in a couple of minutes is out of bounds; similarly with any trig identity beyond sin/cos = tan.  This limit on calculation is born of philosophy -- I never want to do math for math's sake, or I would have gone into teaching math -- and of practicality as well -- if I put a detailed calculation on a test, I can't ask more than one or two questions in a 45 minute period.  (Plus I'd be testing math ability, not physics ability.)  So here's what I teach about collisions:

1. The fundamental meaning of "conservation" of momentum.  Everyone's got to understand the idea of an unchanging quantity; but also how an individual object's momentum can change without violating conservation.  We would test this understanding with conceptual questions like "A ball bounces off a wall.  Did its momentum change?  How is that consistent with conservation of momentum?"

2. Basic computations with momentum conservation in one dimension.  A no brainer.  "A cart moving 30 cm/s collides with and sticks to an identical resting cart.  What is the speed of the carts after collision?"  Carts can bounce, stick, be moving in any direction.

3. Definition of "elastic" collision.  Although some books and teachers split hairs over the precise definition of elastic, most define an "elastic collision" as one in which kinetic energy is conserved.  While momentum is conserved in all collisions, kinetic energy is only conserved in elastic collisions.  Colliding objects may not stick together in an elastic collision, though a collision is not necessarily proved to be elastic just because objects bounce off one another.

4. Calculation to determine whether a collision was or was not elastic.  Note that this is NOT the two-equation, two-variable calculation to predict speeds of two objects after an elastic collision.  No, all I'm suggesting here is that we teach students how to add up the total KE of all objects before a collision, add up total KE of all objects after a collision, and compare.  The question generally takes the form "Was the collision elastic?  Justify your answer."

5. The vector nature of momentum, the scalar nature of kinetic energy.  "Two identical carts move toward each other at the same speed, stick together, and remain at rest.  Does this violate conservation of momentum?  Does this violate conservation of KE?"  Everyone has to recognize that momentum in opposite directions can "cancel out," but that the phrase "kinetic energy in opposite directions" is silly.

6. Ability to consider horizontal and vertical momentum separately in a 2-d collision.  Once again, I would not ask for anything that required multi-variable system analysis.  But we can arrange problems such that the horizontal conservation of momentum is simple to solve; and where vertical momentum was zero before the collision, so must be zero (in sum) after collision.  Usually, such questions will be limited in scope to very simple calculations, or to conceptual questions:  "Calculate the initial vertical momentum of the system before collision.  What is the system's vertical momentum after collision?"  Or, "Is magnitude of the red ball's vertical momentum greater than, less than, or equal to the magnitude of the green ball's vertical momentum?  Explain."

That's about it.  No coefficients of restitution.  No proof of why 2-d elastic collisions always produce final velocity vectors at a 90 degree angle to one another.  All of the types of questions above can be phrased in a straightforward manner, allowing for answers in a couple of minutes.  The list of six ideas allows for both conceptual and calculational questions.  Good.

2 comments:

  1. Is coefficient of restitution really that hard? My son did calculations with it in his 5th grade science fair project:
    http://users.soe.ucsc.edu/~karplus/abe/Science_Fair_2007_report.pdf

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  2. Kevin, it's not that coefficient of restitution is difficult, per se. We use it as a factor in research projects all the time, and my students have no trouble understanding.

    The point is, science projects and research assignments are very different from introductory physics class. In class, my goal is to get everyone to understand the fundamental principles of momentum and energy conservation in collisions. The coefficient of restitution is useless if the basic conservation principles are unclear. That's why I don't consider the coefficient of restitution to be a fundamental, but rather an interesting fact that allows for prediction in more complicated collisions.

    I'd make the same point about most second-year concepts... it's not that they're particularly hard to use. It's that I want to spend my limited class time focused on the calculations and concepts that are more generally relevant to a topic. I'm teaching a survey course that has to get to the basics of electricity, magnetism, and nuclear physics; too much time and too many facts in collision problems mean less time and brain room for later ideas.

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