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.

Honors Physics I: Course Description

As detailed in the previous post, I don't feel like waiting for the AP Physics B redesign.  Woodberry Forest is going to begin teaching according to the principles of the proto-AP Physics 1 and 2 courses right away, beginning in 2011-12.

We're calling our first year college-level course "Honors Physics I."  We're modeling the course structure on the AP program.  That means we're going to commit ourselves ahead of time to a weighted topic coverage list.  That means we have a course exam written which will remain locked away until mid-May, and that will not be changed on a whim.  And, I'm going to arrange for *external* validation of the exam -- I've talked to a few AP readers about a possible "trade and grade," in which I grade a set of their tests or exams, and in return they grade my Honors Physics I exams to the rubric that I send them.

The course topic coverage outline for Honors Physics I can be found here via google docs -- so please forgive any formatting issues.  My goal, approximately consistent with the College Board's goal for AP Physics 1, was to cover about 60% of the current AP Physics B curriculum.  Please remember -- this outline is NOT necessarily related to what the College Board has in development for AP Physics 1.  Nothing about AP Physics 1 topics that has been officially released, because even the people in charge of the redesign have not settled on a final distribution of topics.  My outline represents what I would do if I were solely in charge of the redesign. 

The quick rundown of the six content areas I've included:

Mechanics (40%): Pretty much everything on AP B mechanics, except torque.

Fluids (10%): Static fluids only, i.e. static pressure and buoyant forces.

Thermal Physics (15%): PV diagrams, the ideal gas law, and the first law of thermodynamics.

E&M (15%): Forces due to fields, but NOT the source of E or B fields.  Basic DC circuits.

Waves (10%): Basic properties; sound & light; Snell's law; but not standing waves or optics.

Nuclear Physics (10%): Definitions of particles, conservation laws, mass-energy equivalence.

Also in the linked course description you'll see the exam format.  I've made the exam two hours, so as to fit better into my exam periods.  It's in three sections, but without formal separation; all sections can be worked on at will during the two hours.  Calculators, a constant sheet, and an abbreviated equation sheet will be accessible during all sections.  (Why?  That makes administration easier.  And the calculator won't really help much on the multiple choice, anyway.) 

My major divergence from the current AP exam format is the third "short answer" section.  I'll ask ten brief questions that will usually involve a verbal explanation.  You know how every recent AP free response question includes a lettered part that says "justify your answer?"  Well, these short answer items will each be similar, except in isolation, without the context of a larger problem.  While I have no idea whether such items will appear on the future AP Physics 1 exam, I do know that all formatting options are on the table.  It sounds likely that the current dichotomy of just multiple choice and 10 or 15 point free response items will be adjusted.

Want to use this course?  Go for it.  I'll be happy to send you more materials:  a pratice test that I'll give in November, and hard-copies of the final exam with a rubric next May.  I only ask a couple of things in return:  (1) Collect the exams when you're done, ensuring that they don't get posted online; (2) Report to me how your students did on the final exam, whether they did well or poorly -- I'll keep that info private except for saying globally how everyone did; and (3) Send me a can of Skyline Chili.* 

* or equivalent.  Condition (3) is not mandatory.

If you teach an honors course, this might be just the thing to prepare your students for the AP B exam in 2012-13.  Or, you could use this course and exam to demonstrate the rigor of your non-AP course to parents, administrators, and colleges.  Try it -- I think you'll like it. 

AP Physics 1 and 2 Redesign (as it stands now) and Honors Physics I

So, you may have heard that the College Board has been working on a revolutionary change to the algebra-based AP Physics course.  In its current form, AP Physics B requires an enormous breadth of material.   As it stands, teaching AP Physics B well is as much about organization, scheduling, and pace as it is about presenting the overly-numerous physics topics themselves. 

The College Board's plan, as they have discussed at their annual conference and with readers, is essentially to split AP Physics B into two courses, AP Physics 1 and AP Physics 2.  In principle, each of these separate courses would mimic a semester's worth of college physics, in the style of AP Physics C and its two independent exams.  The overall combined AP Physics 1 and 2 curriculum would allow even more broad coverage of physics; but since the material is intended to be spread over two years, a single course will cover *less* breadth and thus be manageble for a wider student population.

Although the curriculum is still in considerable flux, a few general principles have been released. 

Topics / "Big Ideas": The specific topics to be taught in each year, and the depth to which those topics should be taught, are currently unclear.  Partially this is because the redesign committee has chosen to prioritize "big ideas" of physics that cross topic areas.  For example, Newton's three laws and the relationship between forces, fields, and motion can be applied to more than just blocks on inclines; so, this "big idea" will be revisited in covering static fluids, electrostatic forces, magnetic forces, and so on.  Similarly, conservation laws can be applied across topics, at the introductory level including even (or especially) nuclear physics.  Topics will be chosen to fit the "big ideas" model of learning introductory physics.

Writing:  If you look back at Physics B exams from the 1970s and 80s, you'll see a lot of problems testing algebraic manipulative ability as much as conceptual understanding.  That focus changed substantially in the mid 1990s.  Laboratory-based questions, along with the proliferation of "justify your answer" items became regular features of the free response exam.  Everything I've heard about the new Physics 1 and 2 courses indicates that this emphasis on justifications and explanations will not merely continue, but will dominate the exams.  That doesn't mean that derivations and calculations will disappear, since those are part of physics, too.  However, you can expect that those who consider physics merely as the process of plugging numbers into an equation will be at an even more significant disadvantage than they already are.  Students will need to develop the skill of communicating understanding verbally, and concisely.

The redesign into two separate algebra-based courses provokes ideological struggle amongst physics teachers that sometimes approach Burr-Hamilton levels.  I will not get into the pro and con arguments here, at least not yet.  It's too early to panic or rejoice.  Physics 1 and Physics 2 will not begin for at least three years, and likely more.  The College Board is still in the process of getting buy-in from colleges, designing and norming the curriculum and the exam; then they understand that they need to provide significant lead time so schools can figure out how these new courses fit into widely varying science programs.

My message to teachers is not to worry about the redesign yet.  AP Physics B, in its current incarnation, will continue for a while. 

One great advantage of the upcoming Physics 1 course is the potential to truly serve a broad portion of your college bound population with a first-year AP course.  Those who teach "honors physics" or "college prep physics" will likely find that AP Physics 1 meets their needs beautifully. 

I and my department, we didn't want to wait.  We are teaching "Honors Physics 1" next year.  (Not "AP Physics 1," because that AP course doesn't exist yet, and we can't use the College Board's trademark on an unofficial course.)  In my next post I'll describe my school's course, which is intended to be my own version of what I hope AP Physics 1 might become.  I'll even provide some course materials if you're interested... read on.