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04 June 2017

Deriving expressions in AP Physics 1

Reporting from the AP reading here in Kansas City, where I've discovered that Jack Stack barbecue is excellent, but still no match for Gates.  And, where I've been immersed for days now training people on the rubric for the 2017 AP Physics 1 exam problem 3.  

Based on my experience here, I think it's worth a reminder to teachers about the expectations for "deriving" an equation on an exam.

Introductory physics is all about communication of ideas, and not as much about getting the One True Answer to a problem.  Physics is not a math class.  

Students in my class may whine (early on, at least) about not getting full credit for a poorly presented problem that nonetheless includes the correct answer.  Okay, so your English teacher requests an essay with textual evidence analyzing Shakespeare's characterization of the Romeo/Juliet relationship.  Your entire essay: "He loves her."  You earn a failing grade, of course.  How effective or intellectually honest do you think it would be to whine that your essay deserves an A because the answer is right?  I mean, the answer is in fact right...

A derivation, like any physics problem, is an exercise in communication -- but a derivation requires communication primarily in mathematics.  Just because the answer is right, just because a student knows in her head what mathematical steps she intends to take, that doesn't mean the derivation has served its purpose.

So what SHOULD we expect from students on derivations?

1. Start from first principles, and explain what first principles you're using. That means something from a "facts of physics" list: Newton's laws, Kirchoff's laws, conservation principles, the definition of acceleration or impulse or power... most anything on the AP equation sheet or on my fact sheet will work.

2. Communicate the reasoning for each step.  I think words are best here -- an annotated derivation can hardly fail to earn credit where correct.  Try circling terms and explaining what they mean.  Try telling the reader why you've substituted various terms into the equation you began with.

3. Show enough detail that a strong physics student at another school can understand without asking for clarification.  The audience should NOT be the expert physicist.  I personally don't need to derive an expression for the acceleration of a three-body system connected over a pulley, because I've done so many of those problems that I can write the answer based on memory and instinct.  My students, though... they need to start with Newton's second law for the system, explaining what expression is used for each term and why that expression is relevant.  

4. Use algebra to communicate, not to solve.  I often see students take three steps merely to rearrange terms in an expression, using annotations like "commutative property" and "divide both sides by m."  Assume the audience knows how to do math.  Use the way the math is laid out to highlight reasoning.  For example, if you have energy terms before and after a collision, write all terms clearly in a single line, with before the collision left of the = sign.  Label each term with a circle and a couple of words.

I'm sure readers - both blog readers and AP Readers - may have some further thoughts.  Please post in the comments.  


  1. One question: on Question 2, part b), of this year's AP Physics 1 exam, the students have to derive an equation for the coefficient of static friction. Now, there are a few ways to measure that, but one could use, say, a spring scale and measure some forces directly. I've always tried to hammer in that my students must draw a free body diagram whenever writing equations involving forces, but that wasn't specifically asked for in the problem here. Will it be required for full credit?

  2. Hey, Will... the rubrics will be out in September. I wasn't on that question. I can't imagine that the actual free body diagram would be required, as the question did not ask for it. However, I would expect that at least clear mathematical or verbal statements of equilibrium horizontally and vertically (or parallel and perpendicular to an incline) would be necessary. The first principle on that problem is simply Newton's second law with a=0; start there and communicate, and you should be fine.

  3. I have a question about teaching philosophy. You wrote "with before the collision left of the = sign". All of dynamics is structured as change (final minus initial) equals whatever drives the dynamical change. That naturally leads to final velocity equals initial velocity plus a*t, or final energy equals initial energy plus net work. I encourage my students to stick with that ordering even when the change is zero. Do you have a reason for writing inital plus physics equals final? I have toyed with, and found success with, that reordering for collision and energy problems but would not dare try it with the kinematics equations. As I see it, the conflict is between consistency (which tends to eliminate fundamental sign errors) and linking the equation to a pictograph or textual description of what is happening. The latter tends to run from past to present to future.

    Your thoughts?

  4. Great point, CC... I certainly teach the mathematics of energy conservation by starting with the change in the system energy, setting that equal to the sum of changes in each particular form of energy. Your approach is a good one.

    So how does that square with my discussion above? Perfectly. A typical student would start the derivation with L = L', and go from there. Your student would start with 0 = L'-L. Works for me!

    The AP exam is graded by humans, not robots, for a reason. We look to see if the student clearly communicated the starting point of the derivation -- that is, didn't just plug variables into equations, didn't just write a final answer, but said clearly what fundamental principle was used as a springboard.

    Thus, even the words "conservation of angular momentum" would be sufficient. So would "the rod-disk system's angular momentum can't change."

    And, note the point I made about annotation. I did see a number of papers with odd groupings of terms... often these were clearly labeled with the words "before collision" and "after collision". My overwhelming preference is for a student to communicate in both words and mathematics, in order to make discussions of mathematical order unimportant. :-)


    1. Sorry I wasn't clear, but my context was learning (or perhaps using without error) rather than the grading rubric or getting them to annotate their work. But I also think that equations and description sometimes come into conflict with one another, and that might lead to resistance by students. Let me try again.

      I take them directly from Delta E = W to Ef = Ei + W and then use the latter when doing examples of how to solve problems for only one reason: it mirrors Vf = Vi + a*t and is thus more consistently memorable. Teaching philosopy, or pedagogy if you prefer. Other orderings, whether Vi + a*t = Vf or Vf-Vi = a*t will convey the same physics, but those do not convey the idea that Vf is actually a function V(t), and functions are always defined on the left side of an equation. [I've never seen a textbook present Vi + a*t = Vf as The Equation.] I want them to think of momentum and energy the same way, as an alternative way of understanding the dynamical evolution of a system, and use that standard ordering to get work in the right place every time.

      Order is unimportant in Pf = Pi or Lf = Li problems, and that is about all you get at your level and mine. External impulse problems are rare in intro classes. I don't care how they write it, but I approach momentum consistently just for consistency. Pf = Pi + Favg*t, if you like, even if that is rarely used on a test.

      What complicates all of this is the descriptive process tends to be in temporal order. That is what leads to your statement (and mine) about what happens in that collision. We draw or describe motion diagrams from initial to final, and we should translate that into mathematics in the same sequence, Ei + Work = Ef, which is the exact opposite of how the equations are presented in textbooks! One of the two has to give if we are to push description first, before math, because students take The Book as gospel.