23 September 2011

The OJ Simpson Question: Magnitude and Direction of Accleration

Knowing what an acceleration vector means about motion is perhaps the biggest conceptual challenge in first semester physics.  No matter how many times I say "the direction of motion has nothing to do with the direction of acceleration," this misconception (among many others) remains.

* Speeding up means acceleration is in the direction of motion.
* Slowing down means acceleration in the opposite direction of motion.

However, put these facts in the context of a velocity-time graph, or in the context of specific motion north and south, and heads explode.  And that's really all we can do -- ask about the meaning of acceleration in as many different contexts as possible until the class is sick of such questions.

In the first several kinematics assignments, I've displayed a position-time or velocity-time graph and asked for a description, in everyday language, of the represented motion.  In order to tease out the physical meaning of an acceleration vector, I switch up:  I present a description of motion, and ask students to make a velocity-time graph.

1.    In an alternate universe that still obeys our laws of physics, O.J. Simpson leaves a tollbooth in his white Bronco the morning after killing his wife.  Soon after, he sees a police officer flash his lights.  Hoping to get away, he slams the gas pedal to the floor, but then O.J. hits a concrete barrier and crashes.  (a) On the axes below, sketch a velocity-time graph of OJ’s motion.[1]

[1] Sketch, according to the College Board’s course description, means to “draw a graph that illustrates key trends in a particular relationship, such as slope, curvature, intercept(s), or asymptote(s).  Numerical scaling or specific data points are not required in a sketch.”

Most of the class gets this essentially right on the first attempt; the rest get it after a quick conversation with a friend.  The real point of the problem comes next:

2. Describe in words the magnitude[1] and direction of O.J.’s acceleration as O.J. is leaving the tollbooth.
3. Describe in words the magnitude and direction of O.J.’s acceleration as O.J. is traveling along the road unmolested.
4. Describe in words the magnitude and direction of O.J.’s acceleration just after O.J. sees the officer.
5. Describe in words the magnitude and direction of O.J.’s acceleration while the Bronco slams into the wall.

[1] “Magnitude” in this context means, how much acceleration does OJ have?  Answer relative to his acceleration at other parts of his motion.  No numbers are required, though you are welcome to make calculations if you so desire.

I can tell almost immediately upon reading these responses who understands acceleration, and who does not.  The ones who truly don't get it come in for a consultation, where we work on these concepts.  How can I tell?

Well, the response I'm expecting to parts (c) and (d) refers explicitly to the v-t graph and/or to definitions that we've learned:  "(c) When OJ sees the officer, the slope of the v-t graph is positive (a frontslash), so the acceleration is forward.  The slope of the v-t graph is steeper than when OJ calmly sped up from the tollbooth, so the acceleration has a larger magnitude here.  (d) When OJ crashes, the slope of the v-t graph is much steeper than anywhere else; so the magnitude of the acceleration is highest of all parts of the motion.  OJ's acceleration is backwards, because the car is still moving forward, but is slowing down."

The most common mistake is to state the direction of MOTION rather than of acceleration:  "(c) When OJ sees the officer, he speeds up rapidly.  So his acceleration is moving forward.  (d) When OJ crashes, he bounces back off the wall, so his acceleration is moving backward."  Anytime a student says that the acceleration is "moving," I know that he is conflating acceleration and velocity, so the answer is marked wrong -- yes, verbal skills are part of physics.

A less common mistake is to think that the acceleration must change if velocity changes.  "(c) After OJ sees the officer, his acceleration must change rapidly, because the accelerator pedal is on the floor.  (d) When he crashes, OJ's acceleration changes from a high value to zero."  No, constant acceleration means speeding up or slowing down; this student thinks acceleration must change in order for speed to change.

I do get about half the class writing clear, concise, and specific explanations that refer to the v-t graph or to the definition of acceleration.  I will show a fellow student's good explanation to someone who's struggling, to show the difference in the style of prose.  I'm teaching writing as much as I'm teaching physics, sure.  But the time I spend now demanding clear writing pays off tremendously later in the year, when an AP-style free response test requires one-minute justifications.

GCJ

P.S. Only about half of my class had ever heard of OJ Simpson.  That says something about pop culture in the post-internet era.  What it says, I have no idea.