Mail time: a daily quiz question about average speed

What is the purpose of this question from one of the daily quizzes in your AP class? My guess is that you're highlighting a misconception.

True or false: One method of finding an object’s average speed is to find its speed during one part of its motion, add to the speed during the rest of its motion, and divide by 2.

Yup, you got it!  I'm highlighting this amazingly common misconception.  Usually the misconception appears in response to questions like "a car travels the first 50 km of its trip at a constant speed of 20 km/hr, and the second 50 km of its trip at a constant speed of 40 km/hr.  What is the average speed for the whole trip?"  The answer is NOT 30 km/hr, because more time is spent in the first 50 km than the second 50 km.  But no matter how we try, still a significant portion of the class will say "average speed?  Oh, I know how to take an average!  (20+40)/2, of course!  :-)

When I *do* pose this kind of problem to my (AP) classes, I ask it more conceptually: "a car travels the first 50 km of its trip at a constant speed of 20 km/hr, and the second 50 km of its trip at a constant speed of 40 km/hr.  Is the average speed greater than, less than, or equal to 30 km/hr?" I mean, they could do the full-on calculation of the time spent in each half of the trip, then use the definition of average speed = total distance / total time to get 27 km/hr.  But the simple conceptual approach is to notice more time was spent at 20 km/hr than 40 km/hr, so the average speed will be closer to 20 km/hr than 40 km/hr.  

(In my lowest-level conceptual physics course, the whole concept of "average speed" is out of bounds.  Let's get students understanding basic one-dimensional motion with constant speed, or with speeding up from rest / slowing down to rest.  The whole idea of "average speed" is extremely confusing, even to strong second- or third-year physics students!  So save the complexities for advanced courses.)

Setting up AP questions in the lab: 2016 P1 #2, the bouncy ball experiment

A unique feature of my in-person summer institutes is the final morning of the week, when participants are asked to choose a released AP physics question to set up in the laboratory.  I think of it as "studio time", which functions similarly to my AP class's open-ended lab assignments throughout the month of April.  In studio time, we not only create these setups, we share our ideas and creations.

This is my seventh post based on experimental setups of AP free response questions!  The others:

* Block and cylinder on incline (Milo Jacobs) (2021)

* Student on a raft (Stephen McAliley) (2014 practice exam multiple choice)

* Which battery drains first? (Frank Prost, Tom Mellin) (2017)

* Energy vs. time for a cart on rough surface (Nadia Lara, Joey Konieczny) (2015)

* Bumpy Track (Zach Widbin) (2016)

* Waves on a vertical string (Walter Keeley) (2016)

Today's setup and data comes from Matthew Bourbeau, a participant in my July 2022 Palo Alto High School workshop.  He set up the problem referred to by AP readers as the Black Hole: 2016 P1 #2, the bouncy ball problem.  

I loved this problem when it came out* because it provides no alternative to authentic, creative approaches to lab.  It rewarded students who could look at a situation they'd never seen before, and come up with a simple experimental procedure from scratch.  It emphatically did NOT reward students whose entire laboratory experience, if any, had been following directions on a lab sheet with pre-set equipment.  (The AP laboratory problems since 2016 have been improved in terms of the simplicity of wording, and I appreciate that.)

*(I later grew not to love this problem as much when it came to grading thousands of essay-length responses with 104 of my new best friends.)

The original problem poses a hypothesis about a bouncy ball: that its low-speed collisions are elastic, but its high-speed collisions are NOT elastic.  Students are to design an experiment that could test this hypothesis.

The elegance of this problem is that neither I, nor any test-taker, knows nor ought to know whether the hypothesis is valid or invalid!  This is true science.  Not "is this true," but "how would we know whether this is true?"

Well, it's been six years... but I've finally seen data from an experiment that tests the hypothesis.  

Matthew used the track shown above, inclined such that this lacrosse ball would roll down and hit the wall.  The motion detector at the right read the ball's speed immediately before and after the collision.  Matthew collected data for collision speeds between about 15 cm/s and 90 cm/s.

On the 2016 problem, students who knew what they were doing opted to analyze several different useful graphs.  One common approach was to graph the speed right after collision on the vertical axis, and speed right before collision on the horizontal axis.  The hypothesis predicts a graph that starts along a 45 degree slope, but then drops below 45 degrees for larger initial speeds.  

Matthew chose a less common approach (though one that I saw many, many times): use the speeds to calculate kinetic energies right before and after the collision.  Divide the KE after the collision by the KE before the collision to get an "elasticity percentage".  The hypothesis predicts this value to begin at 100% for low initial speeds, then to drop below 100% as the initial speeds increase.

Matthew's results - shown below - emphatically reject the hypothesis.

The "elasticity percentage" never is close to 100%.  In fact, the elasticity percentage for Matthew's lacrosse ball remains about 50-60% for all impact speeds he measured.  

Now, to be fair to the hypothetical student who posed the hypothetical hypothesis, the exam does state that a "new kind of toy ball" is supposed to behave this way.  A lacrosse ball is not exactly new (though I had never personally heard of the existence of lacrosse until I arrived at college in Philadelphia in 1991).  Yet, Matthew did the experiment with the materials at hand, and got clear results.  Soundly rejecting a hypothesis with experimental data is, in fact, good science.  Thanks, Matthew!