Since being told about the Hudl Technique slow motion video app (free in Android/iOS), an entire world of new labs opened up for my students. Hand-controlled timers are just too inaccurate for most experiments, but a slow motion video that also shows real time to the hundredth of a second is more than adaquate for almost any experiment in an introductory physics class. Shown below is one group setting up their experiment to be recorded by the app.
— Trevor Register (@TRegPhysics) September 25, 2017
My college-prep physics class (non-AP, trig-based, but more conceptual than not) began this lab today. Since the goal of this lab is to have them model the relationship between velocity and time, the follow up to generating the position vs. time graph for the tennis ball is then calculating the velocity of that ball at different times.
I’ve tried having them do this in several different ways over the years, all of which that have been some shade of failure. Most of my attempts have revolved around having them draw tangent lines at different points of the curve (a curve which they hand-drew), calculate the slopes of those lines, and then using that to generate the velocity data. The problem there is that having them hand-draw a curve then hand-draw tangent lines makes the uncertainty in the velocity data skyrocket thus destroying any real linearity with the data.
Additionally, there was just too much for my students to grasp all at once. Even the students that got “good” data still didn’t really understand what they were doing.
I’ve also tried having them use a motion detector to generate velocity data, which they then pick out and graph. But I didn’t like that either as it felt too much like magic hand-waving.
So with this lab approaching, I was dreading how all of it would go again. A lot of confusion, poor data, and blank stares.
One of the primary issues I’ve discovered with the tangent line method (accuracy issues aside) is that it skips a step. A tangent line is the point at which two points on a curve converge (i.e., calculus), and understanding that the definition for instantaneous velocity comes from squeezing two points infinitely close is essential for understanding how the operational definition of velocity (slope of a position vs. time graph) is generalized to non-linear position functions.
Once I realized this, my goal became to figure out a way to take the above into account without a mountain of cognitive overload. Usually this means turning to Desmos and, as expected, Desmos continues to be one of my favorite tools for the physics classroom. Here’s the tool I made.
Students first enter their position vs. time data in the table to the left…
…which then desmos uses to generate the function that best fits the data (all of which is in the WIZARDRY folder):
The back-end of this tool generates a fit for any function .
Understanding Tangent Lines
Once students have entered their data, I then have them talk in their groups about how they can calculate the velocity of the ball. When the topic of “the” slope comes up, I then point out that there is no “the” slope because the line gets steeper and steeper… which makes sense because the ball’s velocity gets greater and greater.
But then how do we calculate the velocity of the ball at any particular point? What points do we choose?
After a few more minutes of them discussing this amongst themselves, I allow groups to throw out suggestions to the class. Often times they won’t come to the “choose two points very close together” conclusion on their own, but one group will say “choose a point before” and “choose a point after.” From there, I lead that into “but how close? should the points be?” Or, an even better question: “But why do they need to be close?”
At this point I put Desmos up on the projector and show them two things:
Note: the tangent line is turned off by default as I think it can confuse things when they’re making their calculations later on. To turn it on, go into the Wizardry folder and click the empty circle next to the very last equation
I point out that the orange and black lines deviate quite a bit for almost all of the graph. However, if you zoom in…
…they overlap perfectly. This is the justifcation for “pick two points really close together.” To wrap up, I ask them to write a few sentences that explain and justify the process of calculating the velocity at a particular point in time.
This also just-so-miraculously-happens (read: completely and totally on purpose) sets them up for calculus later on should they take it.
Generating Velocity Data
At this point, students are ready to start generating their velocity data. I tell students to drag the blue dot (the red follows along) such that they center each of their data points. See the two screenshots above for what that looks like for data points not on/close to the curve. The slope of the line between the red and blue points is the velocity at he x-coordinate of that data point (1 s for the point above). Students generate around 10 velocity-time data points, and then move to graphing their new data.
Because the velocity data is generated from a perfect curve, the velocity data is almost perfectly linear, thus making the further analysis much easier.
If you have any questions about how all of this works or suggestions on how to improve, please let me know!