We just finished kinematics, concluding as we usually do with free-fall, which I treat as a very specific case of constant acceleration. This year, I flipped my classroom, and had the students watch a video about free-fall and take notes. The students reported liking this practice. They can stop the video, and start it again when they have finished taking notes on a particular problem. Students who write slowly have time to listen, understand and then write down what they need to remember. Students who grasp everything the first time you say it can watch the video once and be done.
The individual who created this particular free-fall video solved several sample problems, including one dropped object and one object thrown vertically upward. In one case, the positive direction was chosen as downward, since the ball was dropped, and all motion was downward. In the other example, the ball was thrown upward, so up was chosen as positive, and down was thus negative. I really liked the way the presenter emphasized that if down is chosen as the positive direction, then a = g = 9.8 m/s2; if up is chosen as positive, then the acceleration a = -g = -9.8 m/s2. I liked it because when I use the letter “g”, I intend it to mean the gravitational field strength or 9.8 m/s2 or 9.8 N/kg. Always positive.
But no matter how slowly the students play the video, they always seem to misunderstand. “But Ms. Lietz! The guy on the video said that when the ball is moving down, it’s positive and when it’s moving up it’s negative.” [Their tendency to overuse the word “it” could fill an entire blog entry.] No, sorry. That’s not what he said. But I understand how you came to your interpretation.
I want my students to be flexible, I want them to be able to choose either direction as positive and be able to solve a free-fall problem. Some of them can. Most of them can’t initially. So we had to have a long conversation about positive and negative and what they mean. I know we will have many more to come.
One of my favorite demonstrations to address this particular challenge involves a Vernier motion sensor, Logger Pro and an oversized tennis ball I found at Wal-Mart (roughly the size of a child’s basketball). I hang the motion detector from the ceiling and drop the ball below it, letting it bounce a few times. All their other experiments have used “away from the motion detector” as positive, and toward as negative. I explain to them that I have set the program to “reverse direction” for the motion detector. That means that away from the detector, which is downward, is now negative. Then we “zero” the motion detector when it is pointed at the floor. This has the effect of making the graph look like it would if the motion detector were on the floor, without having to bounce the ball on the motion detector. Up is positive and down is negative. Now it’s time for them to predict what the motion graphs will look like for this motion
In my first class, I asked them to predict all three motion graphs at the same time, before dropping the ball. This led to lots of confusion and all kinds of crazy graphs. So, having the luxury of three more tries, in my next classes I had them predict the position graph first. They had success with this relatively easy task before we moved on to the velocity and acceleration predictions. The velocity predictions were much more accurate once they saw the accurate position graph. And the acceleration graphs are much more accurate as well. I use these graphs to reinforce many concepts. First, the velocity at the top is zero (we are still talking about one-dimensional motion). We look at the graph of velocity vs. time and see that on the way up, the velocity is positive and decreasing, while on the way down, the velocity of the ball is negative and increasing. I point out that in both of those cases, the acceleration, as evidenced by the slope of the velocity graph, is negative, and thus downward, for the entire trip. As the ball moves upward, the acceleration is negative, and as it moves downward, the acceleration is negative. At this point we broke into groups and started working on solving free-fall word problems using the equations of constant acceleration.
I post annotated images of these graphs on my website for them to refer to later. Each time a student needs help solving a free-fall problem, we can look at these graphs and remember that the acceleration is down, for the whole trip. If up is positive, then down is negative.
This strategy requires that the students are first comfortable with the relationships between the position, velocity and acceleration graphs. When this is true, that understanding can be used to make sense of new concepts. Nevertheless, the direction of acceleration remains a challenging concept for students, and I am always trying new approaches, new demonstrations. The ball drop demo, however, is a must. Many students are helped simply by restricting their coordinate system to “up is positive” and always using that rule. The weaker students initially need rules to cling to and things that are “always true.” Others can be flexible. Even in a tracked curriculum, at a school with three levels of first-year physics, the spectrum of abilities is broad. So my repertoire of demonstrations and other pedagogy must be extensive, and responsive to their needs. Of course, my patience needs to be great as well, because as soon as I finished my fourth time through this demonstration in one day, some student got a practice problem wrong, and exclaimed, “But I thought the acceleration was positive on the way down! That’s what the guy said in the video.”