Note: This post details my version of the paradigm activity for what is Unit 6 (2-D Particle Models) of the Modeling Instruction Mechanics curriculum. My introduction to this unit varies from the official materials, and my classes build this as their sixth model (after building models for motion, forces, and momentum transfer).
Due to differences in math comfort and overall motivation, I take rather different approaches to starting this unit in my two classes. In Honors Physics, we typically spend 4 to 6 class periods worth of work on the entire unit. In the regular physics classes, we spend at least a couple of weeks.
Honors Physics Tackles Projectile Motion
These classes are about a 70/30 split of sophomores and juniors with most students in Algebra 2 Honors (and several beyond that).
Since there’s nothing really new in this unit (it’s a mash-up of [CVPM+BFPM] and [CAPM+UBFPM]), we make short work of it. With very little, if any, guidance from me, they start out with a sequence of goal-less problems that I got via Matt Greenwolfe. They are a series of variations on this problem:
- A yellow ceramic ball is dropped (from rest) from a height of 1.5 meters and hits a landing pad.
Since we do this unit right after our Momentum Transfer Model, they tend to spend a while trying to analyze the collision with the landing pad. They soon realize that there isn’t much they can do there with the information that they have, so they settle for qualitative descriptions (IF Charts, velocity-vs-time sketches) and do more analysis of the falling motion.
They grumble a bit about having to use m for the mass of the ball (keeping an unknown mass as a symbol is something they’ve grudgingly done before, but detested), but quickly see the implications of their analysis to that first problem. They ask whether they really need to do the next two problems (An aluminum ball is dropped (from rest) from a height of 1.5 meters and hits a landing pad; A ping-pong ball is dropped (from rest) from a height of 1.5 meters and hits a landing pad.) since everything will be exactly the same, and they quickly turn the page to start attacking the 2-D problems (A yellow ceramic ball moving at 3 m/s rolls off of a horizontal table, falls 1.5 meters, and hits a landing pad; A yellow ceramic ball is launched from the edge of a 1.5 meter table. The initial velocity of the ball is 3 m/s at 20º.)
By the end of that worksheet, they’ve pretty well figured out the basic ideas of the model. We do a ranking task, watch some Cinema Classics video clips (usually A 47-ball falling from moving cart, B 11-vertical and horizontal motion, B 14-ping pong ball model, and B 12-horizontal ballistics) and the hammer/feather drop NASA video.
After the video clips (which we watch with a lot of pausing, talking, arguing, additional question-posing, re-watching, etc), we do some deployment practice with some challenging PMPM problems and are soon ready to move on to building the Energy Transfer Model.
Physics! Tackles Projectile Motion
These classes are almost entirely juniors who are in regular precalculus (how is that a topic?? But anyway…) math classes.
We take a longer, more careful route to building the model in the regular physics classes. Even though there is still nothing new in this unit, it feels new to them anyway, and when I’ve tried to jump to the goal-less problems with them, I’ve lots many students. Enter Dan Meyer’s basketball blog post last (school) year. Ignore the fact that it is supposed to be about drawing parabolas. See the lovely goal-less problem with an object in free fall.
Dan shoots a basketball
So I start out by showing the half-video from one of the takes (not this one, actually, but it was the one I could find on vimeo).
Wait (for them to think, be curious, etc). Point out that there is a still of the shot on the first inside page of the packet (note: the video that I embedded here and the screenshot from my packet are actually from two different shots because I use a different take during class. In class, the video and the image in their packet ar matching—no worries!). Wait.
Well, it’s basically a motion map, right?
Usually, our motion maps have been in a straight line. So let’s break this one down into parts. We can draw in horizontal and vertical lines so that we can see the x– and y–motion of the ball separately (usually I just get them drawing the lines without going into it much because they are interested in drawing lines with a ruler and will see the patterns as they emerge anyway).
So get a ruler. Draw in horizontal lines through each shot of the basketball. Draw vertical lines through each shot of the basketball. You can hold the edge of the ruler against the edge of the paper to keep yourself drawing horizontal and vertical lines.
“The spacing is even!” “No, it’s not!” “You probably missed one. There’s a basketball in the trees.” “Oh, shoot. Now I see it.” ”Part of it has even spacing, but part of it doesn’t.”
Draw all the graphs!
That’s really neat. It looks like a pattern, huh? Or a couple of patterns. Let’s analyze the situation some more and draw some diagrams. There’s plenty of space below the photo for you to make some sketches. What does the free body diagram look like? Let’s just analyze the motion when Dan’s no longer touching the ball.
That’s it? Really? So then some discussion and some arguments—there’s still the remnants of Dan’s force on the ball—and the rebuttal from another student that he’s not crunching the atoms anymore because he’s not touching the ball, so there’s no normal force on the ball. And finally, a discussion-settling system schema.
How about a velocity-vs-time graph? What is that going to look like? Then we get some confusion and discussion about whether it is constant velocity or not based on the lines we drew on the photo. I suggest drawing two velocity-vs-time graphs: one for the horizontal aspect of the motion and one for the vertical aspect of the motion. They usually find that to be a good compromise.
The horizontal velocity graph is easy enough to draw. We decide pretty quickly that the forces are balanced horizontally (since there aren’t any forces at all horizontally) and that the velocity is positive and constant.
The vertical velocity graph is trickier. There isn’t always a quick consensus here. I let them sketch it on their own papers first, then try to get me to draw the graph they’ve drawn on the board. One common path is to progress from an all positive velocity graph to a constant negative slope velocity graph. (Note: they almost always continue the motion on from what is shown in the clip/photo to include the ball coming back down).
A lot of other suggestions might arise about the shape of the y-velocity graph. I draw whatever they tell me to draw, and I let them argue their way through to the correct graph (which always happens eventually).
Once they’ve settled on that correct graph, a nice thing to do is to help them make visual connections between the FBD and the two parts of the velocity graph (the forces are unbalanced in the negative direction the whole time, so the slope on the velocity graph should be negative the whole time). Actually, at this point, they usually think of it as three parts: the way up, when it is “at rest at the top”, and the way down. Connecting the graph parts to the FBD usually helps with the “at rest at the top” business, too.
Very quickly, it becomes clear that the “at rest for a while” bit would mean balanced forces for a while. And it is also clear that there is no way there could be balanced forces for a while (even a very short while) at the top of the arc. So it must look like the first option and only have a velocity of 0 m/s for an instant (while it is turning around) and not for any length of time. It’s never moving at a constant velocity in the y-direction.
One of my classes this year was bitten by the momentum bug and wanted to draw IF charts for the ball (which we did) and even got to drawing a momentum-vs-time graph (which we had never done before) to try and figure out the shape of the y-velocity graph. In both classes this year, having done momentum before projectile motion led to them often including graphs and even doing calculations using momentum principles rather than just using kinematics to solve problems. So cool!
Time to start learning
At this point in the class period, I’m getting anxious to move away from the front of the room and stop talking so that the kids can actually start learning something (remembering that they only learn when they’re doing the thinking and talking, never when I am). At some point along the way, I’ve mentioned that having an FBD with the only force acting on the object being gravity is called “free fall” and noted that they see or experience that case (or approximately that case) quite often. I have not talked about the value of the acceleration or made any motions in any kind of quantitative direction. I want them to figure out all of the interesting details about the acceleration (etc) from the goal-less problems (the same sequence that I described in the Honors Physics section). They need to create and own that idea.
So I basically try to get them to move on and start in on the first worksheet. Usually at some point, they object because they want to know if the shot makes it (though they aren’t always very bothered by it—they sometimes seem to be more interested in delaying the real work by bringing in this “distraction” (a distraction that I’m interested in them pursuing, though they don’t always realize that)). Regardless, the general method is usually just to continue the motion map and figure out whether the ball gets close to the hoop. Or to refuse to do that on the principle of, ”Why would he upload a video of him missing a shot?” (I then show a take where he misses, too, to prove that he didn’t only upload the successes.) When they’ve pretty well decided whether the ball gets close to making it or not, we watch the full take video. In all, this intro can be about one class period.
Finally, they’re ready to start building (or knocking down and rebuilding) their gut feel for free fall. After this first day of the unit, they follow a path similar to the Honors path above (with more time for arguing, a couple of extra video clips, and significantly simpler problems in the deployment worksheet). Another big difference is that they are not ready to use a variable for the mass and come up with their own values (leading most of them to actually do all three of the first problems and freaking out about how they keep getting the same results for every problem until they start to realize that they are multiplying by and then dividing by the same number every time as they calculate the acceleration). In the first couple of days, they usually claim to have entirely forgotten how to solve kinematics problems (well, not entirely, but they’ve forgotten what to do if they can’t solve it in one step). After a little prodding, they remember that they can come up with a system of equations using the slopes and the areas, and then they’re zooming off through the rest of the packet in no time.