One of my favorite tasks for students to do is to create well-coordinated position, velocity, and acceleration vs time graphs for a bouncy ball, where careful attention is given to the moments of contact with the floor. I got this task from my high school physics teacher, but it’s in Arons’ book as well. The future physics teachers have this as a content standard in my class. There are many predictable obstacles, but the real meat I want them to get to is reasoning about the acceleration. This is so not easy for them-partially because they mostly know acceleration through special cases, and partially because they aren’t strong in thinking about vector kinematics. Mostly students say the acceleration is constant. It’s like the bounce isn’t even on their radar when considering acceleration. Part of what I like about it, is that students know enough to get started and the task itself is clear. My job when they ask to assess, is to keep them talking until they notice some inconsistency, and then to help them orient to that inconsistency and how they knew there was something wrong. Then I send them off.
One of the things I’m doing this year is trying to provide more opportunities for deliberate practice (with feedback), while trying to support students in building and articulating explicit strategies for solving problems (not me just telling them strategies), and selecting class problems to work on that can be solved with current ideas but also put us in contact with puzzles that later ideas will help to resolve or bring insight into.
Here is an example from earlier this week:
This week my students are going to be evaluated (by another instructor) on their understanding of projectile motion and Newton’s laws (including circular motion).* Afterwards, we are going to be diving into energy. So, how do I give them more deliberate practice and feedback with projectile motion and forces while also putting us in contact with energy puzzles? Here is what I tried:
First, I asked students to predict which of two ramps would result in a block of ice having more speed at the bottom. Both ramps were from same height, but one was shallow one is quite steep. No numbers were given. They think and vote peer instruction style. We were pretty much evenly split between all possibilities, so they discuss in small groups and then I collect arguments at the front board. The arguments were basically the following:
- The steeper ramp has a greater acceleration, so it will be faster.
- The shallow ramp will provide more time for the ball to speed up, so it will be faster.
- The opposite effects of acceleration and time will balance out so that they take the same speed.
We’ve gotten pretty good at doing this, so I can mostly stand to the side and just write down arguments and do some re-voicing. After hearing the arguments, I have them revote. There were some shifts, but still not near any consensus.
I now tell them that I want to help settle this by applying some of the skills we’ve learned over the past couple of weeks. I add some information to the scenario. Block of ice has a mass of 25 kg. The ramps are angled 30 degrees and 60 degrees. The height is 5m. I split the class in half, half the groups work the 30 degree problem while the others work the 60 degree problem.
Before sending them off to work the problem, however, I tell them to talk strategy with their group–what will you need to figure out to answer the question, what skills and ideas might be useful, what might you do first, second, etc? They talk for a minute or two, and then we collect strategy ideas at the board. They say most of the things they need to–drawing free body diagrams, using Newton’s laws to find acceleration, finding the length of the ramp using trig, using kinematics ideas / equations to determine the final velocity, etc. Now they are off, and the board is there to help remind them of things they can try if they get stuck.
Doing this together makes me free to monitor for progress rather than helping students get started. I’m checking free-body diagrams for bizarre combinations of Normal and Weight fores, if and how they are finding components, whether they are using a rotated coordinate system and using that consistently with forces and kinematics, etc. I point out things that they are doing which are very “physics-y”, like drawing careful diagrams with labels, starting from big ideas rather than launching into equations, etc. If groups finish early, I ask them to solve for other things that came up in our arguments. For example, I might ask students to solve for the time on the ramp to see if its true that the larger acceleration was paired with less time to accelerate, etc. As multiple groups finish, I have them check with each other on their answer and check with people across the room.
Once we are done, I do a quick summary of what we found, highlighting that its odd that both ramps end up giving the block the same speed. I restate the arguments we heard, and I emphasize that the argument for the right answer made it seem plausible that it could balance out, but why it exactly balances out seems like a puzzle to me. It didn’t just balance somewhat, it balanced out exactly.
I tell them that I want to consider another problem where we compare final speeds, but this time not with ramps. In this problem a baseball player throws a ball with same speed. In one case the ball is angled upward, and in the other case the ball is angled downward. The question is about the speed’s of the two ball’s just before impact, and how will they compare. Students vote. This time there is a split between two answers. Most students vote they will be the same, but don’t have good arguments. They are banking on it being similar to the last problem. Intuitively, it makes sense that the one thrown down will have more speed, and I support this argument a bit. If you are throwing it down, in the same direction of gravity, and its got a real direct path to the ground, isn’t it going to be a lot faster when it hits. There are some other really awesome arguments for why it should be the same, about why it must balance out, including consideration of what the one that goes up is like once its on the way back down. The best argument came from a student who had never spoke up in large discussion, so I spent some time re-voicing that argument and giving it space for consideration.
Once again, I turn the conceptual question into two problems to solve, adding angles and heights and an initial speed. We talk and collect strategy at the board. They solve the problems. I monitor progress, give extension questions, ask them to check with each other. Finally, I summarize and make connections at the end. I still try to keep the puzzle open: Why is it that when the two blocks fell through same distance, and ball’s fell through same distance that their final speeds were the same? Our current skills help us to calculate that this is the result that should happen, but it doesn’t help to explain why.
An interesting outcome of asking students questions to compare, and then asking them to compare pairs of questions, is that they start doing more and more comparing. Several groups started re-thinking the shooter-dropper experiments. Looking for connections across phenomena is something I want to promote and this kind of activity seems to promote more of it.
Anyway. So later this week, we’ll revisit these same two problems from an energy perspective, but I’ll also introduce puzzles for us to resolve that further our understanding of energy and kinematics. Namely, this time we’ll do a problem where two balls rolls down the same ramp, one with an initial speed and one from rest. In this case, they will neither end with same speed nor gain the same amount of speed. Rather they will gain the same amount of kinetic energy…
* This used to bother me, having someone else test my students. But I now love it. My relationship with students is not of evaluator or judge. I am a learning coach. Sure, some of the evaluation is not meaningful. Sure, my students are learning things that aren’t evaluated. But my students do well on the evaluations for the most part, and students are constantly getting feedback from me on a broad range of their learning.
In our physics department, every physics major has to serve as an undergraduate TA. Most of them get assignments in our algebra-based introductory physics course. Because of the manner in which most of these students were taught (i.e., find an equation and substitute numbers), they can easily find themselves feeling a bit lost in my class, especially if they think they are supposed to be an expert of the content.
For example, here’s a question discussed in class. A bowling ball is dropped from a height of 45m, taking 3 seconds to hit the ground. How fast is it moving the very moment before it hits the ground? The problem is intended to draw out the following answers and arguments, which we hash out.
10 m/s, because all objects fall at the same rate
15 m/s because you can calculate the velocity as 45m/3s = 15 m/s
30 m/s because it gained 10 m/s in each of the 3 seconds
Other more idiosyncratic answers come up as well, but not with high frequency.
The first answer points to the ways in which students haven’t yet teased apart clearly the meaning of acceleration and velocity. The second answer points to the ways in which students haven’t yet teased apart clearly the meaning of average and instantaneous velocity. The third answers is consistent with the idea of constant acceleration. We hear arguments, and counter-arguments, and at some point I help clarify the right reasoning, and what’s both so tempting and subtly wrong about the other answers.
So, here is the way the TA solved it, before class began.
xf = (vf + vi)/2 * t + xi
0 = (vf + 0)/ 2 * 3 + 45
0 = 3/2 v + 45
-45 = 3/2 v
v = – 30 m/s
While the TA could solve this problem, they didn’t have a rich set of ideas for thinking about. It didn’t seem obvious that 30 m/s makes sense, because of the idea that its 10 m/s/s, or because final velocity sould be twice the average velocity (since it accelerated from rest). For other questions without numbers that we discussed, the TA seemed just likely as students to give answers inconsistent with the concept of acceleration. I’m perfectly OK with that, but my suspicion is that the TAs aren’t prepared for this. They aren’t prepared to be wrong about so many things or confused about so many things. I wonder how I can better position them as learners in the class–learners who just know somethings that the first-time students don’t, but not everything.
Of other interesting note is this. In my physics content course for future physics teachers, the students that have had me for a semester or two are pretty rock solid on having a repertoire of ways of think about kinematics problems, and also for avoiding common pitfalls. The others are pretty much falling for all the pitfalls. The difference is pretty striking. The thing that I like is that the range of expertise we have allows for peer-coaching, but also some, “Hey, it’s OK. We were making those exact same mistakes 4 months ago,” and, “Yeah, get used to it. Brian isn’t too into solving problems by putting numbers into equations.”
A question I’ve gotten a lot of leverage out the past two semesters is the following one:
You toss your keys straight up to a friend, who is 30m above you leaning out over a balcony. They keys leave your hand with a speed of 25 m/s. Will it get to your friend?
Sure this is a standard boring question. What makes it work is how the show is run. We start off by listing our best guesses about whether it makes it up and the top height they think it gets to: Their answers this semester ranged between 19m and 40m.
In my class, I actually work out this first answer for them (because I’m supposed to model a sample problem), but I ask for their help along the way.
First, I draw a motion map showing how the speed changes at 1s intervals, and we talk about the speed going from 25m/s to 15m/s to 5 m/s, etc, and how the time to the top is when v = 0 m/s. We talk about how much time it takes to get to 0 m/s if you are losing 10 m/s each second: it takes 2.5s to lose 25 m/s. We also talk about the average speed during the trip (12.5 m/s, half way in between 0 m/s and 25 m/s). This, of course, all builds on ideas we built up last week when talking about 1D acceleration problems.
The answer is immediately given as 12.5 m/s * 2.5s = 31.25m
The best guess this time was 32m, and kudos were given to that group.
Lot’s of students then want to talk about why it’s not 40 m (25m + 15m + 10m), and we get to talk about what constantly changing velocity means.
Because of class constraints, I typically re-derive the 31.25m in a way that is more typical of how they are expected to do it: Write down your knowns and unknowns and pick an equation or two to plug away with.
I then send them off to work on the next question. How fast are the keys moving by the time they reach your friend’s hand? Our guesses range between 1.25 m/s and 2.5 m/s.
The right answer is 5 m/s. And students are pretty surprised to find out that we all underestimated the speed. Every group got the right answer. Most students solved the problem by plugging away into equations. One group did so, but didn’t believe that 5 m/s was right, and so they took another approach, using two equations instead of one.
One group took this approach:
In the first second, the ball slows from 25 to 15, with an average velocity of 20 m/s. Thus in the first second, the ball covers 20 m. In the second second, the balls slows from 15 to 5, with an average velocity of 10 m/s. Thus in the second second, the ball covers 10m. That’s 30m covered, with a final speed of 5 m/s. That same group realized that for the first 2 seconds, the average speed was 15 m/s for 2 seconds, also giving 30m of travel.
Last semester, I had a group solve the problem by finding the speed of a ball dropped 1.25 m/s, arguing on the ground of symmetry that it had to be the same.
We ended the problem this semester by talking about the last 1/2 second, where the ball has an average speed of 2.5 m/s for 1/2 second, thus covering the final 1.25m, and why our guesses for the speed were so off.
Simple problem, but lots of places for intuition, lots of places for multiple approaches, and lots of opportunities to talk about velocity, distance, average velocity, and acceleration.
This week’s online pre-class question:
An object is dropped from a height of 45m and takes 3 seconds to hit the ground. Explain why someone might think the object’s speed just before hitting the ground is 15 m./s. Then explain why that can’t be correct.
Three responses representing very different places students can be:
“First of all, wow! That’s the exact answer I had in mind and that is because if it’s dropped from a height of 45 meters and it takes 3 seconds to hit the ground you would want to divide the 45 meters by the 3 s to speed per second (15 m/s), but that is if it was going at a constant speed. So you also have to keep in mind that it was dropped at rest/zero so the speed will increase slowly not constant. I’m still confused.”
“Someone might think it is that because they would divide distance (45m) by time (3 s) which would come out to be 15 m/s. But that would be the average speed. To find the final speed you would take the initial speed (0 m/s) and add it to the acceleration (9.8 m/s^2) multiplied by the time (3 seconds). The final speed of the ball before it hits the ground would be 29.4 m/s.”
“Because most people would think just divided 45 into 3 to get 15m/s but we haven’t put in our minds about the acceleration of gravity, which is 9.8m/s that can round up to 10m/s then if you was to times 10m/s by 3s you would get 30m not 45m.”