Kinematics Graph Checker

Here's an applet for kinematics (mostly CAPM) practice that I put together this fall. Students have the option to choose what information they are given, from these choices:

• Initial x, v, and a values
• Position graph, initial x values
• Velocity graph, initial x and v values
• Acceleration graph, initial, x, v, and a values

The length of the time interval considered can also be varied. After getting the given information, students can draw their predictions (either sketching the shapes or drawing quantitative graphs), and then press the "Show Solutions" button to reveal the hidden two or three graphs. I finished this one after students were through 1D kinematics this year, so I don't have any info on how effective students find these. Let me know if you like (or don't like) them for your classroom purposes!

Counter-factual Animations and Energy

Soon after my students began energy, I presented them with a set of five YouTube videos that I made with VPython of a race between two identical balls, launched by identical springs that had been compressed identical amounts.

The five simulations present five different ways that the race could play out; one is physically accurate and, while the other four have some sort of logic, their results are not physically correct. The students have some time in groups to determine which they think is correct and, more importantly, what specific issues they have with the others. I'm challenging them to figure out the laws of physics in these four alternate universes, in essence.

After they've worked for a while, they vote, and then we go through the unpopular ones first, with students giving their reasons against them, debating as disagreement crops up.

It usually boils down to two or three, and they hone in on the correct answer pretty reliably in a peer-instruction-esque way. The discussion has been lively and productive, and I like it as a way to focus their attention on the kinds of things that energy conservation does and does not allow in the world in general. It also is a good review of some kinematics concepts, including average velocity and the velocity/displacement relationship.

The YouTube format isn't the best - the size is a bit too small, unless you want to switch back and forth between fullscreen and smaller. Additionally, the suggested videos pop-up at the end is distracting and annoying.

To that end, I coded the simulations into GlowScript instead; students can now deal with them in-browser, repeating or switching at will. It would be neat to have two windows to select different simulations to run against each other simultaneously, and I may add that feature in the future. In the meantime, this will be a big improvement for students over the previous incarnation!

Circular Motion Simulation Follow-up

I last posted about a new circular motion applet that I was planning on using with my classes as the quantitative part of their UCM paradigm lab. Some reflections:

• When students came up with a list of variables that might affect the size of the centripetal acceleration, the list was: speed, mass, radius (always in that order). The visual accelerometer on a rotating table showed the qualitative effect of speed nicely, and the thought experiment about driving a car around a corner (tight or wide) addressed radius. We couldn't do mass with the given stuff, so I told them to check that out in the applet. A few seconds' work with the slider showed that it's irrelevant.
• The applet is framed in terms of string length (radius) and rotational frequency - instead of speed. This means that students had to confront (and figure out) the relationship between rotational frequency (or period) and speed just to get their data for the acceleration's dependence on speed. I like that.
• The other way that they have to confront it is to control speed while investigating acceleration's dependence on radius - changing the radius but not the frequency would change the speed. The students have the figure out the proper frequency for each new value of the radius in order to keep the speed constant during the second experiment. I like this a lot as well.
• Students still have trouble reconciling their two models ( and ) to determine the complete function of v and r. Even when they have figured out the units of the two constants, the connection is hard for them to make. I'm very open to suggestions of ways to make this go more easily - I don't have a great handle on what the conceptual difficulty is for them here. In the second section, I framed those two models as "OK, so a is proportional to v-squared, and a is proportional to 1/r," and that may have helped.
• Overall, the quantitative modeling went much more quickly, had some good conceptual things to think about, and was good practice with function modeling, so I'm pretty happy about it, at this point. We'll see how things go over the next couple of weeks; did this begin to build lasting understanding?

Circular Motion Simulation

I've been through several variations of circular motion paradigm labs over the years. Lots of approaches, lots of pros and cons.

Here's where I'm landing this year:

• Preliminary investigation: a qualitative exploration, using basketballs and "science hammers" (lab rods with clamps on the ends) - differentiating between the effects of forces parallel to and perpendicular to the velocity. This establishes the conceptual foundation of uniform circular motion, and comes back later in the year during the energy transfer model (work)
• Quick conceptual investigation: using either a visual accelerometer or a wireless dynamics probe/LabQuest, determine the qualitative effects of various variables on the acceleration of an object in UCM. Narrow it down to radius and (linear) speed. Angular speed can be a more natural variable for this (and easier to design an experiment to control for), but I've found that students have a lot of difficulty differentiating between angular and linear speeds at this point, and that they later confuse an angle in a banked turn with the angle "around the circle." 
• While designing a real-world experiment with constant angular velocity is easier, using an applet can make experimental design with linear velocity as a variable just as easy. It's also quicker and more reliable (whirligig experiments can be a little dicey with data quality), putting the emphasis on the data analysis. (Save the whirligig for a practicum later!)

The various circular motion applets that I've used before are pretty much inaccessible now, because of Java's waning usability. So, I wrote one using Glowscript. 

Here's a screenshot - click through to use the applet.

VPython, Energy, and Stability

The content in AP Physics C about stability and its relationship to energy is a pretty thin introduction to a fairly deep idea. Classifying equilibria by looking at the derivative of potential energy can be a quick add-on, or we can make it a little deeper with the help of VPython.

When it first comes up, we go through the stable/unstable/neutral equilibrium discussion and the corresponding shapes of potential energy graphs, but I really wanted a way for students to apply that. I've thought of a few, only three of which I have tried out so far:

• Have VPython plot a potential energy vs. tip angle curve for a box that's standing on its end - compare for different heights/widths of box
• Have VPython graph total potential energy vs. stretch amount for a mass on a vertical spring
• Have VPython graph potential energy for a mass of a spring that doesn't necessarily stay vertical (on y vs. x axes, with U value signified by color)
• Investigate the Lenard-Jones (interatomic) potential, both for meaning and location of equilibrium points and to fit an approximate quadratic to it near the equilibrium point, justifying the use of the ball-and-spring model of matter.
• Later in the year, during rotation, have them create a physical pendulum program with a box on a pivot not through its center - modify this to show the graph of potential energy as a function of angle

For the first and fifth, I have some results using VPython to show. (The fourth is great, too, especially if you're using Matter and Interactions!) For the box tipping, the graph is neat, not least of which because of the discontinuity in the derivative at 0. Calculus wouldn't find this minimum on without intervention from the student, and it's easy to forget to check those endpoints! This gives a nice graphical reminder. My one concern is that the geometry is fairly harrowing - determining the height of the CM as a function of angle isn't super-easy, and requires a high-quality diagram. 

Here's a screen shot:

When you vary the parameters, that central well can become deeper (more stable) or shallower (less stable), and vanishes as the box gets narrower (and becomes a pencil).

For the final one, I think that the program, being a modification of a previous one, isn't too difficult to do, though it does bring in the complication of having a "meta loop," where the whole simulation runs several times, with different placements of the axis. 

Here's a screen shot that links to a video of the program running:

The graph here is incredibly rich. Not only do we see two different kinds of equilibrium (three, if you let the pivot be at the middle for the first run through), but we see how the system is forced away from or towards those equilibrium more or less violently as the pivot moves. This helps to bring together their common-sense understandings of stability and the physical principle that we're trying to investigate.

Drag Graph Checking

I previously posted about a class exercise where my AP C students pair up, pick two random objects, and try to draw qualitatively correct position, velocity, and acceleration graphs for them falling through the air. The idea is to get a qualitative feel for drag graphs and to check qualitative results for terminal velocities.

I had written a VPython script to do this, but it required my intervention to change the values each time, and everyone had to watch all of the graphs. I don't have enough students for that to be super-terrible, but I wanted a way for them to do it themselves.

Enter Glowscript; I've ported the VPython script to there (and made some improvements), and students can now check these graphs for themselves!

There's a screenshot below which links to the simulation - feel free to use it and drop me a line if you do! Definitely let me know if you find any bugs!

Standing Waves/Resonance Applet

The second applet that I wanted to rewrite to save it from Java purgatory was a great transverse standing wave applet by C.K. Ng. I used this principally for a data source for students to explore resonance - it's a lot easier to get reliable data with sufficient amplitude variation using an applet for this than a real experiment. In addition to the standing wave amplitude never being overwhelmingly large with a string vibrator, there are hysteresis effects that will drive the kids crazy. I have them collect this data at home, BTW, so the Java issues have meant that, for the last two years, only a handful of kids have successfully been able to use the applet at home and, without anyone for troubleshooting, they quickly give up.

I'll also say that the approach of summing over the normal modes to find the solution for a given f, L, etc. gives a much neater animation than using a finite element/balls and springs model of the string and waiting for the old waves to damp out. It's idealized, but we're really just looking for the steady-state here anyway - this just gets us there faster. It will make the computer work, though!

The most significant difference here is that I haven't created the draggable ruler, opting instead for more prominent gridlines. I always wanted to measure the amplitude anyway, so the horizontal ruler in the applet didn't help much, but using the grid and some arbitrary 'block' unit should be able to serve both purposes.

Let me know if that is an important feature for you, or if there's anything else that you can think of to add or modify to increase the usefulness of this applet! Click through the photo for the applet itself.

Longitudinal Wave Simulation

In the wake of the big Java security crisis, Java applets have become increasingly inaccessible and/or onerous to use, due to security settings. Add to this issues like Java 7 needing a 64 bit browser in OSX and Linux, and it's rather difficult to get a classroom set of computers, much less a BYOD environment, to effectively run Java applets in class.

I've found that I can just forget asking students to use them at home, given their computer setups and ability to navigate these complications. Because some of my favorite applets seem to be going extinct, I'm going to try to duplicate as many as I can in Glowscript, which is a kind of mashup between JavaScript and VPython. It has much of the readability and ease of VPython, and can run in WebGL-enabled broswers, which covers most situations (maybe just 'many' in the mobile world, at this point). More importantly, that coverage is on the way up, while Java is on the way down.

My first applet is an attempt to replace Walter Fendt's longitudinal standing waves animation. I love this for my students - it's difficult for them to picture particle motion in longitudinal pulses, but nearly impossible for them to visualize what the particles are doing in a longitudinal standing wave. This depiction is obviously idealized, but it can help them get over that hump.

The second thing that I like about this setup is that it shows SW diagrams/graphs of not only particle displacement, but also the change in pressure. At this point in class, we've been merrily drawing standing wave diagrams for waves in tube as if they were waves on strings (or some kind of string that can have an unconstrained end or two). What have we actually been drawing? This helps to clarify that we had been illustrating the change in position of the particles and shows that we can also describe the change in pressure that they undergo. Looking back up at the animation gives students a sense of why the two trends are related the way that they are.

I've decided to leave out (at least for now) the numerical data on the side, as I hadn't generally found much use for it. Perhaps I'll add it - let me know if you see a good reason for including that.

Click through the screenshot for the applet itself - enjoy!