# Category Archives: motion maps

## Unit 2: Constant Acceleration Particle Model

This second unit on the CAPM will proceed along the traditional modeling framework. I expect it to take approximately two and a half weeks. It will begin with ball rolling (or cart sliding) down and incline plane and whiteboard meetings to analyze the results.

From there they will proceed through the modeling materials for the Constant Velocity Particle Model as provided by the Modeling Materials. The first worksheet allows them to analyze additional data sets similar to what the saw in the lab. The second worksheet has the student create motion maps, position-time, velocity-time, and acceleration-time graphs for more complicated ramp systems. Worksheet 3 focuses on analyzing position-time and velocity-time graphs. Worksheet 4 has the student solve quantitative problems. We end with additional problems for review.

The students goals for this unit are:

SWBAT

1. create and interpret graphical and mathematical representations of objects moving with constant acceleration.
2. can correctly differentiate between acceleration and velocity.
3. correctly interpret the meaning of the sign of acceleration.
4. solve kinematic problems involving constant acceleration.

## Modeling Unit 3

Unit III: Uniform Acceleration Particle Model

We began the “Cart on an Incline Plane Lab.”  After talking to Jon and Chris about Brian Frank’s Blog during our lunch break, Jon and Chris altered the first question to begin the cycle by asking “What do you see/notice?”  Again, we moved through “What can you measure?” and “What can you manipulate?” before arriving at the purpose: “to determine the mathematical and graphical relationships between position and time for a cart on an incline plane at a fixed angle.  As a group, we seemed to struggle with this process this time.  I’m not sure if we are taking our role in “student mode” too seriously and confusing ourselves in the process.  We didn’t seem satisfied to look at this relationship and many wanted to add more quantities such as mass and angle into the procedure.  One good suggestion was to focus the students back on the first question, “What do you see?”  By using what is already on the board, we can steer the students to the necessary target of this unit.  I might add that we should include some help to our students to focus on the actual demonstration we are doing (cart rolling down a fixed ramp) and try to dissuade them from altering the set up.  Possibly reminding them that our job is to create the experience that will teach them the necessary component of physics, and we chose this exact one for a reason.

After we were settle on the debate, Jon told us to use the motion detector to acquire “clean data” for the cart.  When asked what he meant, he said, “You’ll see what I mean.”  He put on the board for us to then manually calculate 10 values of “average speed” over the range of our data (he originally put instantaneous, but later corrected it).  He also remembered later that he usually has the file “01a Graph matching.cmbl” from the physics file loaded onto the computers, so that the velocity will not be automatically calculated for the students.  For the average velocity, he tells the students to use the position and clock readings from just before and after the point we are using to calculate the average velocity.  Jon also mentioned that during the acquisition of data, and the subsequent creation of the whiteboards, he would talk to the groups about the significance of the data produced (if you started the cart after the motion detector, what do the initial data points mean?  points after the cart hit the bottom of the track?).

We next analyzed the graphs we made from the cart on an incline plane lab to derive the kinematic equations. Although this only takes about 10-15 minutes, it made this post too long.  So I made a separate post to show the process.  Although most texts give these equations, many omit the entire process.  For the sake of helping your students foster their connection between the graphs and the equations, Jon recommends spending the time to show these derivations.  {My guess is that you could either do this during whiteboarding, or as a mini-lecture (for those that aren’t quite ready to give up the reins, and want to be a sage on the stage again).}

After showing all the derivations, we moved on to Lab Extension: Speeding Up and Slowing Down.  {As I’ve noted at least once before, we were given version 3 of this worksheet.  However, I’m only seeing version 2 on the modeling website.  I guess that’s one more reason you need to go to the workshop and not just read my blog.}

Jon told us that he gives the students all the equipment except for the motion detector.  Jon said that after the individual groups show him the completed worksheet, he provides the motion detector.  After the students have all completed acquiring the data/graphs, he has them white board what they got for a given problem.  Chris does it a little differently.  He never gives them the detector, but rather has the students make the predictions for HW, and then the groups whiteboard their predictions at the beginning of class.  He then projects the actual results after the class has come to agreement for the each given problem. {My $0.02 on this is that I like Chris’s approach better (sorry Jon).} By the way, I had never seen motion maps that showed both velocity and acceleration at the same time. For those like me, you plot the velocity above the displacement vector and acceleration below. Have the points that represent the same time line up vertically. I’ve tried to show what the map for #1 would look like below: The blue vectors represent the velocity and the red vectors represent the acceleration for an object accelerating from rest. {I’m honestly not sure how to draw the first point for the acceleration portion, whether they should be inline with the arrow overlapping the second point, or as shown with the first point slightly above the second. I’m guessing how I have it is correct. And no, I didn’t waste the time to make sure the arrows were to scale. Remember motion maps are qualitative, not quantitative.} A couple points made by Jon and Chris: #3 is the first instance for the students where an object is speeding up even though it has a negative acceleration. You need to socratically question the students (What is happening to the magnitude of the velocity? What then is the sign of the acceleration? Can a negative acceleration increase the velocity?). According to both Chris and Jon, this is a confusing idea, since they are used to describing a negative acceleration as a deceleration (a term you should dissuade the students from using). #4 is a similarly confusing example in that the acceleration is positive but the object is slowing down. Again, use Socratic questioning to lead the students to this idea. #6 Jon omits this problem as changing the origin doesn’t really come up later in the curriculum. He said that it’s up to you and your students. Do you want/have time to spend on this? {My thoughts are that I might leave this out for standard level, buy include it for the honors level of my classes. If I have more than 6 lab groups in honors (which I did this past year (’10-’11)), I might make additional problems with the adjusted origin so each group whiteboards their own problem.} From there we worked on Worksheet 2Worksheet 2a, and a supplementary worksheet. 2a: #3 Jon mentioned that students tend to struggle with all the technical vocabulary in this problem. 2a:#5 Chris asked the group presenting: “I remember a problem from the earlier work, where the negative velocity and it was speeding up. Why is this different?” {Your trying to get the kids to focus on the speeding up when acceleration is in the same direction as motion, (and slowing down when opposite) not based on +/- sign} Wkst III: 1 c&d Jon mention that these problems are very tricky for students. We next moved on to another experiment using a Vernier Photogate and the Vernier Picket Fence. (note: you may need some of the accessories to attach the photogate to a ringstand). We used the “picket fence” file provided by vernier. We were asked to get one measurement of “g” for the picketfence by itself, and one value while a hanging weight was attached to the picketfence. Jon and Chris made this a competition among groups to see who could get the closest value to the accepted (9.80665 $m/s^2$) for each set of measurements. {I’m not sure if I would tell my students the correct value or not. I would probably just calculate the class average and then ask the students to explain our error. One side note, one of my pet peeves is “human error.” To me that is a student being lazy and not wanting to think about what they did wrong. I would push my students to say that the picket fence was rotated one way or another, photogate wasn’t level, etc.} From there, we then did another “competition” lab where we were provided the motion detectors, a rubber ball (similar to traditional dodgeball that could actually leave a mark, not the foam ones given now.) {Don’t get me started on that one.}, and a metal filing shelf (similar to this, only it was one level not two). The shelf was used over the top of the detector to help protect it from the ball. The basic procedure was to toss the ball above the motion detector and have it fall towards the detector. Again, the group with the closest value to “g” received a prize. We used the “ball toss” file provided by vernier. {I think I might introduce video analysis at this point, either have the students do it in their groups, or run this as a demo, videotaping the students tossing the ball. Then I would show on the smartboard how to use video analysis. I would probably use the tool in LoggerPro, however, seeing Rhett Allain use VideoTracker throughout his blog, makes me think it might be worth it to have the students download and use that programHowever it might be worth leaving video analysis until we get to 2D motion.} We ended the day by whiteboarding sections of the assigned reading from the previous night. Instead of giving a summary of the summary, I would just say we read Aron’s book and discussed 2.7 – 2.16 (excluding 2.14). It builds off most of the concepts already discussed yesterday. For those that haven’t read it, it’s a great text that explains many of the students’ misconceptions and strategies to help overcome them. We started today by finishing our whiteboard summaries of Ch 2 from Aron’s book. Since I didn’t go into detail on the Day 5 post, I’ll omit them here as well. From there, we wrapped up Unit III with some feedback to Jon and Chris: What worked: • The worksheet “stacks of kinematic graphs” – we felt that it was a great tool for helping students convert from one type of kinematic graph to another. Chris mentioned that if/when you have students whiteboard this, to make sure that they display the graphs vertically. • Worksheet: Speeding up/slowing down – we liked that this allowed us/students to predict what they thought would occur, then later them seeing the results. • We liked that there were multiple labs that were short, as you could get more hands on time, but not use multiple days to do different activities. • We liked seeing the graphical proof of kinematic equations • We liked the reading from Aron’s book, especially the misconceptions he mentioned, and tools to help overcome them. What didn’t work: • We said that we would like more insight into the “mechanic” of implementation the modeling cycle • What does the day to day flow look like • When do learning objectives come into play (some are at schools that must display the objectives for that day’s lesson at the start of class) • Pacing of course • Some of us that aren’t familiar with the content want more time to complete activities, and we also recognize that we need tools to overcome what we see in the workshop; some people are done with nothing to do, while others are struggling to keep up. • Some asked, “What to do if we don’t have loggerpro/equipment?” • One other thing we liked in the first cycle that didn’t occur here was the division of Labor/variation of control variables. {I’m not sure how you would fit that in, but that’s what came up in discussion} {To those in the workshop (merely reading) that want to see the pacing of a class, one website I found was Mark Schober’s Website. Another great blog that you might find useful is Action-Reaction, one especially nice feature is that he organized his blogroll for different subjects (I’ll try to get to that at some point).} Miscellaneous Questions: • How often do the students need to do formal lab reports, and how do those “Work?” • As already mentioned, what are some ideas for extensions of labs for “faster” students For the first question, Jon referred us to some of the resources at the beginning of the modeling binder (here, here, and here). For the second question, Jon mentioned that he often splits up the groups that are done and have them help the groups that are going slower. One other point that came up, was that if you need help keeping everyone engaged, assign each person in the group a roll. {When I need to do this, I use “Leader,” “Secretary,” “Technician,” and “Gofor.” The leader in is charge of making sure the group is on task. The Secretary is in charge of recording all necessary information/procedures/equipment/etc.. The technician is in charge of running the actual experiment. The gofor (some call it the Yeoman) is the person in charge of “going for” stuff. He/she gets the equipment at the beginning, is in charge of cleaning up at the end, and the assistant for all other jobs.} One other conversation that came up was to make sure that everyone in the group knew one anothers’ names. Jon mentioned that he was surprised how many problems could be avoided if they knew that one simple fact. He makes it a point to quiz students each others’ names at beginning of new lab groups. When Chris got a chance to address the question about objectives, he said he often uses some of the resources from the modeling curriculum to make review’s Kelly O’Shea has a blog that I love, which focuses a great deal on Standards Based Grading. (One oversimplification of SBG is you report grades based on learning objectives of the unit. ) (Here are her objectives for Honors Physics, by the way). When I asked her when she reveals her objectives to her students, she said: Usually try to hand them out at the start of the unit. I would say few students look at them before they are preparing for an assessment. Some probably don’t look at them until they get the test back and look at their scores. Jon mentioned that he often uses the provided Unit Objectives sheet to create a review. Chris said that he often has the 3 ring-binder out when groups are whiteboarding, and often asks questions right out of the teacher notes (post lab discussions especially) This was a lengthy discussion, but some great ideas came out of it. ## FIU Modeling Workshop – Day 3 We started today by finishing our whiteboard discussion for our constant velocity lab (buggy lab). As mentioned the previous post, Jon and Chris recommended to not worry about which variable was on a given axis (just make sure they had thought it through and had a reason). However, they said as the whiteboard session is winding down, begin to force the discussion (with Socratic questions) to which graph ($d$vs$t$or$t$vs$d$) gives more meaningful information (answer:$d$vs$t$since the slope is speed/velocity). They also mentioned to direct the students to think about position and time interval rather than distance and elapsed time, as the former will help with the distinction of speed and velocity (which haven’t been resolved as yet), and the concept of acceleration. They also reminded us, that at this point in the year, the students “know” LoggerPro and scientific techniques learned in the first unit. The supposedly know what slope means (but in reality they know how to calculate it, not what it means). One important line of questioning to pose to the students is, “What does the slope of the$x$vs$t$graph represent?” Slope is defined in algebra classes as the change in$y$divided by the change in$x$.$ \large m= \frac{\Delta y}{\Delta x}$Since the$y$-axis of a$x$vs$t$graph represents position, a change in position relative to a change in time, the slope represents the average speed over the elapsed time$\large \overline{v}= \frac{\Delta x}{\Delta t}$Since$\Delta x$is a distance, it would have units$ \textit m$.$\Delta t$is an elapsed time, so it would have units$\textit s$. Thus the slope of the x vs t graph should have units$\textit m/s$, which is consistent with the units for speed. {Thanks Global Physics Department for introducing me to LaTex!} Next on the agenda was using LoggerPro to create a$v$vs$t$graph for our data. After which, we used the integral function to find the area under the “curve.” Again, we discussed the meaning of this area. From math, we know:$area=(base)(height)$Since the base is time,$t$, and the height is velocity/speed,$v\$, we can show that the area is displacement/distance.  In looking at my notes, one thing that I’ll get clarification on, is when due we, the teacher, distinguish between speed/velocity and distance/displacement during this process.  Have we gotten to that point, and I forgot to note it, or are we not to that stage in the cycle.
Next on the agenda, we were asked to work on Unit II worksheets One & Two.  Each group was again  asked to present one part of this assignment on a whiteboard.  A few comments to note:
1. Jon mention that before beginning these experiments, he has the lab groups perform a vernier experiment using motion detectors and labquest mini interfaces to match their motion to given position vs time and velocity vs time graphs. (I mentioned that I do the same activity as a competition between lab groups, which I find gets the kids very excited.  I’ll probably write about my “Physics Olympics” at some point in the near future.)
2. On wkst 1, question 2a, ask the group “How do you know they are the same?”  Meaning, get them to discover how the could determine the scales were the same given the limited information.
3. On wkst 1, question 2d, ask the group “Can two of your members enact the motion depicted in the graph?”
4. On wkst 2, question 6&7, use Socratic questioning to lead students to drawing dashed vertical lines at the points of discontinuity.  Someone asked about including open and closed dots to show where the object was at the point of discontinuity.  Chris answered that we don’t know, nor do you need to get into that level of sophistication.

We ended the day by discussing the last tool in the modeling arsenal, Motion Maps:

The above examples show two different maps (the first above the red line, the second is below).  Some key features of the map are: the position vector, which shows the origin (X) and the direction of positive motion; the dot (which I couldn’t get to work as a small dot); the arrow on the dot, which represents the velocity of the object at that location and time.

That’s where we ended today.