# Category Archives: kinematics

## 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.

## Unit 1: Constant Velocity Particle Model

This introductory unit on the CVPM will proceed along the traditional modeling framework with only a few additions. I expect it to take approximately three weeks. It will begin with the Buggy Lab and whiteboard meetings to analyze the results. The only change from the traditional progression will be to first complete the “Graphing Practice” worksheet from the Scientific Methods unit.

From there they will proceed through the modeling materials for the Constant Velocity Particle Model as provided by the Modeling Materials. Thus the new worksheet 2 will be a worksheet that focuses on the students converting between the position-time graphs, motion maps, and verbal descriptions. Worksheet 3 will then add velocity-time graphs to the mix. Worksheet 4 brings back data analysis and converting to the other representations. Worksheet 5 does the same, but for slightly more difficult situations. We end with additional problems for review.

For those using Standards Based Grading, the first draft of may standards is as follows:

Students will be able to (SWBAT):

1. design an experiment that properly controls variables
2. report measurements and calculations with proper precision
3. develop a mental model that correctly explains and predicts an event
4. algebraically solve an equation for a given variable.
5. create a scatter plot of independent and dependent data points
6. linearize data points
7. create a mathematical model of a graph.
8. create and interpret graphical and mathematical representations of objects moving with constant velocity.
9. distinguish between position, distance and displacement.
10. solve problems involving average speed or average velocity.

I’m fully aware that this is a long list of standards, and quite possibly too many standards. Any feedback on whether or not the list should be adjusted (and how) would be greatly appreciated.

## Modeling Unit 1

As I noted at the top of my “modeling” page, I’m going to go back and reorganize my blog posts.  I’ve found that organizing by the day of the workshop isn’t very effective.  I plan to try to go back and organize the posts based on the Unit in the modeling cycle.  So, here’s Unit 1 (with the preliminary stuff of the workshop added in):

I’m blogging about my experience at the FIU Modeling Workshop.  Much of this is for me, so that I can remember my experience.  However, maybe this will help someone else to come over to the Modeling Method.  I’m not sure if I’ve mentioned it before, so I might as well state it here, I currently teach Standard, Honors, and AP-B Physics.  I’ve been using the CPO Program, which is a hands-on program.  To me, it’s biggest downfall is that the labs, although well constructed, are cookbook labs.  The students can get caught up in the procedure, and miss the concept.  After joining twitter, I’ve come across several teachers that use the Modeling Method, and have become more and more interested.  Which brings me back to the point of this post, my experience on the first day.  However, before I get into that, I would make the following claim, if this interests you, please go to the workshop, don’t just rely on me.  Even after only one day I can tell that my recount will mean nothing for you without you attending.

Day 1:
We started the day with our leaders introducing themselves (Jon Anderson and Chris Doscher).  They quickly led us through a great introductory activity, that I might very well use with my students.  We each had to come up with 2 truths and 1 lie about our self, and the other people in our small group had to try to determine which is the lie.  After that, each person in the group had to introduce another member from the group to the entire cohort.  To me, it was a fun way to break the ice.

After taking the Force Concept Inventory test, we then got our first taste of whiteboarding.  We were asked to answer the following 3 questions as a group:
1. What are your greatest content-related teaching challenges?
2. What are your greatest instructional teaching challenges?
3. What are your goals for this workshop?

Here are the whiteboards:

Unit 1: Scientific Thinking
After breaking for lunch, we began our first experiment, a Pendulum Experiment.

In walking us through the experience of the lab, we were given a few questions and comments after we completed the task.  (For the sake of brevity, I’ll omit our responses to the questions).

Jon set up a simple pendulum and then wrote the following questions in succession:

What do you observe?
(side note, Brian W. Frank  recommended asking “what do you notice,” rather than “what do you observe.” Here’s why)
• Jon mentioned to try to not give any comments/facial gestures, just write.
• Ask if you need to rephrase for fewer words
What can you measure?
• Don’t comment until at the end.
• Do you need to pare down the list, do to lack of equipment?
• Are any measurements redundant, if so discuss with the class.
What can you manipulate to change the time?
• Edit down after complete based on equipment present
State purpose of lab for students:
To determine the mathematical and graphical relationships that exist between time, length, mass, and angle of release of a simple pendulum.
(Jon told us that the bold part represents the beginning phrase for basically all the lab objectives)

Before assigning the different types of relationships to different groups, Jon told us two important “rules” for labs:
1. Fair Test: manipulate only one variable at a time
2. 8×10 rule: collect at least 8 data points separated by at least a factor of 10
After collecting the data, they then introduced the group to LoggerPro, to analyze the data. We used LoggerPro to analyze our results and then put them on whiteboards to share with the other groups.

During this time, my small group discussed some of the strength and weaknesses with excel vs LoggerPro.  Namely, to us LoggerPro can analyze the data faster, but excel integrates with word docs a little easier.  (We could easily be wrong on this.)

This is where we ended the first day and began the next byfinishing up the “Board Meeting” with the groups that studied length vs period.  For physics teachers, this is obviously the group that was able to show an actual correlation.  One of the most interesting parts of the discussion, to me, was when Jon and Chris recommended not worrying about linearization yet.  They told us to now worry about that battle, as it will come up as you move into the next phase of the cycle.  Just let the kids use LoggerPro to get the mathematical relationship.  They did recommend spending some time to discuss whether or not the data should go through the origin.  In the course of that discussion then mentioned what they called the “5% Rule” which basically states that if the y-intercept is less than 5% of the biggest measured value in the data for the y axis, assume that it goes through the origin.After we finished that discussion, we then moved into the next phase of the modeling cycle in which we worked on linearizing data using LoggerPro.  The worksheethad 6 data sets (we had version 3 of this worksheet, I’ll add that link if I find it), and we had to plot the data and determine how to manipulate the data to create a linear graph that went through the origin.  Jon and Chris mention that they only used the first four problems (which I think are the 4 in version 2) with their classes as they have found that they are sufficient to get the students acclimated to the process. Jon and Chris did recommend to have the students write the regressed equation rather than the proportion shown (ie: equation with slope and y-intercept, not y is proportional to $\frac{1}{x}$).After we had linearized the data, each group was assigned a different problem to put on a whiteboard to share with the cohort.  Again, we were able to get a greater feel for how the whiteboarding process works, and able to ask questions as to how to moderate, when to step in and when to let the conversation go.

After that, Jon and Chris asked for feedback as to how we thought the first unit went.  It’s amazing how well these modeling people all act as I remember Frank Noschese blogging about getting feedback from students more often than just the end of the year (read the post here).

They asked what worked and what didn’t?  To the first we said, we liked learning: how to use LoggerPro (especially for linearization), the linearization summary sheet, breaking up the pendulum lab to finish the lab in less time (made groups take more ownership of work since others were depending on them to get it right), and using inductive reasoning to determine relationship instead of the teacher just telling “us” the answer.  What we didn’t like: some wanted more explicit explanation of the relationships between independent and dependent variables (hopefully I’m remembering that correctly), and some wanted the workshop to move a little faster (I think she was referring to limiting some of the discussion, but Chris rephrased it as getting started quicker/more punctual coming out of breaks.  I’m not sure which was what she meant).

## Derivation of Kinematic Equations

To begin the derivation of the kinematic equations, first start with constant velocity (tumble buggie lab).  The more of this you can get the students to do, the better.

Figure 1 shows a plot of position vs time:

With the graph, guide the students to explain what the slope of the line represents.  You need to get the students to say the slope represents how much the position of the object changes every second, not just speed/velocity (we’re trying to build a concept/model, not memorize a word).  Continue to dialogue by asking questions such as “what does it mean if the line is higher/lower/negative,” and “what does it mean if the y-intercept is higher/lower” will help clarify the concept of velocity.

You can then go on to have the student explain how to determine the average velocity by adding an initial and final time (and corresponding position at said times) as seen in Figure 2.

Have the students start from the the beginning again:

$\large \overline{v}=\frac{\Delta x}{\Delta t}$
$\large \overline {v} = \frac {x_f – x_i}{\Delta t}$

Rewriting in $y=mx+b$ format, you get:

$x_f = \overline{v}\Delta t + x_i$

Also remind the students about the velocity vs time graph for constant motion, as seen in Figure 3.

From there, you can guide the students to the idea that the area under the line represents the displacement, shown in Figure 4:

Get students to explain that the shaded area is a rectangle:

$Area = base*height$

Since the “base” has units of time and the “height” has units of velocity, then:
$\large s*\frac{m}{s}=m$

So, the area has units of distance.  Since that area could be above or below the x-axis (thus positive or negative), the area is the displacement:

$\Delta x=\Delta t * v_o$

From this point, we can now engage the students with the velocity vs time graph for the cart on the inclined track (Figure 5).  Again, lead the students to say that this graph now shows that the velocity is changing with time.  Repeat the same questions about what the slope represents, what does it mean if line is steaper, etc.
{It may seem redundant, but a major misconception is what slope actually is. Most students only think of the definition/equation, not the true, physical concept}

Once again, lead the students to the definition of acceleration {I’ll omit the derivation to save time, I’m surprised you’re still reading this far}

$\overline {a} = \frac {\Delta v}{\Delta t}$
Which, again, can be rearranged into a “y = mx + b” format:

$v_f = v_i + \overline{a}\Delta t$

Which I’ll call Equation 1
{At this point, I’ll let you know that I circle the equation with a red marker in my class.  The only time I use a red marker is for a fundamental equation.  I picked this up from a college professor.  This fits in with the ABC Gum Rule talked about in Day 2.}

From here, we again lead the students to explain what the area under the velocity vs time graph represents.  The catch here is for them to recognize that the area under the curve can be broken into two geometric shapes as seen in Figure 6:

The area for this shape would therefore be the area of the triangle plus the area of the rectangle:

$\large \Delta x = v_o \Delta t + \frac {1}{2}\Delta v \Delta t$

Which, after distributing the second term for $v-v_o$ and combining like terms, simplifies to :

$\large \Delta x = \frac{1}{2}\left(v_f + v_i \right)\Delta t$

Which I’ll call Equation 2, which also gets the red encirclement.
From there we can algebraically combine Equations 1 and 2, to get equations which are easier to use in common situations.
If you know acceleration, but not final velocity, rearrange Equation 1 so it is explicit for $\Delta v$ and substitute that into the unsimplified form of  Equation 2.  That looks something like this:
$\Delta v=\overline {a}\Delta t$
$\large \Delta x = v_o \Delta t + \frac {1}{2}\Delta v \Delta t$
$\large \Delta x=v_o\Delta t+\frac{1}{2}\left(\overline{a}\Delta t\right)\Delta t$
$\large \Delta x = v_o \Delta t + \frac {1}{2}\overline {a}\left( \Delta t \right)^2$
Which is circled in red and called Equation 3.
If the time interval is not known, you can rearrange Equation 1, so that it is explicit for $\Delta t$ and then substitute into Equation 2.
$\large \Delta t = \frac{\Delta v}{\overline{a}}$
$\large \Delta x = \frac{1}{2}\left(v_f + v_i \right)\Delta t$
$\large \Delta x =\frac{1}{2}\left(v_i +v_f \right)\left(\frac{v_f-v_i}{\overline{a}}\right)$
$\large \Delta x=\frac{1}{2\overline{a}}\left(v_f +v_i\right)\left(v_f – v_i\right)$
$\large \Delta x = \frac{v_f^2 – v_i^2}{2 \overline{a}}$
Which would be Equation 4, and also get the red box.  With that, you have the four equations of kinematics:
$v_f = v_i + \overline{a}\Delta t$
$\large \Delta x = \frac{1}{2}\left(v_f + v_i \right)\Delta t$
$\large \Delta x = v_o \Delta t + \frac {1}{2}\overline {a}\left( \Delta t \right)^2$
$\large \Delta x = \frac{v_f^2 – v_i^2}{2 \overline{a}}$