Update 2013-06-07: These are some notes I wrote over a month ago but never published. Since this work got me thinking about a how to align my instruction with SBG objectives and a later post I’m about to publish, it seemed like now or never.

In trying to get better at teaching graphical representations of uniformly accelerated motion, I tried something similar to Kelly O’Shea’s paradigm laboratory for the Constant Acceleration Particle Model (CAPM). Being not so brilliant at teaching myself, I think I injected a little too much of myself into it, but it was clear from a “pre-test” (review of Constant Velocity Particle Model graphs going into this unit) that students were confusing velocity and position, didn’t remember much about how to analyze velocity-vs-timer-reading graphs, so we needed to do some review. Part of this is the insanely long time between CVPM and CAPM, since we went ETM⟶CVPM⟶MTM⟶BFPM⟶CAPM⟶UBFPM. If I do this again, I will need to include more model-based reasoning problems that incorporate CVPM throughout the previous units. Thus, I found myself trying to come up with a list for us to make to summarize our large number of different scenarios in the CAPM paradigm (class) lab. For instance, in one of my periods, we had (among other things):

Velocity-vs-timer-reading graph for a cart on a ramp

I had the students annotate each segment of each graph with how the speed was changing (“v↑” or “speeding up”, “v↓” or “slowing down”), which direction it was going (“↑r” or “up the ramp”, “↓r” or “down the ramp”), and what was going on (“” for nothing, “pushed”, “stopped”, etc.). We made important notes besides the graphs that students drew in their lab notebooks, such as “☆The mass doesn’t seem to affect the graph very much.” Then we tried to summarize what we could tell with a table like:

Student-driven exploration of the velocity-vs-timer-reading representation
Feature of the $\vec{\boldsymbol v}\text{-}t$ graph Feature of the motion
point on the graph given by a pair $(t,\vec{\boldsymbol v})$ a data point, i.e. snapshot, of an object moving with a velocity $\vec{\boldsymbol v}$ at a timer reading $t$
horizontal position of a point on the graph timer reading $t$
vertical position of a point on the graph velocity $\vec{\boldsymbol v}$
vertical distance of a point from the timer reading ($t$) axis (the $\vec{\boldsymbol v}=\vec0$ line) how fast it is moving (speed $v$)
position of a point above(+)/on(0)/below(-) the timer reading ($t$) axis (the $\vec{\boldsymbol v}=\vec0$ line) which direction it is going
steepness of the $\vec{\boldsymbol v}\text{-}t$ graph’s slope in the neighborhood of a point how fast the velocity changes
sign of the $\vec{\boldsymbol v}\text{-}t$ graph’s slope in the neighborhood of a point which direction the velocity is changing (somewhat artificial)
the $\vec{\boldsymbol v}\text{-}t$ graph’s slope in the neighborhood of a point is moving [away from (+), parallel to (0), or toward (-)] the timer reading ($t$) axis (the $\vec{\boldsymbol v}=\vec0$ line) speeding up (+), maintaining a constant velocity (0), or slowing down (-)
point on the timer reading ($t$) axis (the $\vec{\boldsymbol v}=\vec0$ line) where the slope crosses the axis from negative to positive or vice versa. changing direction
? ?

Some classes were able to come up with their own entries. In others, I had to debase myself by suggestion them. In one class, it worked rather well to express my frustration that no one was saying anything, put a student in charge, and tell them that I would be silent while they figured it out. They put much wrong on the board, but they were jumping back to correct things they realized were wrong when trying to identify how to tell some of the other kinematic features; then the bell rang! I tried a similar approach in another class but didn’t give them enough awkward silence before going into silent mode myself. That class didn’t bother checking whether they hypothesized connections actually worked and weren’t given enough time to find out. I jumped into it with a few minutes to go and proceeded to ask them questions to test their statements, destroying all of them. I felt like rain on their parade. I had a hard time even convincing them that their statements were wrong because they could not tell me, given two points on the graph, which was moving faster. I intend to ask more questions like this next year during CVPM. I wanted to cry for them, and I was angry at all (including partly myself) who failed teaching them how to read a graph. Every year we say, “These are smart kids. They should be doing better on the science part of the ACT.” Now we know why. (Thus, I added the first three lines of the table above.)

I know this can’t be the best way to teach this. Engagement was low, and the whole paradigm lab had a demonstrative feel.

Position-vs-timer-reading graph for a cart on a ramp

# Calculating velocity and acceleration from (timer reading, position) data using CAPM

I’m collecting some calculations I find useful for evaluating student data and for more advanced students to process their own data. CAPM says that we think acceleration is basically constant, gives us a mathematical model for how position and velocity depend on time, and allows us to make predictions. If we believe that CAPM holds, then we only need three points to calculate velocity and acceleration, one data point at the spacetime event in question and one on either side of the event in time.

## Assumptions

1. Acceleration is constant. Thus $x(t)=\frac12 At^2+Bt+C$ and $v(t)=At+B$.
2. The data point in question is $(t_0,x_0)$, and the points on either side are $(t_0-dt_-, x_0-dx_-)$ and $(t_0+dt_+, x_0+dx_+)$.

## General case

• $v(t_0)=\frac{\frac{dx_+}{dt_+}\cdot dt_- + \frac{dx_-}{dt_-}\cdot dt_+}{dt_-+dt_+}$
• $a(t_0)=\frac{\frac{dx_+}{dt_+}-\frac{dx_-}{dt_-}}{\frac12(dt_-+dt_+)}$

## Breadcrumb method case ($dt_-=dt_+=dt$)

• $v(t_0)=\frac{dx_-+dx_+}{2dt}$
• $a(t_0)=\frac{dx_+-dx_-}{dt^2}$

## Split time method case ($dx_-=dx_+=dx$)

• $v(t_0)=\frac{dx(dt_-^2+dt_+^2)}{dt_-dt_+(dt_-+dt_+)}$
• $a(t_0)=\frac{2dx(dt_--dt_+)}{dt_- dt_+(dt_-+dt_+)}$ (Note: This needs some reinterpretation.)

## Summary for use

With N data points, one can easily calculate N-2 velocities and accelerations at the same timer readings. Of course, one can calculate N-1 velocities at intermediate timer readings, but this can make it difficult for students to make spreadsheets for velocity. Using the method above, one can calculate both instantaneous velocity and instantaneous acceleration at the mesh of timer readings by focusing on three data points at a time. This makes it easier to test the validity of the Constant Acceleration Particle Model (CAPM).

I like to throw the formula into a spreadsheet that I use when looking at student data.  When they whiteboard in class, I type in their data to see how consistent it is, helping me to assess where students are going wrong and whether their graphs make any sense.