Summary of the Constant Acceleration Particle Model
Scenario: A low-friction cart moving up or down a ramp, with increasing or decreasing change in position (displacement) for each unit of time.
Motion Map: Dots showing position have consistently increasing or decreasing spacing. Velocity arrows starting at each dot gradually increase or decrease in length, showing direction of motion and speeding up or slowing down. Acceleration arrows of constant length above or below each dot show the amount and direction the velocity arrow will change between this time snapshot.
- Quadratic (parabolic) position time graphs, with vertical intercept at initial position, and steepening slope for speeding up or shallowing (flattening) slope for slowing down.
- Linear velocity-time graphs, with the vertical intercept representing initial velocity, slope representing the rate of change in velocity with time (acceleration), and sign (quadrant of the graph) representing direction (towards more positive positions or towards more negative positions). Adding up the area under this line between two times gives the displacement (change in position) that occurred during that time.
- Constant acceleration-time graphs, with the value of the constant acceleration showing the amount the velocity changes each second. When the sign of the acceleration is the same as the velocity, it represents speeding up in that direction. When the sign of the acceleration is opposite to that over the velocity, it represents slowing down.
- delta x = 1/2 a t^2 + v0 t -- relates displacement to acceleration, time elapsed, and initial velocity; corresponds to the area under the velocity-time graph.
- v = at + v0 -- relates current velocity to acceleration, time elapsed, and initial velocity; corresponds to the equation for the line on the v-t graph.
- vf^2 = v0^2 + 2 a delta x -- relates final velocity to intiial velocity, acceleration, and displacement; allows those quantities to be related to each other without using time elapsed.
How will I use this next year?
This unit is the first I ever tried to (imperfectly) nick from the modeling materials, because there's a big gaping hole in the FME curriculum I've been using where acceleration and Newton's Second Law should be.
For the paradigm lab, I don't have ticket tape timers, but I've experimented with using motion detectors and carts, first on the ramps of the hallways at my school (before I got my PASCO tracks), and this past year with the PASCO carts and tracks. I love the idea of hand marking the position according to the metronome every beat on a very shallow ramp -- I think it'll give my 9th graders (and also my 11th graders) a much more concrete connection to the acceleration motion maps and x-t graphs, before eventually bringing out the motion detectors. I will definitely be adopting the term "slope-o-meter" with my students.
I also loved part two of the lab, where students calculated the velocity at the midpoint of each x-t pair "by hand" (or by excel). I have definitely not been this careful about bridging from the position-time data to the calculated velocity-time data, and I think it's a lovely set-up for students as they prepare for thinking about derivatives and approximations of them with finite data in Calculus. It was also super satisfying to see how lovely the v0 terms coincided on our x-t and v-t fits, even though we had a weird overall fit. I think it's good to make the distinction between instantaneous velocity and average velocity and average speed, but I also think it was overly belabored in our in-workshop discussions. That would be vastly abbreviated for both of my classes, I think.
I very much appreciated seeing how the constant acceleration kinematic equations were developed from the data without just saying, "look here are these equations that match what you observed" (which is what I have tended to fall back on, especially when faced with imperfect data). I will do my best to copy this next year with the 11th graders. With the 9th graders, I haven't typically tried to get quantitative with the constant acceleration equations. I'll need to think about the time requirements and how I spend my time next year -- I think I can do it if I jettison my buoyancy unit, and probably my waves and sound units (which already were sacrified on the altar of snow days this year). But that means I'll need to make sure to be more careful about weight vs. mass when I get to forces unit, and think about whether / how to work in some thoughts about density and maybe pressure at some point. (Do those show up anywhere in the modeling curriculum, or are students supposed to be clear on them already when they come in?)
I loved the acceleration direction dance, and seeing all the work that went into developing it. Figuring out whether an acceleration means slowing down or speeding up is perpetually a source of confusion for all my students, and I'm hopeful that I'll do a better job of helping students clarify this for themselves next year using the Ramp Extension and Stacks of Kinematic Curves (which I have used before, but not to their fullest potential).
I haven't typically used the Unit 3 Worksheet 4 more standard "physics problems" before. With the ninth graders I've been rushed and not aiming for quantitative relationships between acceleration, velocity, time, and displacement. With the eleventh graders, I've typically reverted to textbook plug-n-chug problems at this point. It was fascinating, in student mode, to experience solving all of those problems graphically, rather than with the suvat plug-n-chug approach with which I was "raised". With the 11th graders, I will give this a shot, and hope to use it to help them connect all the different representations in a quantitative problem-solving mode.
I loved the practicum labs again, both for their sense of fun and challenge and open-ended problem solving and also because they so neatly extended the constant velocity practicums. I will be stealing these wholesale.