The APT 1 students just recently finished building and testing (deploying) the Constant Acceleration Particle Model. As stated in an earlier post on studying velocity vs. time graphs for a constantly accelerating particle, the students used these graphs to study how the velocity changed, but also how the graph gave the students a way to calculate the change in position of the particle.
Position vs Time Graphs and “Fitting The Function”
Using the Vernier Motion Sensors, the students then collected position data for a cart that was accelerated across the table at a constant rate (using a modified Atwood’s machine setup). This revealed what some of the students had expected – the graphs were curved!
At this point, I could have introduced the process of linearizing data as some other Modelers have done, and perhaps I will do that next year, but this year I went ahead and just asked the student to use the “function fit” tool in LoggerPro. This tool fits a mathematical model to the data, and then displays that model on the screen. I can see why some teachers feel that it is better to get the students to go through the process of linearizing the data because depending on the tool in LoggerPro can lead to some misunderstanding and it encourages the students to depend on the computer to “give them the right answer”.
What Do “A”, “B” and “C” Mean?
The reason I do introduce this tool is because we are going to use it in the future, and it gives me a reason to talk about the coefficients of the fitted function. So, immediately after the students see the fitted function, I direct them to desmos.com to have some fun with quadratic functions – and (re)learn how the coefficients in the quadratic function change the curvature of the graph and the location of the vertex. This allows us to get into a good discussion about how these coefficients are related to physical changes. I ask the students to consider the questions – “How would the graph have changed if the cart had a greater acceleration?” and “What if the cart had accelerated in the opposite direction?” and “What if the cart had started from a different location?”
Once they have a solid grasp of this function, and how it can be used to predict the position of an accelerating particle, I set them off on a deployment task – cart jousting!
The students set up two carts a distance of about a meter away from each other. Each cart was on its own track. Once again, a weight hanging on string that passed over a low friction, low mass pulley was connected to each cart. I gave the students different masses for each cart so that the carts would accelerate towards one another at different rates. The students were also asked to attach a pencil to the front of the cart so that the cart would visibly push the other cart as it passed. The students used LoggerPro one more time to get the slope of the velocity vs time graph so that they had the acceleration of each cart. Using the function that outputs the position of an accelerating particle, given a specific time value, the students were able to predict the point of collision.
When I have the students do this, as suggested by modeling pedagogy, it is really important to have the students explain the process by which they established their prediction. This is done on whiteboards. I don’t however have a class discussion unless I think it is going to help certain groups. Otherwise, I take on the role of Socratic inquisitor and allow the teams that are confident, to proceed with the experiment. This allows me to coach certain teams that I know might just otherwise wait to see what other teams have done, and simply follow them.
Students then have fun taking video and sharing the video to me. I’d like to set up some way for the students to post to a social media site, but I am a bit concerned that I would need to be the filter – something I have not yet found.
Well, now its time for us to turn our attention from strictly kinematic models to causal models that describe and represent causal relationships for answering the question – “why do things move the way they do?”