aka How to kill .
Now that we’re pros (Wheaties box, here we come) at drawing velocity-vs-time graphs, we need to be able to turn those graphs into equations if we want to use them as tools for solving problems.
One big stumbling block in solving constant acceleration (CAPM) problems is that, very very often, we are looking at situations where the object starts from rest (in no small part because the algebra can get a lot more involved for situations where the initial velocity isn’t 0 m/s, and we (or I) don’t necessarily want them spending their time solving a bunch of quadratic equations rather than playing with more graphs and physics problems. But anyway.). So the velocity graph they are drawing for the problems that they encounter almost always looks like one of these two:
And the area we care about, from the start of the object’s motion until some particular time (or through some particular displacement) is, indeed, .
Pretty soon, though, that () is the automatic response from a student when they want to find the displacement— even when the velocity does not start from 0 m/s. And pretty soon after that, they can’t solve any problem if the object doesn’t start from rest.
So. How to go about fixing this issue?
Annotations for the win
At my school (and maybe elsewhere, too?), the students are very familiar with the idea of annotating a text. When I would tell them to “label” their graph, though, that meant to them that they should draw dozens of tiny tick marks and unit-less (aka naked) numbers along the axes. That’s not what I meant at all! So I shifted to the language that they already hear all the time in their other classes (annotate) and gave them some guidelines on what it looks like to do a good job of annotating a graph.
The first step to solving a problem is definitely having a qualitative, but annotated, graph. Once you put in the annotations, you create a language you can use to talk about your graph. You hear this again and again when you listen to students going from bumbling explanations about how they are solving a problem to eloquently talking about the area as they add in the annotations.
Write an equation in symbols first
Now that we have a vocabulary, we can put it to use in writing a sentence to describe the area on this graph. So here’s where the intervention needs to happen. Many students were making beautiful annotations, then completely ignoring their graphs as they wrote down (what else?) . When you are standing next to them and watching them do this, you can actually see that they don’t look at the graph at all while they write that equation. They have just memorized it now!
Enter the new worksheet. I am thinking about adding this to my CAPM packet next year, and I will be auditioning it during our exam review this January. I’ve done this exercise with a few kids one-on-one as they’ve come to see me about solving CAPM problems and it has been really useful, so I have high hopes for this working well with the larger class.
Here’s an example of what I will expect them to do (but they just need to use one of the methods for finding the area):
Pretty simple and quick exercise. But if the kids aren’t writing the equations for area using their graphs, they must just not know how to do it (or what I mean when I say it). These guys definitely know how to find areas of triangles and rectangles (and therefore trapezoids), so I think they will be able to augment this skill really quickly and easily. Get a little consistency going and—BAM! Then they can solve any CAPM problem with (relative) ease.
Extra note: In case it is confusing why I would have my students go to all of this trouble— we actually don’t derive any of the usual kinematics equations. We use the graphs to solve problems. All year. Some of the kids start noticing patterns and figure out for themselves all the equations that they’ll see in a book the next time they take physics, but I ask them to solve everything graphically in this first-year high school class.