Velocity Graphs into Equations

aka How to kill \Delta x = \frac{1}{2} \Delta v \Delta t.

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, \Delta x = \frac{1}{2} \Delta v \Delta t.

Pretty soon, though, that (\Delta x = \frac{1}{2} \Delta v \Delta t) 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.

Here's a classically "labeled" graph. The endless tick marks make solving the problem difficult because they often don't even want to commit to really drawing the graph since they can't just plot out points without already knowing all of the info. They never add symbols to the graph for the quantities that they don't know, and then they predictably have trouble finding out those unknown quantities.
This graph is more on track with drawing the qualitative shape of the graph, using symbols, and doing legit annotations. It's also displaying one of the strange habits a lot of my students have. Once she solved the problem, she felt she needed to go back to the original graph she drew and add in the numbers that she now knows (vf and ∆t in this case). Some of them really don't like putting symbols on 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?) \Delta x = \frac{1}{2} \Delta v \Delta t. 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.

Links to the document: PDF | DOC | PAGES

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.

13 thoughts on “Velocity Graphs into Equations

  1. I guess that my question is about why you’d build .5 delta v delta t = delta x in the first place, since it’s only coincidentally correct for cases with v_i = 0. Why not delta x = v bar delta t, where you can make great graphical connections to how v bar relates to v_i and v_f, and it’s always always true? The first thing in the brain is usually the stickiest, so why make it something that’s not really true? I’m maybe missing something here in the genesis of .5 delta v delta t, since that’s where your posts begins.

    1. Okay, yes, so that’s my point to them (well, not the average velocity part because I don’t think most would get why that was true and would just be memorizing another equation that happened to work all the time, though they don’t know why). It only works for a few cases. I never tell them that equation. We just build the idea that the area on a v-t graph represents ∆x, and then after they do a couple of problems where ∆x = (1/2) ∆x ∆t is true, they think that it is always true. Even when they keep getting problems where it is not true. To them, they have just discovered “the equation for ∆x.” I think part of the origin of this misconception is evidence of them noticing patterns. When I ask them to write equations for slope and area on a graph, they notice that the slope equation is always the same (a = ∆v / ∆t), so it follows naturally that the area equation is always going to be the same…

  2. Hope you don’t mind. I gave this link to my students. They know you from me talking about you. I told them they might learn something, I do modeling and try to add some things. I like Moving Man from Phet, and I liked your blog.

  3. First, I love the graphical approach, too. But one question: How much do you emphasize problems that, when using the graphical approach, requires the use of simultaneous equations? For example, given acceleration and distance, find time and final velocity. Students would need to set up a slope equation and an area equation, each with two unknowns. These type of problems come up frequently in free-fall.

    1. Nice post, Kelly! I agree that this is the way to go with 9th/10th grade physics. Heck, maybe all grades! I gave in during free fall and reminded my students that we derived the x=1/2at^2 earlier in the course. We used it sparingly during our problem practice sessions and instead focused on solving graphically. I even wrote a goal for solving problems graphically. Most did well on the formative for it.

      To my horror, less than five of my 84 students used a graphical method to solve two word problems:
      1. Find the acceleration given delta v and delta t
      2. Find the displacement given acceleration and time.
      The majority of students got the first one correct and the second incorrect.
      I am not too happy with the results and am wondering what will help.
      I am thinking of starting to use goal less problems before problem solving.

      However, perhaps the response to frank should be we should not pose problems that are not solvable graphically. To me, a graphical solution (with a marked up graph) shows an understanding of the physics and the problem much more so than a correctly memorized algorithm.

      I guess each class is individual as to how far you can take the equation usage. I think my 9th graders probably just proved to me they don’t really understand how the equation relates to the graph.

      1. Hey Jim,

        Thanks! I’m pretty convinced that graphical is the way to go for all grades (in high school at least… don’t know much about teaching this in college).

        It sounds like although you showed them the graphical approach, you also maybe derived some general equations at the same time? When I used to do that, I got what you got. About 1 or 2 kids who would do it graphically and the rest plugging and chugging. Last year I delayed introducing equations until three units after the constant acceleration unit. I got tons of graphical solvers that way. This year I’m basically not even mentioning the fact that there’s a way to derive general equations. I’m just letting them (10th and 11th graders) do it graphically. It’s basically just as fast as using equations once they’re good at it, but a lot more successful for a lot more students. And many of them will start seeing the patterns of the general equations on their own after a while.

        I think another key is that they will do what you do. So if you always approach problems in front of them by drawing and annotating diagrams before you ever try calculating anything (given that we didn’t learn physics this way, it takes a while to train yourself to always do that and not slip into the more comfortable habits of old!), then the kids will do something like that, too. But if you only draw cursory diagrams or often none at all when you’re in front of them, then they will see that and get the message that the diagrams are optional and that it is better to solve problems by plugging things into equations. In a lot of units, after they have tried some problems on their own, talked in groups, and done a little whiteboarding, I will let them sit back and watch me solve a problem (almost always one that they have actually just whiteboarded and presented themselves) so they can see how I approach things and how I organize my work. They won’t learn any physics understandings from watching me, but they will start picking up my habits of organization, labeling, etc. It makes a big difference and makes them feel good (they love seeing me work problems even though it doesn’t result in increased understanding… so the compromise is to only work problems where they’ve already done the thinking).

        Also, I think any of these problems can be solved graphically (see my response to Frank below). The equations only come from the graphs anyway, so if it can be solved with the equations, then it can be solved graphically. I haven’t found a problem my kids can’t do (eventually)! 🙂


    2. Hey Frank,

      All the time. In fact, if I give the Honors kids a problem that _doesn’t_ need two equations, they comment on how this was “a really easy one.” I harp on the idea that you can always use the velocity graph to write two equations: one for slope and one for area. The slope equation is always the same, but the area equation is the one that requires thinking and really looking at the graph.

      My CAPM packet has the bear and the honey problem, and any situation where they end up needing two equations usually gets referred to as a “bear and the honey” situation… 🙂 Until they start just writing both equations.

      The harder part is getting them to write equations for slope and area when they know the numerical value of the slope and the area. They think they should only write equations when they don’t know the thing they’re writing the equation for (sorry for the grammar there). But once they do it once (Honors) or four times (not Honors…) they start to realize that it can be helpful to write the equation from the graph regardless.


  4. I just stumbled upon this method and am really excited to put it into practice with my CP physics kids next year. In doing 2 D problems….would you just develop a v-t graph for each dimension (horizontal and vertical)? Thank you for sharing this!!

  5. Kelly,
    Do you continue to use this type of graphical approach when you get to 2D motion? I get better success with the graphical problem-solving every year, but I fail to make it work when I get to projectile motion. As you said, I wasn’t taught this way myself, and it is so much easier for me to fall back into equations when we get to 2d. Thank you for your awesome blog.

    1. Ah—I’m clearly really behind on responding to comments. I saw your more recent ones (and just responded), and it seems like you’ve worked through projectile motion this way now. Yes, we definitely keep it graphical. I don’t introduce the equations at all, but do nudge the more mathematically-comfortable students to notice patterns in their work (and some do figure out general equations from that sort of pattern hunting).

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s