Reading the physics in a graph

In sciences we often represent a relationship on a graph. We've chosen a screwdriver as our icon for reading the physics from a graph since a screwdriver often is what you need to open something up to see what's inside it that makes it work.

When you learn to create graphs in math classes, you are often showing the graph of a function — a relationship between two variables (usually $x$ and $y$) and that's pretty much the end of it. But in science we graph many functions and many relationships so often we have many different graphs representing the same physical situation.

As a result, graphs can give you multiple views of the same situation. Each graph highlights a different aspect of what's happening. It's like picking up a real object and looking at it from different angles, rotating it, turning it over, and looking at it with different lighting.

Looking at a situation using different graphs help make our understanding of what's happening richer, more coherent, and make more sense.

Blending the reading of graphs with the story tool is particularly useful to see how and what parts of the physical story a graph tells us.  Graphs can be valuable tools to help analyze and make sense of a complex situation.

Here are a few hints how to keep your graphing tool sharp and polished:

• Draw your graphs carefully
• Label your axes
• Remember how to plot and read individual points
• Use professional conventions
• Learn to read slopes and areas from a graph

Draw your graphs carefully

When you work an assigned physics problem, you'll often be asked to "draw a graph." In my experience, many students view this as an end rather than a beginning — something the instructor asked them to do so they want to complete it. But often, the instructor is asking you to draw the graph as a part of building complex and rich reasoning about a physical situation. When you draw your axes, be careful about making them straight and  perpendicular. Sketch your graph lines to correctly represent the salient features of the problem: Where are the maxima and minima? Where does the graph cross the axis? Which parts of the graph are straight lines and which are curved?

You don't have to pull out a ruler and protractor every time you want to sketch a graph to help you understand something, but be careful about drawing reasonably straight lines, making right angles look like right angles, and marking scales evenly and appropriately. (Of course if you are creating a graph to be shared, you do want to use a ruler, protractor, or graphing program!)

The point is that some patterns in a physical situation are MUCH easier to pick up on when you see them in a graph or another visual representation. In a carefully drawn graph, the pattern you are seeking may just jump out at you. In a sloppy one, it will be hard to find, and drawing the graph will just be another tedious task instead of a way to a solution.

A carefully drawn graph is a valuable tool to help you understand a physical situation.

Label your axes

In math, you often only dealt with a single graph of the form, $y =f(x)$. There were two related variables — $x$ and $y$. Even in science classes, you would often be shown or create for yourself a single graph to represent a relationship or a data set. It was therefore OK to refer to the horizontal axis as "the x axis" and the vertical axis as "the y axis".

In this class, that's going to cause you a mess of trouble. In this class, we will often use multiple graphs of different things at the same time to describe a physical situation. You might be plotting the horizontal displacement of an object as a function of time. In that case, the horizontal displacement might be the vertical axis and the time the horizontal axis. You then could find yourself saying such confusing sentences as "We have x on the y axis and t on the x axis." And you might be trying to compare graphs of x vs t, v vs t, a vs t, F vs t, F vs x, and more, all at the same time. (See, for example, Sticky carts ADD LINK.)

So we'll call the traditional "x axis" the horizontal axis and the traditional "y axis" the vertical axis. (The real technical terms for these are "the abscissa" and "the ordinate" respectively. We will NOT use these terms.) To keep the graphs you draw as useful tools, be sure to label your axes!

We're going to have so many graphs for the same situation that not labelling your axes will get you totally confused! (Besides - not doing so might damage a personal relationship, according to Randal Munroe.)

Randall Munroe, xkcd.com/833/

Remember how to plot and read individual points

Remember that what each point on a graph is doing is showing a relationship between two variables. For example, if we are plotting the velocity ($v$) as a function of time ($t$), to plot the velocity 5.0 m/s at a time 7.5 s, we go along the time axis until 7.5 and draw a perpendicular straight up (dashed line). We then go along the velocity axis until 5 and draw a perpendicular straight over (dotted line).  Where they meet is the point on the graph that represents $(v,t)$ = (5 m/s, 7.5 s). Of course, if we have a point on a curve, we can run this procedure backwards to find the coordinates of a point on a curve.

While this may seem obvious, I have seen students fail to call on this knowledge time and time again when asked to draw or read a graph. Of course, drawing careful perpendiculars and marking the scales are necessary for this to work.

Use professional conventions (or class conventions)

There are two conventions for graphs that can cause confusion:

• Arrowheads
• Signs

Arrowheads

For some reason, a few decades ago school mathematics teaching picked up the convention of drawing arrowheads on the ends of axes to indicate that they go on forever. Sometimes this convention was extended to the curves that represented functions — arrowheads were drawn at the ends of the curve to indicate that it goes on forever. If you have picked up this habit, please break it. There are three reasons to do this:

• The professional convention used in scientific publications does not use arrowheads on the ends of axes or graphs.
• Many of our functions (and even our coordinates) do not go on forever, but only represent a limited range of variables.
• In this class, we want to use a different convention for to emphasize something other than "goes on forever": signs (namely, which direction has the positive sign, since directions will often be important).

Our class convention will be to ONLY use arrowheads when we want to emphasize with side of an axis represents the positive side.

Signs

For some students, learning to handle signs is challenging. In this class, we use signs to specify directions and to distinguish changes in various quantities, differentiating gains from losses. Sometimes, when we are plotting a graph, we will want to emphasize which direction is the positive direction on our axis. Although usually these will be "up and to the right", we will learn that we are free to choose our axes and that it is sometimes convenient to choose them at an angle or upside down. We will sometimes put an arrowhead on the positive end of an axis (never on the negative end!) to remind you which way is +. Look for it! (When you start creating graphs for published papers or professional presentations, you'll use the professional convention and drop the arrowheads entirely.)

Learn to read slopes and areas from a graph

In your calculus class, you learn that the derivative of a function is represented by the slope of its graph and that the integral of a function is represented by the area under its graph. In this class we will use slopes and areas on graphs as powerful tools to relate position, velocity, and acceleration or potential energy and force, for example. Check out the follow-ons at the end of this article for details.

Joe Redish 12/27/17