Physics Data and Graphs

Much of our time in this class will be used in collecting and analyzing lab data. Almost all of the time, we will use a graph to analyze the data. Different types of graphs are appropriate for different situations; below is a brief summary of the graphs we will use in physics (and in most sciences).

Histogram

This type of graph is very unusual in a physics class. It is usually used to compare amounts in various categories that do not make a sequence. For example, the histogram here shows the types of cars driven by various types of teachers. There is no way you could make a sequence out of the categories, so we use a histogram.

I only put this graph here because some of you will want to draw one. I cannot think of a time in our class when we will use a histogram. Don’t do it.

Line Graph

While common in economics classes, line graphs are unusual in physics, except during the first month of the course. We will use these graphs to show idealized distances, speeds, and accelerations when doing kinematics problems. Draw this type of graph when you have a simple kinematics problem that gives idealized data. You will not draw this type of graph to analyze data taken from an experiment.

Scatter Plot

This is the most common type of graph used. 99% of the time, when I say to draw a graph, this is the type of graph I mean. Physicists and other scientists use these graphs as a tool to analyze their data, not just as a display method for their results. This type of graph has data points individually marked on the graph, without any connecting lines. This way you only show the actual data points you took during the experiment. (In physics, time is never on the vertical axis of a graph. Time is the ultimate independent variable because we have no control over it. So when time is one of the values you are graphing, it always goes on the x-axis.)

Notice that the points do not all line up on a simple line. Whenever you do an experiment, there is some amount of uncertainty. But we always assume that there is a simple relationship underlying the complex data. To get at that simple relationship, we use the graph to determine a best-fit line. The equation of the best-fit curve then gives us what we take to be our simple relationship.

There are many ways of determining a best-fit line for a collection of data. The simplest way is to just use a ruler to “eyeball” a straight line that comes as close to as many of your data points as possible. Then you can use the line on your graph paper to calculate the slope and y-intercept for your best-fit line. You can also use computer or calculator programs to mathematically figure a best-fit line. I’ll show you some of those programs as we move through the year. In any case, your new graph will look something like the second graph here.

Once you have the best-fit line and the slope/intercept equation, you are not finished. The most important part of your analysis is to translate from the math equation into a physics relationship. That means to replace the x and y variables with the physics quantities they represent, and then figure out a physical meaning for the slope and y-intercept. In the graph above, the equation given by Excel is y = 1.3x + 0.64.

  • Step 1: Replace x and y with physical variables (we’ll use v for speed and t for time).

v = 1.3t + 0.64

  • Step 2: Figure units for the slope and the y-intercept. The units for the y-intercept will be the same as the units for the y variable. In this case, the units will be m/s, meaning the y-intercept (0.64) has to be some kind of speed. For the slope, which was calculated by using rise over run, the units will be m/s over s, or m/s/s.

v = (1.3 m/s/s)t + 0.64 m/s

  • Step 3: Come up with a physical interpretation of the slope and y-intercept. The y-intercept is usually easiest; it’s the value of the y variable when the x variable is zero. In this example, the y-intercept (0.64 m/s) is the speed of the object at time = 0, or when you started your stopwatch. In other words, it is the initial speed of the object. (We’ll call that v0.) For the slope, it’s usually some kind of rate of change. In this case, it’s the rate of change of speed with time. We can give it a name if we want, or we don’t have to. In this case, let’s call it a. So our final relationship is

v = at + v0

where the initial velocity v0 is 0.64 m/s and the rate of change of speed a is 1.3 m/s/s.

Suppose, though, that your graph is obviously not a straight line (like the one shown here). How can you analyze the graph then? The trick is to change one of the axes of the graph so that your new graph does look like a straight line, then analyze it just as we did above. In the graph shown here, the data points are curving upward, meaning that the distance is increasing faster than the time. In other words, when the time doubles, the distance more than doubles. In this case we need to increase the time axis somehow so that it “keeps up with” the distance. You might have to try several graphs with different time axes, but eventually you’ll find that if you graph distance vs. the square of the time, the line will come out straight. Then we analyze that graph as usual, remembering that the x-axis is not time in this case, but the square of time.

A useful tool for trying out different axes is a web-based graphing tool available through the course website. Go to and click on GraphFit.