# Kinematic Graphs

## Graphs for the eye, graphs for the mind

We've talked about how we quantify space and time and how we create a kind of an abstract "map" of space by creating a spatial coordinate system that allows us to tell where something is. The track of a motion in a spatial coordinate system provides a graph for the eye, as it represents the track of what the eye would actually see. The picture below shows a spatial coordinate system used to describe the motion of a ballet dancer performing a grand jeté (big jump). The track of green dots is the position of her eye in subsequent frames of the movie. This is what the phrase "graph for the eye" means (our eye — not hers). The track shows the actual position of the object as it moves through space.

But often we are interested not just in the path, but in how the path evolves against time.  In this case, we would construct mathematical graphs using one of the position coordinates (or a velocity, acceleration, or force) and plot it against time. If we plotted the x data of this graph as a function of time we would get the graph at the left. (Note that there is a suppressed zero on the vertical (x) axis.)

This does NOT look like what your eye sees directly. To make sense of it, you have to translate — interpret what the graph is telling you into physical meaning. It says that at the start of the film (taken to be t = 0 s), her eye is about 2.7 m to the right of the 0 of the x axis, and as she moves, the x-coordinate of her eye decreases (she is moving to the left — towards the origin) at approximately a uniform rate.

Often, when you draw graphs in math classes you are graphing some function of an independent variable called x. So the equation graphed is "$y = f(x)$" and your horizontal axis is called the "x axis" and the vertical axis the "y axis". This is very bad practice, as you can see from the graphs above. That notation works OK for the first graph (the spatial coordinate system) but does not for the second — since the "x axis" is really "t" there and the "y axis" is really "x".  Calling the horizontal and vertical axes "x and y" is going to be immensely confusing since only in a small number of cases will we actually be plotting "x and y".  Get used to calling them the horizontal and vertical axes. (The correct technical name for the horizontal and vertical axes is "abscissa" and "ordinate" but these terms are not used very often anymore.  Note that for "x-y", "horizontal-vertical", and "abscissa-ordinate", each pair is in alphabetical order. That's a useful memory trick.)

Note that we've chosen the origin of the coordinate system to be at the lower left so the dancer is always in the first quadrant and her x and y coordinate values are always positive. And we've chosen the x axes to be horizontal and the y axis to be vertical. But we are free to choose our coordinate system arbitrarily, with any origin and any orientation of the axes. Think about how our graphs would change if we made different choices.

Joe Redish 7/25/11

Article 311