Working content

In this activity, we will use the example of a ballet dancer performing a grand jeté ("big jump" in French) to explore how the motion of an actual object in physical space is modeled with mathematics through graphing both with spatial coordinate graphing -- graphs for the eye that show the path you see -- and graphing various quantities against time -- graphs for the mind that require interpretation to make physical sense.   We will take data from the video of Carrie Imler, from the Central Pennsylvania Youth Ballet, doing the jump. You can see the first frame of the video at the right. Carrie is just about to begin her jump. Note that a scale is shown.   This is typically carried out as a demonstration in a class as an introduction to taking data from video.  If you want to do this yourself, you need the program LoggerPro from Vernier Software. If your class is using this software in lab, you are entitled to have the program installed on your computer.  If not, you can download a 30-day demonstration version from the Vernier website.

You can do this yourself by right-clicking on the two files below and downloading them to the same directory of your computer.  If you then run Grandjete.cmbl it should imbed the video in it automatically.

• GrandJete.cmbl
• GrandJete.m4v

Setting up the coordinate system

In mapping a physical system to a spatial coordinate system, we have to go through the following steps:

1. Choose an origin
2. Choose an orientation of our coordinate axes (and positive directions)
3. Choose a measurement scale.

In the picture below, I've chosen the origin to be at the lower left of the screen, I've chosen the horizontal axis to be the x-coordinate with the positive direction to the right, I've chosen the vertical axis to be the y-axis with the positive direction upward, and I've chosen the scale to be meters. This is what it looks like. (The yellow dot identifies the "x-axis" and in the program you can grab it to change the orientation of the coordinate system.)

The movie is in the upper left, the data table that the program generates is right below it, and the graph of the data is to the right.  For now we have it set to just show one graph -- the spatial graph. When we take data by putting a point on the dancer's eye in each frame and transfer the set of data of her x-y positions to the graph, we get the figure below.

Note the table under the video.  The frame number has been converted into a time (using the frame rate of the camera) and the x and y positions are taken from the coordinate system that we set up.  (The fourth and fifth columns are the x and y velocities.  We won't talk about that yet.)

Note that the graph shows the same pattern and shape as we see on the picture. This is what we mean by "a graph for the eye".  But our graph doesn't tell us anything about how the motion took place in time -- just where she was, not when she was. If we want to understand more about the motion we need to make time graphs.  Answer the following question before you add these graphs to the program.

1. What do you think the graph of the y position vs time will look like?  From looking at the figure above, can you figure out what is the range of time you will need to allow for plotting and what is the range of y position you will need?  Discuss why you think it looks the way you do and how you figured out the relevant ranges.

2. Do the same for the x position.

Joe Redish 7/28/11