Falling

Purpose: to observe Galileo’s law of falling, also called constant acceleration.

Method: use video to find position as a function of time. Analyze with straight line plots.

Galileo spent years trying to model how things fall. In his final, correct, formulation, he realized that the distance H fallen in time t from rest is

He discovered that this vertical motion takes place in exactly the same way if the falling object is also drifting along horizontally. In this case the object falls in a parabolic arc. The special case of dropping straight down is the case in which the horizontal component of velocity is zero. Using video and image analysis, we should be able to see exactly what this means.

More generally, Galileo’s description means that the parabolic trajectory of a falling object in terms of horizontal and vertical Cartesian coordinates (x,y) and the time t is

Here the significance of the constants x0 and y0 is just the position of the object at the time t=0. We can, if we like, choose the origin of coordinates to be the location of the object at t=0. We will see below how to do this in practice. By this good choice we make x0=y0=0. If we do this, we expect the superficially simpler behavior

How can we tell if the data support this prediction?

It is easy to test the first one, which says x is proportional to t. Just graph the data and see if they fall on a straight line (through the origin). If they do, then the relationship is verified, and the slope of the line gives the constant of proportionality, v0x, which is the horizontal component of velocity.

The second relationship is not linear, however. If we graph the data, they will suggest a curve, but how are we to know if this is a quadratic curve, as predicted, or some other curve? There is no easy way to tell. By contrast, it is easy to tell if a line is straight. Therefore we consider taking the data and forming the combination y/t (for all the data points after the first, since 0/0 is undefined). The relationship above is the same as

And this is a linear relationship! It says y/t is a linear function of t, with slope –g/2 and intercept v0y. Thus it is easy to check experimentally. Graph y/t vs t and see if it is straight. If it is, then the slope of the resulting line determines g, a constant that is the same for all falling objects.

(Note: although y/t has the dimension of velocity, it is NOT the vertical component of the velocity of the ball. It is just a convenient combination for data analysis. The intercept in your plot of y/t vs t is the velocity at time t=0, so that particular value of y/t has a meaning.)

The main steps to doing this are

(1) Use the image analysis program Tracker to get (x,y) vs t data.

(2) Transfer the data table to the program Excel for manipulation and graphing.

(3) Create the data column y/t in Excel.

(4) Plot x vs t and y/t vs t.

(5) Fit y/t vs t with the best straight line and find its slope.

Details on how to use these two programs are found below. Excel, in particular, is a general purpose computational program that is worth learning. Tracker is more specialized, but it is easy and fun to use. We will use it at least once again in the future.

Open tracker. http://www.cabrillo.edu/~dbrown/tracker/webstart

On the home page of Tracker, click BallToss.

A clip should load of someone’s hand throwing a small white ball in front of a long black banded white stick. The green arrow plays the clip. If there are marked green positions already on the screen, from someone else’s work, get rid of them by clicking the green ball in the Track Control, then unchecking the “Visible” box. The green marks should disappear.

Now we have to make some choices to make the data analysis easier. In particular we have to be sure the clock says t=0 when the motion starts. In the lower right hand side of the screen there are four buttons. Use the two arrows that look like the play buttons on a DVD player to advance through the clip frame by frame. Find the frame where the ball has just left the person’s hand. Now go to the button on the far right that looks like a piece of film, and click on it. This opens Clip Inspector. Enter the FRAME NUMBER for this frame where it says START FRAME. (The frame number can be found at the lower left of the video window.) To be sure that we don't skip frames, make the STEP SIZE=1, and not some bigger integer. Finally, to be sure the data taking doesn't end prematurely, make STEP COUNT=100, although we won't actually take that many steps. The time assigned to this frame of the video will be now be t=0, important for what we do later. Click OK!

In Track Control, click on the button labeled New and select Point Mass. You will see a new button that says Mass A (or perhaps B or C depending on how many times you have done this).

You may have already noticed the yellow box in the lower left hand corner that says “Shift-click to mark mass A” (or whatever you named it). The box is telling you to find our object of interest (the white ball), put the cursor over it, hold down the shift key and left click as close to ball’s center as possible.

Once you have finished zooming in on the clip, shift-click on the center of the white ball. When you shift-click, a colored shape will appear and you will be advanced to the next frame. Continue shift clicking until the center of the ball is no longer in view. If there are more frames past that point, ignore them. Notice how the ball has fallen in a nice parabolic shape, as it was claimed in Equation 1.1.

If you were at all concerned as to where the data is, look carefully at the right side of the screen. The edge has those grey dots that generally mean there is something hiding there. Position the mouse so that you see the double ended expansion arrow, and drag open the window. Hopefully a graph and a table of data are appearing. Once the window is large enough for your comfort, take a quick look at the data. The time column is in seconds, while the x and y columns are in meters. Check to be sure the first point is at time t=0.

IMPORTANT: We must choose axes so that the initial position of the ball is the origin! Recall that we are testing the law of falling in the simpler form above, Equations 1.2 and 1.3, where this choice has been made. For this purpose we must set the axes. Move back up to the tool bar, but this time click on the two red lines that are crossing each other forming axes. The axis will appear on the main screen, but not in a convenient way. You must move it so that when t=0, x and y also equal 0. You can drag the axis around by grabbing the center of it with the mouse and holding down left click. Move the axis so that the first row in your data table contains only 0’s. Putting the axes in this particular spot reduces x₀ and y₀ to 0, so that the theoretical equations of motion can be reduced to the simplified form, as shown with Equations 1.2 and 1.3.

(2) Open up a new Excel file. You’re going to paste the data from Tracker into Excel. Select the data in Tracker's Data Table using the mouse. Next right click on the Data Table, and choose Copy Data. Then go to your Excel file, right click and choose Paste. You should see three rows that are identical to the ones in Tracker. In the t, x, and y headers add in the units (seconds and meters). Always keep track of your units. It seems superfluous now, but if for any reason you have to retrace your steps, you will be very glad you left yourself notes.

(3) Create a new column of data; label it y/t with units of meters/second. You don’t have to go through and divide y by t by hand. Excel will do it for you, if you ask it right. If you don’t already know the magic words, see the Calculations section of the Excel handout.

(4) Once you have a column with y/t, you need to make three plots. The first will be x versus t, then y versus t, and finally y/t versus t. If you need directions on how to make a plot, see the Plotting section of the Excel handout.

(5) The experimental proof that the path of the ball is a parabola is that y/t versus t is a straight line. If it came out straight, fit a line to the y/t versus t plot and find the slope. In this way you can determine the acceleration due to gravity, g. How to fit a line in Excel is also in the Excel handout, in the “Putting a fit a line on your graph” section.

In order to check that the units are right, Tracker has conveniently placed a white meter stick in the video, with bands every 10 centimeters.

In the upper left toolbar there is a blue two headed arrow with a 10. That will create a Ruler. Click on it, and line up the Ruler with the white stick with black bands. The ruler will be horizontal at first, but you can grab the arrow tips and pull them up to your points of interest and the orientation will adjust accordingly. Measure from the first black band to the last black band. That distance is 1 meter. Make sure that it really is 1.00 in Tracker's units.