Goals

1. to investigate the acceleration of a ball in falling a long distance through air
2. to compare the motions of two balls, differing only by weight, when dropped at the same time from the same height

Prelab:  Do the prelab assignment indicated in the Weekly Schedule.

Submitting Files

In the following, you'll be creating two electronic files. At least one of these will be submitted to the instructor. We'll provide instructions later on how to do this. You must name your files according to the convention given in the instructions. With this convention, the instructor will know--simply by looking at the filename--the lab being submitted and the person submitting it.

Always use the proper naming convention for files that you submit to the instructor.  This helps the instructor keep track of your electronic work. If you don't use the convention, you may receive a penalty when the teacher asks you to resubmit your file. For future reference, the file-naming convention is given here as well as on the Weekly Schedule.

Software

This lab uses Tracker and Graphical Analysis 3.4. You should have already installed these programs on your computer.

About the instructions below

We're providing very detailed instructions, and it's important that you read carefully and follow the steps.  We do this for two reasons:  1)  to teach you how to use software that is probably new to you, 2) to teach you how to use a standard analysis technique in physics.  In future labs, we'll be less specific and directive about these things, because we'll expect you not only to know how to use them but to know when to use them.

Marking the video clip

These instructions were written for Tracker 2.0.

1. Open Tracker by double-clicking on Tracker.jar on your computer. To open your movie clip, go to File -> Import. Navigate to the movie falls99_17c.mov that you downloaded for the prelab and click OK. The first frame of the video clip should appear inside a window.

2. View the movie by pushing the play button on the left of the bar at the bottom of the movie window. Several frames may be played at the start before the ball starts to drop. Rewind the movie by dragging the slider at the bottom of the window all the way back to the left. Also try using the forward and backward step buttons to view the video frame-by-frame. Note the button in the lower left of the movie window that indicates the frame number that is currently displayed in the window.

3. Now you'll need to clip off the frames of the movie at the beginning before the balls appear at the bottom of the trap door and at the end after the first ball has hit the ground.  Step through the clip as needed to decide on the first and last frames that you want to use.  In order to trim the movie to those frames, click on the clip settings button (looks like a small film strip) at the lower right corner of the movie window.  Enter the Start and End Frame numbers.  Make sure the Step Size is 1 and the Step dt is 0.033 s. Play the movie again to make sure it's trimmed the way you want it.  If not, you can change the clip settings.

4. Now you'll analyze the video clip by marking the positions of the falling objects in each frame of the video. On the menu at the top of the screen, go to Tracks ->  New -> Point Mass and create a point mass. Change the name from mass A to something more descriptive of the object that you will mark such as mass left. You can also change the marker symbol and color.

5. Create another point mass for the other ball, and change its name appropriately.

6. To increase the precision with which you can mark the video, enlarge the video by right-clicking in the movie window and selecting Zoom. Zoom as much as possible, without blurring the images so much that it is hard to see the features that you want to mark. Think carefully about what part of each object you should mark.  Develop a consistent technique.

7. To mark an object's position in a given frame, first click on the object in the Track Control window. Then hold down the control key. A symbol for marking the object's position will appear. While holding down the control key, use the mouse to move the symbol to the location you want, and left-click to place the marker. Once you have placed the marker, you can drag it around the screen if you want to adjust it. Mark the positions of one of the balls in all of the frames.  You may have to move the vertical slider down in order to mark all the frames. Then click on the other object in the Track Control window, rewind the movie to the first frame, and mark the positions of the other ball.  The marker color should change when you switch to marking another object.

8. Before going on, save your data.  It's a good idea to do this frequently so that you don't lose all your results in the case of a program crash.  In the main menu, select File, Save As. Give the file the unique name L04-WAusername.trk, where you replace WAusername with your WebAssign username. The latter is unique to each student and therefore eliminates the possibility of two students submitting files with duplicate names.

9. Zoom out to see all the points that you've marked.  (Remember, Zoom is in the right-click menu.)

10. If you can't see your data table right now, drag the right edge of the movie window towards the left, and Data Table and Plot windows should become visible. The data table should contain time, x-, and y-coordinates. Make your coordinate axes visible by going to Tracks -> Axes -> Visible in the main menu. Move your origin to a meaningful spot (for example, the initial position of the left ball) by left-clicking on the origin and dragging it.  Zoom back in so that you can position the origin accurately.

11. Notice that the space coordinates are in pixels, not meters. You need to create a scale for your video. To do this, make your tape measure visible by going to Tracks -> Tape Measure -> Visible in the main menu.  Zoom out so that you can see the entire video frame.  To the right of the right ball, you'll see a vertical series of white points that don't change position as the balls fall.  These are markers placed 1.00 m apart for the purpose of providing a standard distance scale.  Drag the ends of the tape measure in turn to the highest and lowest markers.  Zoom in to position the ends of the tape measure more accurately.  Now create the scale by clicking on the tape measure's magnitude in the middle (this may be difficult to see) and entering the actual distance in meters. Now your data should have meaningful numbers. Look at the Data Table window to check this. Are the X-coordinates nearly constant? Do the Y-coordinates increase from 0 up to a distance that makes sense? Is the total time of fall reasonable? 

12. Once you're satisfied with your results, save your trk file one more time.  However, keep the Tracker program open for later use.

Transferring the data to Graphical Analysis

These instructions are for Graphical Analysis 3.4.  You're expected to use this version of the software.  If you're using Logger Pro, see the note at the bottom of the page.

1. Although Tracker has some limited analysis options, Graphical Analysis for Windows is better for this. Open Graphical Analysis (GA). Now go to Tracker. Make sure the right mass is selected at the top of the data table. Click and drag the upper boundary of the table up so that you can see all the data without using the slider. Then click the first entry in the time column of the data table and drag all the way down. Copy the column either by right clicking and selecting Copy Data or by using CTRL-c.

2. Back in GA, click on the first cell of the X column of the data table.  Then right click and select Paste (or use CTRL-v).  The time data should be copied into the table. Now go back to Tracker and copy the y column of data. (We're not using the x column.) Paste the y data into the the Y column of the data table in Graphical Analysis.

3. GA isn't smart enough to know how to label the column headings.  It uses generic X and Y labels that don't correspond to your X- and Y-coordinates.  For example, GA's X is actually time.  You have to change the names and assign units and significant figures.  In order to do this, double click on the X column heading.  Enter the full name of the variable (Time), the short-hand symbol (t), and the units of measurement (s).  Then click on the Options tab.  Under Displayed Precision, select the appropriate number of decimal places.  For time, this number is 3, since the frame rate of a video clip is known that precisely. 

4. Now change the properties of the Y column. The variable name must distinguish it from the other ball. For reference in the instructions to follow, we'll be using the name Y-Right. The shorthand could be, say, Yr. (What should the Displayed Precision be?)

5. Create an additional column in the GA data set as follows:. Select Data -> New Manual Column.  This column will be for the left ball y-position data. Therefore, enter the names and precision appropriately.

6. Go back to Tracker, change to the left mass, and copy the y data. (You don't need to copy the time data again, because that's the same for both balls.) Now paste that data into the corresponding column in GA.

7. Save your Graphical Analysis file the same form as for the Tracker file:  L04-WAusername.ga3.  The only difference is that the extension will be ga3 rather than trk.  Save your work frequently hereafter.

Interpretation: Fitting the data with a free-fall model

You may close Tracker if you wish.  You'll be using Graphical Analysis for the remainder of the analysis.

1. A graph of Y-Right vs. Time for the right ball should already be plotted. If you need to change the variable plotted on an axis, simply click on the axis label and select the variable you wish.  Note now that the data points are connected with lines.  Connecting the dots like this is a poor scientific practice, and it's surprising that this otherwise excellent program uses that as a default.  Always remove these connecting lines.  Here's how:  Double click on the line on the graph.  In the window that appears, uncheck Connect Points.  While you have the Graph Options window open, give the graph a title:  Y-Position vs. Time for the Right Ball.  Make sure the title identifies the object.

2. Now consider what relationship you would expect the data to obey.  If the ball is falling freely (negligible influence of air friction), the physics equation y = y_o + v_ot + ½at² should apply.  That's a quadratic equation, so it would make sense to apply a quadratic fit to the data.  In order to do this, go to Analyze -> Curve Fit.  Under General Equation, select Quadratic.  Then click Try Fit.  The coefficients of the fit will be displayed.  Click OK, and the fit results will be displayed on the graph.  A line will also be drawn.  This line is acceptable, as it is a graph of the equation of fit.  Rather than simply connecting the points, it displays the quadratic function that is the best fit to the data.

3. A box on your graph displays the equation of fit in algebraic form and also gives the values of the numerical coefficients.  There is also something called RMSE or root-mean-square error which gives you a way to determine how good the fit to the data is.  The smaller this value, the better.  However, we'll see that's there's a better way to examine the goodness of fit. Something that Graphical Analysis doesn't do is give the units of the coefficients or round them to the appropriate number of significant figures.  That's up to you, and it's necessary that you do those things whenever you fit data. 

In order to help with that process, let's make a connection between the equation of fit and the physics that you learned in Chapter 2.  First, the algebraic form of the equation of fit is y = At² + Bt +C.  We've put these symbols in the matching table below in the Math column.  What is the corresponding Physics equation?  We already gave it in step 3 above:  y = y_o + v_ot+½at².  y_o corresponds to the coefficient of the t⁰ (=1) term.  That must be c.  v_o is the coefficient of the t¹ term.  That must be b.  We have to be careful on the next one, because the symbol a is used in two different ways.  In the math equation, it's the coefficient of the t² term.  But in the physics equation, the coefficient of the t² term is ½a, where a represents the acceleration.

Math	maps to	Physics	Value (rounded)	Units
y	-->	y	variable	m
A	-->
t	-->	t	variable	s
B	-->
C	-->

This should give you enough introduction to allow you to complete the table above.  Fill in the empty cells in the Physics column.  Then enter the values of the coefficients of fit taken from your graph, but be sure to round them appropriately.  You'll need to use some judgment based on what the coefficients represent and what data determines them.  For example, y_o represents the initial position of the ball.  Positions were measured to the nearest millimeter or 0.001 m.  So that should tell you how far to round off the c coefficient.  b is the initial velocity of the ball.  Velocities are ratios of position and time.  These values are given to 3 or 4 significant figures in your data table.  So you would expect velocities to have 3 significant figures.  Similar reasoning applies to the coefficient of the t² term.

After entering the rounded values of the coefficients, enter the units.

You're now prepared to write the complete physics equation of fit.  You'll do this in the notes window in GA.  This is the white space below the data table.  Write your name at the top of the notes window.  Type in your matching table as best you can.

Tech Tip.  You can use keyboard shortcuts to copy and paste.  Here's how to do this in Windows operating systems.  Select the item you want to copy. Hold down the Control key and type c. To paste the item that you've just copied, first position your cursor where you want to paste the item. Hold down the Control key and type v. Here's more information on keyboard shortcuts.

As you make additional entries in this window, double-space between entries to make it easier to read your work. Now copy and paste this equation under your name in the notes window:

y = y_o + v_ot+½at²

Double-space and then paste the equation again.  In this second equation,  replace the coefficients with rounded values and units from your matching table.  It will look something like this when you're done:

Y-Right = (# w/units) + (# w/units)t+(# w/units)t²,

where of course you replace "# w/units" with specific values and units.  This completes your equation of fit. You've related it to the real world by putting it in physics form and using numbers acquired from measurement. 

Whenever you obtain equations of fit from data in the future, we expect you to follow a procedure like the one you just carried out.

4. You're now in a position to determine the acceleration of the right ball from the coefficient of the t² term.  This coefficient is ½a, where a represents the acceleration.  Knowing the value of ½a, what then is the acceleration?  Write your result under your fit equation in the notes window and tell how you determined it.

5. Now you'll examine the goodness of fit. Create a new calculated column for the equation of fit by using the Data -> New Calculated Column command. Enter the name Yr fit and shorthand name the same. Next type in the equation. Note that GA uses the term equation loosely. What you type in the window is just the right side of the equation, which will be C+ B*"Time"+A*"Time"^2 with C, B, and A replaced with the numerical values (without units this time). Use the symbol * for multiply and ^ to raise to a power. The variable is the time. You can enter it into the equation using the Variables button. When you select Done, the new column should be created. Check to make sure the values make sense. They should be close to the measured values of Y-Right.

6. Next create one more calculated column for the difference between the measured Y's and the calculated Y's: Y-Right - Yr fit. These are called residuals. Label them Res Right.

7. Insert a new graph window using Insert -> Graph. If the graph of Res-Right vs. Time isn't automatically plotted, select the appropriate variables by clicking on the variable names on the graph. Use the Page -> Auto Arrange command to arrange the windows in your display so that they all show.

8. Save your file before continuing with the data for the left ball.

9. Insert a new graph and plot Y-Left vs. Time. Now jump back up to step 1 of this section and repeat the entire analysis for the left ball. When you're done, auto arrange the page so that all objects can be seen without overlap. Your page should have the following:  1) data table, 2) notes with matching table, equation of fit, and acceleration for each ball, 3) position vs. time and residuals vs. time graphs for both balls. Double check that all columns and graphs are titled appropriately and connecting lines (but not lines of fit) are removed from data points.

10. Go to Page -> Page Options and name the page Data. Save your file before continuing with the interpretation.

Discussion

Create a new page in GA using the command Page -> Add Page.  For the title, write Discussion and type OK.  A blank page will appear.  Note how you can switch between the two pages you've created by using the drop-down box in the menu bar.  Back to the Discussion page, select Insert ->Text.  A new notes box will appear.  Resize and drag it to fill the screen.  Here's is where you'll type the answers to some questions and write a summary.  Number your answers the same as the questions.

1. Estimate the uncertainty in your measurements of position. Justify your choice. [Here's a strategy: Take another look at your Tracker file. How closely (to the nearest pixel) were you able to position the cursor to mark the position of the ball? Note that you can see the individual pixels when you zoom in all the way on the video. How big is a pixel in meters? You can estimate that by comparing the side of a pixel to the diameter of a racquetball.]

2. Examine the residual graphs for both balls. Generally, a fit is considered good if 1) the magnitudes of the residuals are comparable to the uncertainties in the measurements and 2) the residuals show no pattern but rather are randomly oriented on either side of the horizontal axis. Using the residual graphs and your estimate of uncertainty as evidence, discuss and compare the goodness of fit for the two balls.

3. What is the evidence from your analysis that supports the statement that the right ball falls with uniform acceleration?

4. Is the value of the right ball's acceleration what you would expect? Explain.

5. Does the left ball fall with uniform acceleration? Explain your answer.

6. Do you think the conclusion in #3 would hold up if the right ball were allowed to fall a much greater distance?  Explain.

7. In creating a scale in Tracker, you used the highest and lowest markers. That was recommended because there's usually greater accuracy in larger measurements. In order to demonstrate this statement with numbers, let's suppose that the uncertainty in positioning the markers is 5 cm = 0.05 m. What percentage is 0.05 m of the total distance of 14.0 m? If you measured the distance between 2 adjacent markers, what would the percentage be?

Conclusion

Below your Discussion, write a conclusion for the experiment.  Recall the characteristics of a good conclusion:

"This is where you summarize what you did and state what you found out.  In summarizing what you did, give an overview of the method used to obtain and analyze data.  Include a description of the methods used for acquiring and analyzing data.  Always make it clear whether or not you achieved the goals of the lab.  Be very specific in stating what you found. This may include final numerical values."

Submitting your file

Before submitting your file, make sure that all your labels are clear and unambiguous and that all your work is present. The instructor should only have to toggle between two GA pages or scroll windows in order to see all your work.  When you're satisfied, submit your Graphical Analysis file according to the instructions provided by the teacher. Do not submit your Tracker file at this time. The instructor will ask for the file if it is needed to corroborate your data.

Note about using Logger Pro

While you may use Logger Pro for Graphical Analysis, it's essential that you save your files in GA3 format. Here's how to do that. From the File menu, select Save As. For filetype, select Graphical Analysis 3 (*.ga3). Here's the really important part: When you type in the filename, append .ga3 to the end. If you do not do this, Logger Pro will append the extension .cmbl. In the latter case, you will be required to resubmit your file, and this will result in a penalty.