A Graphical Analysis Exercise: Analyzing the Motion of a Projected Marble

dropped and projected marblesGoal

Investigate quantitatively the horizontal and vertical motions of a projected marble.

Video Analysis

Download the video clip and view it. Then insert it into Logger Pro.

You'll do everything in Logger Pro. For convenience, here are the functions associated with Logger Pro video analysis.

Displays video analyis tool menu to right of video
Play, pause, step buttons
Set scale factor
Set axes
Mark points
  1. Open your Logger Pro file. Advance the movie to the frame that has a meter stick. Use this to set a scale factor. The numbers on the meter stick are centimeters.

  2. Advance the movie two more frames until the red and white marbles overlap. Now set the axes. Set the vertical axis to coincide with the string and the horizontal axis to pass through the bottom of the white marble.

  3. Starting with the frame in which the red and white marbles overlap, mark the positions of the white marble. (Ignore the red marble.) Mark the bottommost point of each image of the marble. (This point is picked because it's easier to locate than, say, the center of the marble.)

  4. When you're finished marking points, you'll have a data table and a graph of horizontal (X) and vertical (Y) position vs. time.

  5. Delete the Vx and Vy columns from the data table. Those columns will not be helpful.

  6. The default time scale assumed by Logger Pro is that of a typical 30 frames per second movie. However, this movie has a frame rate of 40.0 fps. Here's how to change the frame rate to that value. In the upper menu, click on Options -> Movie Options. Click Overide frame rate and change the number to 40.0. Click OK, and you'll see all the times in your data table adjust to the new value.

If Movie Options is grayed out on your computer, here is an alternative way to change the time scale:

  1. Select Data -> New Manual Column from the menu.
  2. Enter an appropriate name such as Time-act (for actual time) and units of s.
  3. Select Generate Values.
  4. For Start, enter 0, and for Increment, enter 0.025. For End, multiply 0.025 by (number of data points - 1). Click Done to generate the column.
  5. On your graph, click on the Time label on the horizontal axis and select your new time variable.
  6. Finally, delete the original Time column in your data table.
Another possibility is to upgrade your Logger Pro version in case you don't have the most recent version.
  1. Save your file. Then continue with the graphical analysis.

Graphical Analysis

  1. Do a quadratic fit on the x vs. t data (Analyze -> Curve Fit). When you do the fit, be sure to click the box, Create Calculated Column. Then, after clicking OK, immediately change the name of the new column in the data table to X-fit and enter the units. You can bring up the appropriate dialogue box by double-clicking on the column header.

Reflect on what you just did: You may think that a linear fit would have been more appropriate, since we expect the horizontal velocity component to be constant. By doing a quadratic fit, however, you put yourself in a position to check that assertion. You can look at the coefficient of the t2 term to see how close it is to the value we would expect.

  1. Repeat step 1 for the y vs. t data.

  2. Create the residuals for the x vs. t fit. Click on Data -> New Calculated Column. Enter the name, Y-res, and units. Then enter the formula to calculate residuals: "X"-"X-res". Click on Options and change the number of decimal places to 4. Then click Done to add the column to your data table.

  3. Repeat step 3 for the y vs. t fit.

  4. Click Insert -> Graph. This should insert the y-residuals. If it doesn't, click on the axis label to make the appropriate selection.

  5. Click Insert -> Graph again. Change the vertical axis label to X-res.

  6. Save your file. You've completed the fits and the residuals and are ready to interpret your results.

Interpretation

Click Insert -> Text. Then select Page -> Auto Arrange. Give answers to the questions below in the text box. Number them as given.

  1. Now you'll extract some numerical information from the fits. Which of the fit coefficients can you use to determine the vertical acceleration of the marble? Determine the value of the vertical acceleration.

  2. What is the accepted value of the vertical acceleration? Calculate the experimental error.

  3. What evidence do you have from the quadratic x vs. t fit that the velocity is constant? We're not looking for a statement that the line is straight. We're looking for evidence from one of the fit coefficients.

  4. Which of the fit coefficients gives you the horizontal velocity of the ball? What is the horizontal velocity?
  1. Examine your residuals. What is the largest residual (in absolute value)? How does this compare to a typical position measurement, say one about halfway through the clip?

  2. Do the residuals appear random? Based on your answers to that question as well as item 5, are the fits good?