Freefall - The acceleration of gravity

Gravity is the force of nature we are most aware of.  One can argue that other forces, such as the electromagnetic force, which holds molecules together in solid objects, or the nuclear force, which determine the structure of atoms, are more important, but these forces are less obvious to us.  Near the surface of Earth the force of gravity on an object is constant and points straight down.  If we can neglect other forces, then we have motion with constant acceleration g = 9.8 m/s2 downward.

Observation:

Hold a tennis ball at about your height and then let go.  Observe the motion of the ball.

 Describe the motion as the ball is falling. Estimate how long it takes the ball to reach the floor. What can you say about the speed of the ball as a function of the distance it has already fallen? If you drop the ball from about half of your height, does it take approximately half the time to reach the floor?

Experiment:

It does not take the ball a long time to reach the floor.  It is hard to get detailed information about its motion without using external measuring instruments.  In this experiment the instrument is a video camera.  You will analyze a video clip.  The clip shows a person dropping a ball.  You will determine the position of the freely-falling ball as a function of time by stepping through the video clip frame-by-frame and by reading the time and the position coordinates of the ball off each frame.  You will construct a spreadsheet with columns for time and position and use this spreadsheet to find the velocity as a function of time.  The slope of a velocity versus time graph yields the acceleration of the ball.

Procedure:

To play the video clip or to step through it frame-by-frame click the "Begin" button.  The "Video Analysis" web page will open.  Choose one of the ball_x.mp4 video clips.

## Begin

"Play" the video clip. When finished, "Step up" to frame 0.  In some browsers you have to click "Pause" first..

In the setup window choose to track the y-coordinate of an object.

Click "Calibrate".  Then click "Calibrate Y".
The video clip contains calibration markers.  Each black and white stripe is 10 cm = 0.1 m wide.  Position the cursor over top end of the highest black stripe and click the left mouse button.  Then position the cursor over the top end of the lowest black stripe, 80 cm below, and click the left mouse button again.  This will record the y-coordinates of the chosen positions.  Enter the distance between those positions into the text box in units of meter.  For the example positions, you would enter 0.8 into the text box.  Click "Done".

Click the button "Click when done calibrating".  A spreadsheet will open up.  Click "Start taking data".

Start tracking the ball.  Position the cursor over the ball.  When you click the left mouse button, the time and the y-coordinate of the ball will be entered into the spreadsheet.  You will automatically step to the next frame of the video clip.

Repeat for each frame in the video clip as long as the ball is moving downward.  Then click "Stop Taking Data".
Highlight and copy your table.  Open Microsoft Excel, and paste the table into an Excel spreadsheet.  Your spreadsheet will have two columns, time (s) and y (m).

Right-click column B and chose to insert a column.  Label the new column time^2.
Into cell B2 type = A2^2.  Copy the formula into the other cells of column B.  The entries in column B now are the squares of the entries in column A.  (To copy a formula, position your cursor in the cell that contains the formula, choose copy from the menu bar, highlight the cells that will receive the formula, and choose paste from the menu bar.)

Produce a graph of position versus time.
 On the Excel menu bar click Insert, Chart, XY (Scatter), and pick one of the subtypes. Right-click the chart and choose Select Data, Add. Position your cursor in the X-Values text box, and highlight all the entries in the time column. Now position your cursor in the Y-Values text box, erase any entries in this box, and highlight all the entries in the y (m) column. Type Position vs Time into the Name text box. Select your chart, chose Layout or click the + button next to the chart, give the chart a title, and label the axes.  The label for the x-axis should be "Time (s)", and the label for the y-axis should be "Position (m)".

Right-click the data in your position versus time graph and choose "Add Trendline".   Choose Polynomial, Order 2. and click "Display equation on chart".  An equation of the form y = b1x2 + b2 x + b3 will be displayed where b1, b2, and b3 are numbers.  For motion with constant acceleration we expect that y changes as a function of time as y = x0 + v0t + (1/2)at2, where a is the acceleration.  So we expect that the coefficient b1 from the trendline fit is equal to (1/2)a.

We now want to find the uncertainties in the fitting parameters.  On the menu bar click click data, data analysis, regression.  For the input y range choose the data in column C.   For the input x range choose the corresponding cells of columns A and B. Under output options check new worksheet, and under residuals line fit plots. Click OK.

The regression function finds the best fitting polynomial of the form y = b3+ b2x + b1x2 for your data.  Under SUMMARY OUTPUT, Intercept, you will find the coefficient b3.  Under SUMMARY OUTPUT, X Variable 1, you will find the coefficient b2, and the standard error in this coefficient from the fit.  Under SUMMARY OUTPUT, X Variable 2, you will find the coefficient b1, and the standard error in this coefficient from the fit.  The errors are due to uncertainties in the measurements and are computed using statistical analysis.
For motion with constant acceleration we expect that y changes as a function of time as y = x0 + v0t + (1/2)at2, where a is the acceleration.  So we expect that the X Variable 2 from the fit is equal to (1/2)a.  We expect the magnitude of 2 times X Variable 2 to be equal to that of the gravitational acceleration g within experimental error.

Open Microsoft Word and prepare a report using the template shown below.