## Physics Laboratory 2

### Free-fall - 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 nuclear forces, 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 of mass m equals Fg = mg.  It is constant and points straight down.  If we can neglect other forces and the net force is approximately equal to Fg, then we have motion with constant acceleration g.

Open a Microsoft Word document to keep a log of your experimental procedures and your results.  This log will form the basis of your lab report.  Address the points highlighted in blue.  Answer all questions.

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).  If you followed the instructions above, the the y-axis points down.

Produce a graph of position versus time.  Label the axes.

• Describe your graph?  Does it resemble a straight line?  If not, what does it look like?
• Was the ball moving with constant velocity?  How can you tell?

Let us find the velocity of the ball as a function of time.  We find vy =  ∆y/∆t by dividing the difference in successive position by the difference in the times the ball was at those positions.

• Type 0 into cell C2.  The ball starts from rest, its initial velocity is zero
• Type =(B3-B2)/(A3-A2) into cell C3.  Copy the formula into the other cells of column C, down to the second-to-last cell.

Produce a graph of velocity versus time. Label the axes.

• Describe your graph?  Does it resemble a straight line?  If not, what does it look like?
• Right-click your data in the velocity versus time graph and choose "Add Trendline".  Choose "Type, Linear" and "Options, Display equation on chart".  An equation y = ax + b will appear on your graph, where the number a is the slope and the number b is the y-intercept.  What is the physical meaning of the slope of the velocity versus time graph, if the graph is a straight line?
• Paste your velocity versus time graph (with trendline) into your log.
• What value do you obtain for the acceleration of the ball?  How does your experimental value of the magnitude of the acceleration compare to the accepted value of the magnitude of the acceleration of a free-falling object?
Reminder: percent difference  = 100%* |accepted value - experimental value|/accepted value
• What factors do you think may cause your experimental value to be different from the accepted value?  In other words, what are some possible sources of error?

For motion with constant acceleration we expect that y changes as a function of time as y = x0 + v0t + ½at2, where a is the acceleration. For an object accelerating at a constant rate g we have y = y0 + v0t + ½gt2, so y as a function of t is a polynomial of order 2 (a section of a parabola).  We can reduce numerical errors in finding the acceleration of the ball by fitting our position versus time data directly with a polynomial of order2.

• Right-click the data in your position versus time graph and choose "Add Trendline".   Choose Polynomial, Order 2 and under options 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.  Since we are plotting y versus t, the number b1 is the best estimate for g/2 from the fit.  Therefore the value of the acceleration determined from the fit is g = 2b1.  Since our y-axis points downward, we expect a to be close to g = 9.8 m/s2.
• Paste your position versus time graph (with trendline) into your log.
• Does the polynomial of order 2 fit the data well?  What value do you obtain for the acceleration of the ball from this fit?
• 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 also 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.
• The uncertainty in g from statistical analysis is 2 times the uncertainty in the coefficient b1. What value did you obtain for the uncertainty in g?