## Studio Session 4

### Motion under the influence of a constant force

Newton's second law, F = ma, predicts the future motion of an object if we know its initial motion and the net force that is acting on the object.  Today you will explore this motion when the net force is approximately constant.

Equipment needed:
• Tennis ball

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

### 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.

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 1

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 ball being dropped.  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.

• "Play" the video clip. When finished, "Step up" to frame 1.  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 a meter stick.  Position the cursor over bottom end of the stick and click the left mouse button.  Then position the cursor over the top end of the stick 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 1 into the text box.  Click "Done".
Make sure the video frame stays fixed in the browser window between the two clicks.  You may have to scroll after the clicks to get to the buttons.
• 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.  Make sure the video frame stays fixed in the browser window while you take data.
• Repeat for each frame in the video clip until the ball reaches the bottom end of the meter stick.  Then click "Stop Taking Data".
Highlight and copy your table.  Open Microsoft Excel, and paste the table into an Excel spreadsheet.  Depending on your browser, you may have to use "Paste" (Edge) or "Paste Special, Unicode Text" (Chrome).  Your spreadsheet will have two columns, time (s) and y (m).  If you followed the instructions above, the the y-axis points up.

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 = y0 + 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 upward, 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?

### Projectile motion

Experiment 2

A ball is moving in two dimensions under the influence of a constant gravitational force.  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 ball being thrown.  You will determine the position of the ball in two dimensions 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 x and y component of the velocity as a function of time.

Procedure:

To play the video clip or to step through it frame-by-frame click the "Begin" button.

• "Play" the video clip. When finished, "Step up" to frame 1.
• In the setup window choose to track both coordinate of the object.
• Click "Calibrate".
• Click "Calibrate X".
The video clip contains two meter sticks.  Position the cursor over left end of the horizontal stick and click the left mouse button.  Then position the cursor over the right end of the horizontal stick 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 1 into the text box.  Click "Done".
Make sure the video frame stays fixed in the browser window between the two clicks.  You may have to scroll after the clicks to get to the buttons.
• Now click "Calibrate Y".
Position the cursor over bottom end of the vertical stick and click the left mouse button.  Then position the cursor over the top end of the vertical stick 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 1 into the text box.  Click "Done".
Make sure the video frame stays fixed in the browser window between the two clicks.
• 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 x- and y-coordinates of the ball will be entered into the spreadsheet.  You will automatically step to the next frame of the video clip.  Make sure the video frame stays fixed in the browser window while you take data.  When the ball is caught, click "Stop Taking Data".
• Your table will have 3 columns, time (s), x( m), and y (m).
• Open Microsoft Excel, and paste the table into an Excel spreadsheet.

Produce graphs of the x and y components of position versus time.  Label the axes.

• Describe the graphs?  Does one of the graphs resemble a straight line?  If yes, what does this tell you?
• Right-click your data in the x(m) 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 this graph?
• Right-click the data in your y(m) 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.  What do the coefficients b1 and b2 tell you?
• We can view the motion of a projectile as a superposition of two independent motions.  Describe those two motions.