## Physics Laboratory 3

### Projectile motion

Projectile motion is motion in two dimensions under the influence of a constant force.  An object moving through the air near the surface of the earth is subject to the constant gravitational acceleration g, directed downward.  If no other forces are acting on the object, i.e. if the object does not have a propulsion system and we neglect air resistance, then the motion of the object is projectile motion.
In this lab you will analyze the motion of a ball executing projectile motion.

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.

Exercise

The New York Giants are tied with the Chicago Bears with only a few seconds left in the game.  The Giants have the football and call the place kicker into the game.  He must kick the ball 52 yards (47.5 m) for a field goal.

If the crossbar of the goal is 10 ft (3.05 m) high, and the maximum speed with which the kicker can kick the ball is 25 m/s, which range of angles (in deg) will allow him to score the field goal?

Solve this problem using a spreadsheet.  Produce a spreadsheet with 5 columns.

 v0 (m/s) Angle (deg) t (s) x (m) y(m)

The first two columns each contain one entry each.  Construct your spreadsheet so that when you change these numbers, the numbers in the next three columns will be updated.  The time steps in the t(s) column should be on the order of 0.1 s or smaller.  The entries in the x(m) and y(m) columns are calculated as a function of time using the kinematic equations.

In one full sentence answer the question above.

Experiment

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.

#### Activity

In this virtual laboratory you will investigate projectile motion with and without air resistance.  You will find the maximum range of a baseball with given initial velocity and the launch angles required for it to hit a target.

Procedure:

Open the Projectile Motion simulation.  Click "Intro".

Position the canon 20 m to the left of the target.  Make sure the cross on the canon is at the same height as the target.  Choose the baseball projectile and give it an initial speed of 20 m/s.

• (a)  Given this initial speed there are two launch angles for the baseball to hit the center of the target.
What is the smaller of these launch angles (within ~±1o)?
Note:  You may have to raise the canon by 1 meter to get to the small angle. Since you cannot raise the target, maye you can use the ruler to mark where the target would be if it also was raised by 1 m. That should let you find the small angle.
• (b)  Given this initial speed there are two launch angles for the baseball to hit the center of the target.
What is the larger of these launch angles (within ~±1o)?
• (c)  Find the maximum range of the projectile along the horizontal.  For this maximum range note the launch angle and the hang time of the baseball and estimate the maximum height of the baseball above the ground during its flight.
Do this twice, once neglecting air resistance and once with air resistance with a drag coefficient of 0.35.  Fill in the table below.  Compare your results with and without air resistance.  Paste your table into your log.
Projectile speed: 20 m/s
Maximum range Launch angle (deg) Maximum height (m) Hang time (s)
no air resistance
drag coefficient 0.35