LAB 8: TWO-DIMENSIONAL MOTION (PROJECTILE MOTION)

Please list the members of your group:

OBJECTIVES

OVERVIEW - PROJECTILE MOTION

So far we have dealt separately with motion with a constant velocity, and motion with a constant acceleration. The focus of this lab is to describe the motion close to the surface of the Earth that occurs when an object is allowed to move in both the vertical and horizontal directions. Examples are the motion of a baseball or tennis ball after being tossed or hit. This type of motion is commonly called projectile motion. To understand this motion, it is helpful to review motion with a constant velocity and motion with a constant acceleration separately and then consider how they might be combined.

The lab begins with a review of the classic kinematic equations that describe the relationships between instantaneous position, velocity, and acceleration of objects that move in one dimension. In some cases objects move with a constant velocity (zero acceleration). In others, such as the motion of an object tossed straight upward and pulled by the constant gravitational force, the motion is with a constant acceleration. You have already examined examples of these types of one-dimensional motion in Labs 2 and 3.

In this lab you will focus on the simultaneous vertical and horizontal motions of objects with a horizontal component of velocity while acted on by the downward gravitational force of attraction of the Earth in the vertical direction.

NOTE: Investigation 1 will be done with the IOLab and Lesson Player software. Like Labs 1-4, your work will be saved within Lesson Player. Investigation 2 will be done using Lesson Player and a separate piece of software called Tracker. You will need to save your results from Investigation 2 in Lesson Player, and also submit one Word document with copies of your Tracker graphs, with some explanation and reference to the slide in the lab, at the same time when you submit your Lesson Player file.


Copyright © 2018 John Wiley & Sons, Inc.

INVESTIGATION 1: REVIEW OF ONE-DIMENSIONAL MOTION

You’ll begin by reviewing the one-dimensional motions of a cart.

Prediction 1-1: Suppose that you push the IOLab along a horizontal table at a constant velocity and measure its position and velocity as functions of time using the Wheel sensor. Select the position-time graph below that would correctly represent this constant velocity motion.

A B C D E

Prediction 1-2: Select the velocity-time graph below that would correctly represent this constant velocity motion.

A B C D E

To test your predictions, you will need

Activity 1-1: Motion of the IOLab Rolling Horizontally with a Constant Velocity

  1. Collect graphs for the motion of the IOLab along the tabletop as you push it with a constant velocity in the positive y direction (as indicated by the axes on the IOLab).
  2. Repeat if necessary to get clear graphs. When you have a good set of graphs, adjust the axes to display the results as clearly as possible. (As in previous labs, use +Add Run to collect a new graph, and Remove to remove the graphs you don't want.)

Activity 1-1: Motion of the IOLab Rolling Horizontally with a Constant Velocity

Question 1-1: Describe the shapes of your graphs. Did they agree with your predictions?

Question 1-2: For this motion with a constant velocity, what would the graph of acceleration vs. time look like? Explain.

Activity 1-2: The Motion of the IOLab with a Constant Acceleration

Prediction 1-3: Now suppose you give the IOLab a push up an inclined ramp, allow it to move up in the positive y direction, reach its highest point, reverse direction and come back down. You stop it with your hand when it returns to its original position. Select the position-time graph below that would correctly represent this motion of the IOLab from the moment it leaves your hand until just before you stop it.

A B C D E

Prediction 1-4: Select the velocity-time graph below that would correctly represent this motion of the IOLab from the moment it leaves your hand until just before you stop it.

A B C D E

Prediction 1-5: Select the acceleration-time graph below that would correctly represent this motion of the IOLab from the moment it leaves your hand until just before you stop it.

A B C D E

Activity 1-2: The Motion of the IOLab with a Constant Acceleration

To test your predictions, you will need

  1. Incline the ramp as shown below to an angle of at least 20°, as measured with the protractor.
  2. Collect graphs of position, velocity and acceleration for the motion of the IOLab along the inclined ramp as you give it a short push up the ramp (in the positive y direction), release it and then stop it when it returns to its original position. Repeat until you get a clear set of graphs.
  3. Adjust the axes to display your graphs as clearly as possible.

Activity 1-2: The Motion of the IOLab with a Constant Acceleration

Question 1-3: Describe your graphs. Did they agree with your predictions?

Question 1-4: Would you describe this motion as having constant velocity or constant acceleration? Explain.

The motions of the IOLab examined in Activities 1-1 and 1-2 can be represented as functions of time by a series of mathematical equations called the kinematic equations. In the following activities, you will identify these equations.

Activity 1-3: Kinematic Equations for Motion with Constant Velocity (Zero Acceleration)

Below, you will find some equations that might represent the position, velocity, and acceleration of an object as functions of time,

Question 1-5: Use your results from Activity 1-1 to determine which set of kinematic equations could be used to represent the motion of the IOLab along the horizontal table top where the x-axis is assumed to point along the line of motion. Explain your choice based on the graphs you obtained. Assume that t = 0 is the time when the IOLab begins to move.

Question 1-6: Use your graphs from Activity 1-1, and the kinematic equations you chose in Question 1-5 to determine the approximate values of x0 and v0. Explain how you found these values and what they represent. Again assume that t = 0 s is the time when the IOLab begins to move.

Activity 1-4: Kinematic Equations for Motion with Constant Acceleration

Question 1-7: Use your results from Activity 1-2 to determine which set of kinematic equations could be used to represent the motion of the IOLab up and down the inclined ramp. Explain your choice based on the graphs of your data. Assume that t = 0 is the time when the IOLab begins to move.

Question 1-8: Use your graphs from Activity 1-2, and the kinematic equations you chose in Question 1-7 to determine the approximate values of x0 and v0. Explain how you found these values and what they represent. Again assume that t = 0 is the time when the IOLab begins to move.

Question 1-9: Use your graphs from Activities 1-1 and 1-2 to determine the values of the acceleration for (a) the motion of the IOLab along the horizontal surface and (b) the motion of the IOLab moving up and down the inclined ramp. In each case, explain how you found the acceleration.

(a) Motion in Activity 1-1:

(b) Motion in Activity 1-2:

INVESTIGATION 2: PROJECTILE MOTION

The world is full of phenomena that we can know of directly through our senses—objects moving, pushes and pulls, sights and sounds, winds and waterfalls. A vector is a mathematical concept—a mere figment of the mathematician’s imagination. But vectors can be used to describe aspects of “real” phenomena such as positions, velocities, accelerations, and forces. Vectors are abstract entities that follow certain rules. For example, in figure (a) below, the velocity of an object is represented by the vector, , which is drawn relative to two different sets of coordinate axes in (b) and (c).

A vector has two key attributes—magnitude and direction. The magnitude of a vector can be represented by the length of the arrow and its direction can be represented by the angle, θ, between the arrow and the coordinate axes chosen to help describe the vector.

In earlier labs you drew vectors in only one dimension. But vectors are especially useful in representing two-dimensional motion because they can be resolved into components. In the middle figure above, the velocity vector is resolved into x and y components. Added together, these components are equivalent to the original vector, but they can be analyzed independently of each other. This is one reason it is convenient to use vectors.

INVESTIGATION 2: PROJECTILE MOTION

If a cannonball is shot off a cliff with a certain initial velocity in the horizontal (x) direction, the two-dimensional motion that results is known as projectile motion. The ball will continue to move forward in that direction and at the same time fall in the vertical (y) direction as a result of the gravitational attraction between the Earth and the ball. There is no force on the ball in the horizontal direction and a constant force in the downward vertical direction. This is characteristic of projectile motion.

In this investigation you will examine the motion of a tennis ball that is tossed into the air so that it is moving in both the x and y directions. The toss of the ball and its trajectory are shown in the photos below.

Because the motion of the ball is in two dimensions, it is not possible to make measurements using the IOLab. Instead, you will use the method of video analysis to examine the motion and to determine the mathematical representations of the horizontal and vertical components of the motion. First some predictions:

INVESTIGATION 2: PROJECTILE MOTION

Prediction 2-1: Select the graph below that represents your prediction for how the x-coordinate of the ball will vary with time. (Assume that the positive x direction in the photos is to the right.)

A B C D E

Prediction 2-2: Select the graph below that represents your prediction for how the x-component of the velocity of the ball will vary with time.

A B C D E

INVESTIGATION 2: PROJECTILE MOTION

Prediction 2-3: Select the graph below that represents your prediction for how the y-coordinate of the ball will vary with time. (Assume that the positive y direction in the photos is upward.)

A B C D E

Prediction 2-4: Select the graph below that represents your prediction for how the y-component of the velocity of the ball will vary with time.

A B C D E

INVESTIGATION 2: PROJECTILE MOTION

To test your predictions, you will need

Installing and Using Tracker

  1. Go to http://www.opensourcephysics.org/items/detail.cfm?ID=7365
  2. Download the Tracker version for your system: Windows or OS X. If you used the integrated Windows installer, it should already be installed.
  3. You can find Help with installing Tracker at http://physlets.org/tracker/installers/installer_help.html
  4. You can find information on Getting Started with Tracker at http://physlets.org/tracker/help/frameset.html
  5. You can find a Tutorial on how to use Tracker at https://www.youtube.com/watch?v=La3H7JywgX0
  6. Important Note: Tracker requires Java 1.6 or higher. Tracker also supports QuickTime 7 (Windows/Mac only). Important note for Windows users: do not install QuickTime 7.7.6 or later as it will UNINSTALL QuickTime for Java! Instead, use the QuickTime 7.7.4. installer which you can download here.

Activity 2-1: Horizontal Motion of a Projectile

  1. Open Tracker.
    Use Tracker to open the video, Ball_Toss.mov, located in the IOLab-Workfiles folder in the activities folder.
  2. Play the video by clicking on the play arrow at the bottom left, and observe the motion of the ball. Return the video to the first frame by clicking on , next to the play arrow.
  3. Note: If you ever need help with Tracker, search the Help menu and select “Getting Started.”

    Question 2-1: Describe the shape of the trajectory of the ball.

  4. Set the scale of the measurements by clicking on the calibration menu and by choosing New>Calibration Stick. Drag one end of the blue stick to the bottom of the pile of books, and the other end to the top. (Be sure that the stick is vertical—90°.) Type in the height of the books in meters (0.62) given in the first frame of the video in the length box at the top and hit Return (or Enter if you’re using a PC).
  5. Click on to create a set of coordinate axes. These can be placed anywhere, but for simpler calculations, it is best to click and drag the origin to the center of the ball. Be sure that the angle from horizontal is 0.0°
  6. To record the positions of the ball in all frames, select New>Point Mass from the Track menu. Now, hold down the Shift key, position the cursor on the center of the ball, and click. Repeat this for all frames of the video. (The video will advance frame-to-frame automatically.)
  7. When you are done, return to the first frame by clicking on , and play the video again to see the trajectory of the ball traced out as the video plays.
  8. To find the graph for x vs. time, click on the label of the vertical axis on the graph, and select x: position x-component. Save a copy of the graph.

Activity 2-1: Horizontal Motion of a Projectile

Question 2-2: Does the graph for x vs. time agree with your Prediction 2-1? Explain.

Question 2-3: Does the graph for x vs. time represent motion with a constant velocity or constant acceleration? How do you know? Refer back to your observations in Investigation 1, if necessary.

Activity 2-1: Horizontal Motion of a Projectile

  1. Choose the kinematic equation on slide 8 that describes x vs. time for this motion, and model it in the software to find the values of the parameters in that equation, e.g., v0 and x0. Here are the steps in Tracker:
    1. Pull down the Track menu, and select New>Kinematic Particle Model.
    2. Near the bottom of the window, you will see columns for Name and Expression. Double click on the Expression box next to x.
    3. Now, enter a mathematical expression to represent x as a function of time. For example, set 1 on slide 8 gives x = x0 + v0t. You must determine x0 and calculate v0x based on your data for Mass A. Note: multiplication must be stated explicitly; so to model this equation you might write 0.5 + 4*t, and adjust the values of 0.5 and 4 until you get a model that fits the graph you collected.
    4. Select a color different than Mass A by again pulling down the Track menu, selecting Model A and then Color, and choosing a different color.
    5. You can display the graphs of Mass A (the ball tracked in the video) and Model A on the same graph axes by right clicking (cntrl click on a Mac) the plot area, selecting Compare with on the pull-down menu, and then selecting Mass A.
    6. Compare the two graphs, and try to determine what needs to be changed in the Model A equation to get its graph to match as closely as possible to the Mass A graph measured from the video. Select Model A and then Model Builder from the pull down Track menu, double click on the equation and change it.
    7. Repeat (f) until the Mass A and Model A graphs match very well.

Question 2-4: Based on your mathematical model, what is the kinematic equation for x vs. t? What are the values (from your mathematical model) for v0 and x0? What are the meanings of these two values?

Activity 2-1: Horizontal Motion of a Projectile

  1. Use to return to the first frame of the video. Play the video and observe both the motion of Mass A and the x-motion from Model A plot out.

  2. Question 2-5: Based on what you observe with Model A, describe the x-motion of the ball in the video. Is it with constant velocity, constant acceleration or some other motion? Explain.

  3. Now, display the graph of vx vs. time for Mass A by selecting vx: velocity x-component from the vertical menu on the graph. (Be sure that Mass A is selected next to Plot at the top of the graph.) There may be a lot of noise (spikes) on this graph. Adjust the vertical axis to something like 0 to 5 m/s to better see the overall trend of vx vs. time.
  4. Save a copy of the graph

Activity 2-1: Horizontal Motion of a Projectile

Question 2-6: Why was there so much noise on this graph? What does it have to do with the way the positions of the ball were determined in each frame of the video?

Question 2-7: Does the graph for vx vs. time agree with your Prediction 2-2? Explain.

Question 2-8: Does the graph for vx vs. time represent motion at a constant velocity or constant acceleration? How do you know? Refer back to your observations in Investigation 1, if necessary.

Question 2-9: What is the kinematic equation for vx vs. time? Give the values of all parameters taken from your graph.

Activity 2-2: Vertical Motion of a Projectile

  1. Display the graph of y vs. time for Mass A by selecting y: position y-component from the vertical menu on the graph. (Be sure that Mass A is selected next to Plot at the top of the graph.) You can get rid of the Model A graph from the previous activity by right-clicking (cntrl click on a Mac) on the plot area, selecting Compare with on the pull-down menu, and then clicking on Model A.
  2. Save the graph.

  3. Question 2-10: Does the graph for y vs. time agree with your Prediction 2-3? Explain.

    Question 2-11: Does the graph for y vs. time represent motion at a constant velocity or constant acceleration? How do you know? Refer back to your observations in Investigation 1, if necessary.

    Question 2-12: What is the value for y0, the initial y coordinate of the ball? Why does y0 have this value?

  4. Display the graph of vy vs. time, and save it.

  5. Question 2-13: Does the graph for vy vs. time agree with your Prediction 2-4? Explain.

    Question 2-14: Does the graph for vy vs. t represent motion at a constant velocity or constant acceleration? How do you know? Refer back to your observations in Investigation 1, if necessary.

Activity 2-2: Vertical Motion of a Projectile

  1. Rather than more or less guessing to see what looks good, you can also do a fit to the data with Tracker. In order to find the parameters e.g., voy and ay in the equation:
    1. Right click on the graph area (cntrl click on a Mac), and select Analyze from the menu. The graph will appear in a new window.
    2. Click Analyze in the upper lefthand corner, then Curve Fits, and below the graph axes, select Line for Fit Name.
    3. You can change the color of the fit line by clicking on the near the top of the table on the right in the Analyze window, and selecting a new color.
    4. You can click the data line on and off to make the view of the fit easier by checking and un-checking the lines box in the vy column.

Question 2-15: What is the kinematic equation?

Question 2-16: What is the value of v0, the initial y-component of velocity of the ball? What is the meaning of this?

Question 2-17: What is the value of the y component of the acceleration of the ball, ay? Is this value what you expected? Explain.

Question 2-18: Use your observations in the two investigations of this lab to justify the statement that projectile motion is a combination of horizontal motion at a constant velocity (zero acceleration) and vertical motion with a constant (gravitational) acceleration.

ALL DONE!

Please remember to edit the report (insert your name - and if necessary your partners), export the report and submit it on D2L.

Please also submit the graphs you saved from Tracker in Investigation 2 in one file on D2L.

Now do the homework associated with this lab.


Copyright © 2018 John Wiley & Sons, Inc. and David Sokoloff, Erik Jensen, and Erik Bodegom.
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