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.
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.
Prediction 1-2: Select the velocity-time graph below that would correctly represent this constant velocity motion.
To test your predictions, you will need
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.
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.
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.
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.
To test your predictions, you will need
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.
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.
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:
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.
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:
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.)
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.
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.)
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.
To test your predictions, you will need
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.
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.
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?
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.
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.
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?
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.
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.
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.