PARABOLA! Objective Level Prerequisites Platform Instructor's Notes Credits

 
Objective: The objective of this activity is to demonstrate how the graph of a parabola arises from the simple physical process of rolling a ball up an inclined plane. A ball is given a push up an inclined plane and the distance from the starting point as the ball rolls up and then back down is recorded by a motion sensor. A graph of elapsed time vs. distance from the sensor is developed and displayed.

Level: Mathematical modeling, college algebra or precalculus.

Prerequisites: Students should have a basic knowledge of algebra and graphs of quadratic equations. For students in a modeling class knowledge of parabolic regression can be used to handle real data from experiments.

Platform: This demo can be performed using either a motion sensor interfaced to a computer or a calculator with auxiliary motion detector equipment, like a TI-calculator with CBR^TM.

Instructor's Notes:

• Equipment:
  > A motion sensor with calculator or computer interface for measuring/recording elapsed time and distance from the sensor. Physics departments may be willing to loan out computer and sensor equipment or a TI-calculator with CBR^TM may be used.
  
  > A large ball; a playground ball, bowling ball or basketball works well. A heavy ball seems to work better.
  
  > An inclined plane; a board or table about five feet long, that can be elevated at one end. (The length of the inclined plane can vary.) For a temporary set-up books can be used to raise one end.
  
  > An edging of some sort to keep the ball from rolling over the edge and moving directly in line with the motion sensor. For edging thin strips of wood, yardsticks, or meter sticks will work when taped together and affixed to the inclined plane. (Duct tape works well.) An eight foot channel that has a U-shaped cross section donated by the maintenance department was used successfully eliminating the need for edging.
  
  
• Preparation:
  Position the board in a stable position where the motion sensor can be placed at the foot of the board. If you are using a sensor with computer interface, make sure the students can see the monitor. It is highly recommended that you practice with the set-up, unless you are truly an optimist. (Care must be exercised to prevent the sensor from getting bumped by the ball.)
  
  

   

• The Demonstration:
  Start the ball at the foot of the inclined plane near the motion sensor which is activated. Give the ball a push up the inclined plane so that it travels most of the way. (A good reason to practice!) When it rolls back down, catch it before it hits the sensor. The computer monitor should show a nice graph of time vs. distance away from the sensor which is easily recognized as a parabola.
  
  
• Comments:
  This demo has been used as a "quickie" with students studying quadratic functions. It makes quite an impression on some students; one said that now she saw that parabolas are real.
  
  For calculus students one might show the velocity and acceleration graphs which can be displayed simultaneously with some software.
  
  We use a large ball so that the motion sensor sees it and nothing else. We had trouble with small balls. Light weight balls are vulnerable to irregularities in the board and tend to bounce and skip a bit.
  
  
• Associated Activities:
  After the physical demo some problems where the student has to 'read' information off a graph like that generated by the motion sensor can reinforce the mathematics.

  For example we put the following graph and questions on a work sheet.

  
  
  

  Directions: For the questions below, use the graph in Figure 1. The graph illustrates the position of a ball rolling on an inclined plane as a function of time. This graph could have been produced by a motion sensor like that in the demonstration. (Positions are measured in feet and time in seconds.)

  1. For what times is the ball moving with positive velocity?

  2. For what times is the ball moving with negative velocity?

  3. When does the ball reach its farthest point along the inclined plane?

  4. What is the total distance the ball travels along the inclined plane?

  5. How long is the ball beyond 5 feet from the bottom of the inclined plane.

  6. The general equation of a parabola is 

  y = ax² + bx + c. 

  Determine the values of the coefficients a, b, and c for the parabola depicted in Figure 1.

  
   Auxiliary Resources and Approaches:
  
  > In Getting Started with CBR^TM (published by Texas Instruments Incorporated, 1997) one of the classroom activities described is similar to that described above but with the sensor at the top of the inclined plane. 

  

  In this case half of a parabolic curve is generated. The suggestions accompanying this activity are varied and can be used for inquires in a number of directions including the effect on the coefficients in y = ax² + bx + c when the angle of the inclined plane is changed. There is an accompanying activity sheet that can be copied for classroom use.
  
  > In a mathematical modeling class a lab activity based on this demo can easily be developed where teams of students use multiple sets of equipment, different balls, and set the inclined plane at different angles. A comparison of team results can initiate an interesting class discussion or be used as a writing assignment.

  > In Exercise 6 on the sample worksheet above it is intended that students recognize that the three points (0,0), (6,0) and (3,9) lie on the parabola. Then they can construct three equations in the three unknown coefficients a, b, and c and solve this system of equations. However, in a modeling class where real data from an experiment involving a motion sensor is used an alternate approach to determine an equation that approximates the data using parabolic regression on their calculators. 

> For classroom use the introductory animation to this demo can be downloaded by clicking on parabanimation.zip which contains an animated gif file and a QuickTime file.

> In the DEMOs with POSITIVE IMPACT project at URL

http://mathdemos.gcsu.edu/mathdemos/family_of_functions/famfuncs.html

are two galleries (Polynomial & Rational Functions and Conic Sections) that contain animations and programs that can be downloaded which illustrate how a parabola changes when its coefficients are varied. For a sample animation illustrating changing the coefficients of a parabola based on an Excel program click here. 

> At URL

http://www-gap.dcs.st-and.ac.uk/~history/Java/Parabola.html

is a java applet for exploring graphs of parabolas.

> At URL

http://www.ies.co.jp/math/java/conics/focus/focus.html

is a java applet for illustrating the focus of a parabola using parallel light beams.

Credits:  This demo was submitted by 

Dr. Joanne Darken
Department of Mathematics 
Community College of Philadelphia

Dr. Martin Ligare
Department of Physics 
Bucknell University

and is included in Demos with Positive Impact with their permission. 

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