The Bouncing Ball and Graphing Activities
Activity Description:
Students will use a CBL and a TI-83 Plus graphing calculator to construct the 'height versus time' graph of a ball as it bounces. Using this 'height versus time' graph, they will analyze one bounce of the graph finding the equation of the quadratic function that best fits the data, and will construct the graph of velocity versus time (the first derivative) and acceleration versus time (the second derivative) with appropriate equations.
Content Standards: Number & Operations, Algebra, Geometry, and Data Analysis
Process Standards: Reasoning & Proof Standard, Connections Standard, Representation Standard
Mathematics: graph-sketching, Regression, linear and quadratic functions
Mathematics Thinking: analyze graphs, predict, and compare.
Technology: TI-83 Plus graphing calculator and CBL.
Pre- Assessment: Give each student a paper and pencil. Ask them to do the following activities.
1. Imagine dropping a rubber ball from 5 feet above the floor and letting it bounce until it stops. Draw a sketch of the 'height versus time' graph, the ‘velocity versus time’ graph and the ‘acceleration versus time’ graph.
2. Actually drop a rubber ball from 5 feet above the floor and let it bounce until it stops. Draw another sketch of the 'height versus time' graph, the ‘velocity versus time’ graph and the ‘acceleration versus time’ graph. Compare and discuss the similarities between this sketch and your previous sketch in Task 1.
Data
Collection:
3. Drop the ball and let it bounce as in Task 2. As you do, use the CBL and the TI-83 Plus graphing calculator’s APPS program to record the heights of the ball as it bounces.
Setting
Up:
1. Make sure the ball bouncing area is free of obstructions and
contains a level, hard smooth floor.
2. Work with a partner. Have a sturdy chair for your partner to stand
on.
3. Attach the CBL to the TI-83 graphing calculator using a link cable.
4. Turn on the calculator.
Collecting Data using CBL-2:
Student A: Student A stands on the chair and holds the CBL about 6
feet above the floor.
Student B: Student B prepares
the calculator for data collection (see the Ball Bounce program below)
Student C: Student C holds the ball 1.5 feet directly below the CBL. After directing Student B to press the ENTER button, Student C releases the ball to let it continue to bounce under the CBL while the CBL collects data.
Tips
for data collection:
1. Be sure you hold the ball at least 1.5 feet below the CBL so that the
CBL will be able to detect the ball.
2. Be sure there are no objects close by so that the CBL does not pick up
extra vibrations.
3. Be sure the sensor is not at a slant but is directly facing the floor.
4. After pushing the ENTER button, hold the ball under the sensor for a
second or two before dropping the ball.
5. Be sure the ball bounces straight up and down underneath the CBL.
Real time No Time 15 Display Dist Begin On Enter Smoothing None Units Meter Real time yes Time 15 Display Dist Begin On Enter Smoothing None Units Meter
To run the Ball Bounce
program:
1. From the home screen in TI-83 Plus, press the APPS button on your
calculator.
2. Using the down arrow button, select 2: CBL/CBR and press ENTER twice.
3. Using the down arrow button, select 3: Ranger and press ENTER twice.
4. From the MAIN MENU, select 1: SETUP/SAMPLE
It saves D vs T, V vs T, and A vs T at One time
5. Move the cursor to the “start now” (to the top right), press enter
6. Run the motion program data collection, view the graph; press ENTER
Follow the directions on the screen.
Below is a screen shot the TI-83 graphing calculator's 'height versus time' graph of a bouncing ball.
4. Observe the 'height versus time' graph produced by the calculator. Discuss the accuracy of this graph as a representation of the heights of the bouncing ball. What could you do differently to get a more accurate representation?
5. If you are not satisfied with your graph, then, using your ideas from Task 3, repeat the BALL BOUNCE activity until you get a graph that is a more accurate representation of the heights of the bouncing ball.
Analyzing the graph:
1. Select one bounce to analyze. Press ENTER to return to the PLOT TOOLs. Select 1: Select Domain.
a. Press the right arrow until the lower left side of one bounce is reached. Press the right arrow once more. Press ENTER to select the point as the LEFT BOUND.
b. Press the right arrow until the lower right side of one bounce is reached. Press the left arrow once. Press ENTER to select this point as the RIGHT BOUND.
c. Sketch the Distance-time graph for this bounce.