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Chapter 2 Project
Analyzing Elementary Functions
When a ball is bouncing up and down on a flat surface, its height with respect to time can be modeled using a quadratic equation. One form of a quadratic equation is the vertex form:
In this equation, y represents the height of the ball and x represents the total elapsed time. For this project, you will use a motion detection device to collect distance and time data for a bouncing ball, then find a mathematical model that describes the position of the ball.
COLLECTING THE DATA
Setup the Calculator Based Laboratory (CBL) system with a motion detector to collect time and distance readings for 4 seconds or setup the Calculator Based Ranger system (CBR) using the Ball Bounce application. See the CBL/CBR guidebook for specific setup instructions. Hold the motion detector or CBR approximately 5 to 6 feet above the floor and parallel to it. Hold the ball approximately 2 feet below the detector and prepare to release it when the unit begins to click. Activate the system to collect the data. The data table below shows a portion of the sample set of data collected as the ball bounced beneath a CBR.
| Total Elapsed
Time (seconds) |
Height of the
ball (meters) |
Total elapsed
time (seconds) |
Height of the
ball (meters) |
| 0.688 | 0 | 1.118 | 0.828 |
| 0.731 | 0.155 | 1.161 | 0.811 |
| 0.774 | 0.309 | 1.204 | 0.776 |
| 0.817 | 0.441 | 1.247 | 0.721 |
| 0.860 | 0.553 | 1.290 | 0.650 |
| 0.903 | 0.643 | 1.333 | 0.563 |
| 0.946 | 0.716 | 1.376 | 0.452 |
| 0.989 | 0.773 | 1.419 | 0.322 |
| 1.032 | 0.809 | 1.462 | 0.169 |
| 1.075 | 0.828 |
EXPLORATIONS
- If you collected motion data using a CBL or CBR, a plot
of height versus time or distance versus time should be shown on your
graphing calculator or computer screen. Either plot will work for this
project. If you do not have access to a CBL/CBR, enter the data from
the table above into your graphing calculator/computer. Create a scatter
plot for the data.
- Find values for a, h, and k so that the equation
fits one of the bounces contained in the data plot. Approximate the
vertex (h, k) from your data plot and solve for the value of a algebraically.
- Change the values of a, h, and k in the model found above
and observe how the graph of the function is affected on your graphing
calculator or computer. Generalize how each of these changes affects
the graph.
- Expand the equation you found in #2 above so that it
is in the standard quadratic form:
.
- Use your calculator or computer to select the data from
the bounce you modeled above and then use a quadratic regression to
find a model for this data set. (See your grapher's guidebook for instructions
on how to do this). How does this model compare with the standard quadratic
form found in #4?
- Complete the square to transform the regression model
to the vertex form of a quadratic and compare it to the original vertex
model found in #2. (Round the values of a, b, and c to the nearest .001
before completing the square if desired.)
FURTHER EXPLORATIONS
- Repeat the analyses done in #1 through #4 for the data
in Table 2.3 on page 174.
- Compare your model with the model found in Example 9
on page 174.
- Use the data in the table on page 274 to write a model
for the vertical velocity of the ball.
- Use the model you found in #2 to estimate the speed of
the ball for each time value in the table on page 274 for which the
ball is rising.
- Use the values from #4 to determine a power function
model for speed versus distance traveled, as was done in Example 6 on
page 188.