Notes from Mrs. Caso

  This project will demonstrate how parabolas and quadratic equations are seen and used in the real world. It will be split in two different applications. Everything you will need is listed at the left.

Lesson Plan

First we're going to look at how the water stream from a typical drinking water fountain forms a parabola. Then we are going to create the quadratic model for the picture of one water fountain. This is not going to be an easy task, but it can be done.  You will plot the parabola between two x values: the minimum and the maximum. I want you to use 66 for the minimum and 420 for the maximum. Your goal for this part of the project is to use the formula

f(x) = ax2 + bx + c

to find the best model for the parabolic curve. Follow the directions on the screen to test your choices for a, b, & c. When you have found a curve that fits the water flow, show the teacher and print the screen that shows the graphed curve. Then close that window and print the screen showing your equation.
(Hint: a is negative and smaller than half of 0.01.
          b is between 1 and 10.
          c is negative and between 50 and 100.)

Use this simulator to see what happens when you change the a, b, and c in the equation (SIMULATOR)

Cuckoo Clocks:

For this part of the project, we're going to look at how cuckoo clocks keep time. I just happened to have bought one while in Germany in 2003 and I was interested in how it worked. So, I read the instructions on how to set it up and how to adjust the timing so that the clock would keep time accurately (not run fast or slow). Interestingly, to adjust the timing, you move the leaf on the pendulum up or down until the clock seems to be keeping correct time. Since it's not a mechanical part, I thought that the leaf was just a decoration. Apparently, I was wrong. So, I did what the instructions said and it worked.

You can go to HowStuffWorks online and it will explain how "Pendulum Clocks" work.

The length of the pendulum regulates the timing of the clock. Since the leaf acts as the weight on a pendulum, its distance from the top of the pendulum pivot determines the speed at which it swings. So, how far up or down do I have to move the leaf if the clock is running 15 minutes fast each day?

Our goal is to study the relationship between the length of a pendulum and the speed at which it swings. To do this, we are going to experiment with a virtual pendulum and collect data for the time it takes to swing 10 times at different lengths. Then we are going to plot our data on a scatter plot and fit a quadratic curve to the data. We'll use a computer to do the graphing and curve fitting. Then we'll try to answer the question above for Bonus Points.

Okay, let's get started.

When done, go to the Quadratic Regression Applet.

  1. Follow the directions below the applet to enter your data in the x and y data boxes..
  2. Change the x label to "Length".
  3. Change the y label to "10Swings".
  4. Click on the PLOT button to create the graph.
  5. Click on the ANALYZE button to get the equation.
  6. Print the page and write your name(s) on the top.

On the back of the lab sheet, answer the following questions:

  1. You should notice that it doesn't look like a full parabola. Why doesn't it?
  2. Do you think that we could, in reality, make the pendulum long enough to make the graph turn back downward? Why or why not?
  3. What is the Model for the relationship between Length of Pendulum and Time for 10 Swings? (Write out the equation using the information on your graph.)
  4. Which number in the Model tells you that the parabola will open downward if we could see the whole parabola?
  5. According to the Model (equation), how long will ten swings take when the length is zero? Does that make sense in the real world? Why or why not?
  6. If my cuckoo clock pendulum is supposed to swing once every second to keep correct time, how far down from the top should the leaf be? Use your Model and the quadratic formula to find the length. (Hint: 10Swings = 10 Seconds).

When you have answered the questions, staple your Pendulum Lab Sheet to the back of the Graph Sheet (graph showing on top) and turn it in.

Answer the Bonus Points question on another piece of paper and turn that in within one week. (10 Points Possible)

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