**Bungee Barbie & Kamikaze Ken**

**Class**: General Math through Calculus (each at an appropriate level of analysis)

**Materials:** Doll action figures, graph paper, meter sticks, calculators, two pound box of uniform rubber bands. (A project extension requires a scale for weighing figures.) Ask students to bring in action figures and/or dolls.

**Goals:** Students will work in teams, gather and graph data, fit a line to data or determine equations using data, make predictions from the model and gain an appreciation for simulations.

**Time required:** Two to five class periods.

**Background:** Students should know how to plot points on a graph and determine equations of lines.

**Setting: **Team members have been hired to work for the Acme Daredevil Entertainment Company. This company provides rock climbing, sky diving, "extreme skiing", and cliff diving adventures to the public. However, to keep up with the market, the board decided to add bungee jumping to its list of offerings. As part of the first assignment, the board decided that the teams should undertake the task of working out the details of the new venture. The company has several sites planned for bungee jumping and each site is at different height.

**Problem:** Initially each group works with just one action figure (doll). The task is to determine the ultimate length, or the number of rubber bands that can be used with your action figure at any given height and not cause any type of injury or fatality--but that allows your action figure to come as close to the floor as possible (for maximum thrills).

**Solution Requirements:**

1. Teams will need to be able to predict the length of rope for any given height to them

by the teacher. Predictions will be checked through actual experimentation.

2. List any assumptions that were made.

3. Include graphs and tables.

4. Present methods of solution and justifications of conclusions.

5. Include possible sources of error.

**Evaluation:** Students will work in groups and presentations should include graphs, tables, equations, assumptions, error bounds, conclusions, and predictions. Written and oral reports may be used in addition to the actual jump length prediction results.

**Extensions: **This project could be repeated using rubber bands with different strength coefficients or other types of elastic materials. Dolls of differing weights can be used in discovering what effect that has on the band. Problems in the text that refer to Hooke's Law can be modeled by the students.

**Teacher Notes: **Students should make at least three trials when dropping Barbie from a specified height and use the average. Test drop several times to practice taking readings. Drop Barbie with 1 band and measure; drop Barbie with 2 bands and measure; continue doing this until you are using at least 6 rubber bands. Data can be entered on a graphing calculator or graph paper. Use the statistics features of a TI-82 to fit a curve, or the table function to made predictions from the calculator or by extending the line and reading the results off the paper. Consider dropping from a balcony, stadium, gym bleacher or platform ladder. Do not tell where test jump will be made.

If weight is being considered, then use this to find the stretch coefficient: Amount of stretch / weight = stretch factor

Buy plenty of rubber bands (2lbs box from an office supply store; size should be about 4in in length unstretched). Be aware that after several uses the rubber bands will permanently deform or stretch and that this may affect the problem. Let students discover and cope with this complication in any reasonable way (perhaps using new rubber bands frequently or for the final test jump, or pre-stretching rubber bands).

Sketchy Sample Solution

Problem: Develop a model to predict the number of rubber bands needed to drop a Barbie of our choice when given a height.

**Assumptions: **1. The rubber bands are identical to one another.

2. Height and number of rubber bands are related linearly.

3. Barbie will start from a standing position and will free fall

head first.

**Procedure:** Collect data by measuring the height of a Barbie=s fall with various numbers of rubber bands. Repeat each trial three times and average the results.

**Data:** Height in cm.

__Number of Rubber Bands__** **__Jump1__** **__Jump2__** **__Jump3__** **__Average__

1 31 33 30 31

2 52 55 51 53

3 72 74 75 73

4 93 92 94 93

5 116 115 120 117

6 133 137 140 137

**Model:** X = # of rubber bands; Y = cliff height (CM)

Y = 21.2 X + 9.8 X = (Y - 9.8)/21.2

Equation obtained by linear regression on TI-82 calculator and has a correlation

coefficient of .9997.

**Prediction:** We were given a test height of 595cm. Our model gives a prediction of 27.6 rubber bands. To be on the safe side we will use 27 rubber bands. During the actual test we came within about 16cm of the floor.

**Reflection:** We could have used another one (or possibly two) rubber bands, and still have had a safe jump. Possible sources of error include consistency of rubber bands, linearity assumption at extreme heights when rubber bands are stretched to a greater length, and the

permanent deformation of bands after the first jump -- should always use fresh bands for data collection trials and the test jump.

Students:__ __

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**BUNGEE BARBIE & KAMIKAZE KEN**

**Setting: **Team members have been hired to work for the Acme Daredevil Entertainment Company. This company provides rock climbing, sky diving, "extreme skiing", and cliff diving adventures to the public. However, to keep up with the market, the board decided to add bungee jumping to its list of offerings. As part of the first assignment, the board decided that the teams should undertake the task of working out the details of the new venture. The company has several sites planned for bungee jumping and each site is at different height.

**Problem:** Initially each group works with just one action figure (doll). The task is to determine the ultimate length, or the number of rubber bands that can be used with your action figure at any given height and not cause any type of injury or fatality--but that allows your action figure to come as close to the floor as possible (for maximum thrills).

**Report:** A short oral report, explaining methods used, will be given before the

actual demonstration. Then each group will perform a live demonstration

at a new height to test the accuracy of the prediction method. Each group

will also submit a written report.

**Guidelines for Bungee Reports**

A. Oral Report

1. Explain basis of model ( visual aids are encouraged).

2. Give model.

3. Give prediction.

B. Written Report

1. Entire paper is neat and legible.

2. Entire paper is free of spelling, grammatical, and usage problems.

3. Entire paper is well organized.

4. Title page with all names of group members is included.

5. Clear statement of the problem as you understand it is made.

6. Report lists all pertinent assumptions.

7. Report clearly explains the procedure followed in gathering data.

8. Data and graphs are well organized.

9. Report should clearly explain how group arrived at prediction method.

10. Report should clearly explain how well the prediction method worked.

If the group finds their prediction is not as good as hoped, then

adjustments should be made and also described in the paper. Possible

sources of error should be discussed.

*Funded in part by the National Science Foundation and Indiana University 1995*