CSC #5-Culminating Project

Sleuthing Galileo

Name: SelectThisAndTypeYourFullName

Section: SelectThisAndTypeYourSectionNumber

(1=Kreider, 2=Lahr)

What should you do with this worksheet?

This worksheet contains activities that you can carry out at the end of a first course in single-variable calculus, a course like Math 3. Follow the instructions in the worksheet, completing the tasks and discussions as appropriate. You will then turn in a printed copy of the completed worksheet that contains your responses to the activities and questions. The completed worksheet will constitute your report on the topic. The report will be the take-home part of the final exam in Math 3 this fall worth 50 of the 100 points on the final. The other 50 points will come from the multiple choice part of the final, a test that will be designed for one hour but that you will have two hours to complete. The CSC is due at the final exam; bring it with you when you come.

What you should turn in .

(a) You should have typed your name and section where indicated above.

(b) You should type your answers to the questions and directives given below (e.g. express, show, write an equation). Simply type them where the instructions appear, as they arise in the text. In some cases you will complete or create an input cell and execute it.

(c) You should turn in a printed copy of this worksheet with your results (the work you generate below) included. Remember that the worksheet is to be a self-contained report, but before printing close all cells that do not contain your responses to save on paper.

(d) For your own records, be sure to save an electronic copy of the worksheet on your hard disk with a name such as CSC #5 - Dwight Lahr. Although you will not be turning in an electronic copy, it is good practice to save your work as backup.

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Sleuthing Galileo

Background: Galileo's Experimental Data

[Reference-This Case Study was adapted from:

Teaching Statistics with Data of Historic Significance: Galileo's Gravity and Motion Experiments, by David A. Dickey and J. Tim Arnold, Journal of Statistics Education v.3, n.1 (1995)]

The big breakthrough of the seventeenth century was an understanding of motion. We have seen how powerful the concepts of rate of change, derivative, and integral are in the field of study that emerged. And although there are many applications of calculus throughout the sciences today, the beginnings of the subject can be traced back to Galileo (1564-1642) and his rolling ball experiments at the end of the sixteenth century.

Galileo was interested in solving practical problems, many of them coming from the field of gunnery and ballistics. To do this, he had to have a good understanding of the behavior of cannon balls in flight. Therefore, Galileo conducted many experiments to simulate in his laboratory the field conditions of a projectile of the day. We already have discussed a Galileo-like experiment in CSC #2, Part 1. There we concluded that the acceleration due to gravity is constant; with modern methods of measurement we now know that is about 9.8 meters per second per second or 32.2 feet per second per second.

In this CSC, we are going to consider some of Galileo's actual data from one of his experiments, and try to make sense of it in light of a mathematical model that we develop to describe the situation. We will be acting as modern day detectives attempting to reconstruct what went on in his lab and what motivated his thinking. We will be analyzing the problem, making conjectures, drawing conclusions, and giving possible interpretations of his data. We will be trying to get inside Galileo's head and think some of his own thoughts. Because he initiated many of the ideas of calculus as they relate to motion, his thinking and ours should go along similar lines.

Galileo was a 31 year old lecturer at the University of Padua when he began experimenting and writing a paper on the path of projectiles. He was 42 years old when he reported on these experimental studies and gave an explicit mathematical formulation of the motion of falling objects.

In the experiment we are going to consider, Galileo used an inclined ramp that ended in a narrow horizontal shelf at the lower end of the ramp. A groove was etched into the ramp, so that a ball released on the ramp would roll down following the groove all the way to the shelf. When it hit the shelf, which we will take to be the edge of the table on which the ramp sits, the ball would roll horizontally to the end before leaving the edge and falling to the floor some distance below.

The velocity of the ball, as it rolls down the ramp, has two independent components: one horizontal. the other vertical. (See Figure 3 below.) When the ball hits the table, the velocity of the ball in the vertical direction is set to zero; the velocity of the ball in the horizontal direction is equal to the horizontal velocity with which it leaves the ramp. One of Galileo's findings, later stated by Newton as his First Law of Motion, is as follows: An object without outside influence continues to move indefinitely with the same speed and direction that it has originally. Thus, because there are no influences on the ball in the horizontal direction, the horizontal speed is constant from the time the ball leaves the ramp until it hits the floor.

When the ball leaves the table, the distance it travels before hitting the floor depends on the height at which it is released to roll down the ramp. We can see this from the table below of data that Galileo recorded. (You will also explain this observation later on theoretical grounds, so start thinking now about why it is true.) Let be the release height of the ball above the table, and let be the horizontal distance traveled along the floor, measured from a point on the floor directly below the edge of the table. (See Figure 1 below.) Then Galileo recorded five measurements of and of as follows, where the units of measurement are punti ( points in Italian):

Release Horizontal

Height Distance

Above Table Traveled

1000 1500

828 1340

800 1328

600 1172

300 800

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Using the tools of calculus we have learned this term, we will develop a model of the ball's motion and then try to interpret Galileo's data in light of this model.

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Objective

Your task is to make sense of Galileo's data, and to formulate some ideas about how this particular experiment might have contributed to a calculation of , the acceleration due to gravity.

Here is what you have to do in specific terms: Write a report

1) modeling the rolling ball experiment starting from a conservation of energy argument,

2) verifying that Galileo's experimental data are consistent with the results of the theory, and

3) discussing any inconsistencies and/or open questions, and in particular the role of time measurements in the determination of .

The outline of the report is contained in this worksheet. You will be filling in the details, giving explanations, and addressing questions posed along the way. The completed worksheet will constitute your report. Have fun.

Setup

Your Setup description can, and should, be brief, but remember that anyone reading your setup should be able to understand what kind of mathematical facts you are looking for and what methods you expect to be used. The Setup should provide a guide to what will be done in each of the parts below. Parts 1-4 constitute the Thinking and Exploring activities of the report. It is best to write the Setup description after you have completed them since by then you should understand the mathematical issues and techniques.

Part 1: The Mathematical Model

Part 2: Galileo's Data

Part 3: The Total Travel Time and The Gravitational Acceleration

Part 4: The Path of the Ball