## Quadratics

### 1.4 Galileo’s Discovery

Lesson materials located below the video overview.

It was believed since the time of Aristotle (ca. 350 BCE) that heavier objects fall through the air with faster acceleration than lighter ones.

Italian mathematician and physicist Galileo Galilei (1564-1642) questioned this. He reasoned that if this were true, two objects of identical shape and volume but of different masses dropped from the top of the Leaning Tower of Pisa would touch the ground at different times. (The heavier object should land first.) But if we were to tie a very light string between the two objects and thereby make them one connected entity, both the individual masses should somehow “know” they are now one object and land at the same time. This doesn’t make sense.

He deduced that all objects must, instead, fall through the air in unison. Moreover, he suspected they do so with constant acceleration. (Air resistance may affect objects differently, however. This is why a piece of paper is slower to fall than a paper-clip.)

Letters and biographies about Galileo written at and near the time say that he considered going to the top of the Leaning Tower of Pisa to drop and time falling objects, so as to verify that their accelerations were constant. (There is no evidence, however, that he actually did this, only that he talked about possibly doing it.)

Whether or not he performed the experiment, he was right: it is true that acceleration due to gravity is the same for all objects moving the air, either from being thrown or from being dropped.

**WHAT GALILEO PERHAPS HOPED TO SEE:**

If Galileo had the tools to properly time falling objects with accuracy, he most likely would have hoped to collect a table of data along the following lines:

Here \(t \) is the time passed (measured in seconds) and \(h \) is height of object (measured in feet). We start by dropping the object from a height of 100 feet. (COMMENT: Don’t trust this table. I concocted these numbers for the sake of describing this story!)

What does this height data reveal about acceleration?

Well …

*Velocity* is the rate of change of position (or, in our case heights).

*Acceleration* is the rate of change of velocity.

Since we have worked with constant time differences, velocity from data point to data point can be computed via the first differences (change of position).

And acceleration, from data point to data point, is linked with differences of velocity:

And Galileo hoped to see constant acceleration by noting constant second differences.

**GALILEO’S REALIZATION:**

Galileo deduced that if we measure the height of an object moving through the air over constant time intervals, the second differences should be constant. And according to the mathematics of this section on the book, this means that the height of the object changing over time must be given by a formula of the form:

\(at^{2} + bt + c\)

Double differences constant means a formula involving squares and less!

Any formula involving the square of a variable (and smaller powers) is called a *quadratic* and these types of formulas have been studied by mathematicians for centuries. This online course is all about that work.

An object flying through the air does two things:

i) It moves horizontally at a constant rate,

ii) Its vertucal height changes according to a quadratic formula.

The U-shape curve one sees in watching a tossed object (at least in ideal conditions without air resistance, and wind effects, and the like) is the curve of a quadratic formula:

Now that we have meaning and context, we’re now primed to learn all there is to know about the mathematics of quadratics!

But first … check out the final lesson in this section of the course!

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