Freely Falling Objects

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As we said, these lectures are about that wonderful unification of the two questions:  How do objects fall? And how do the Moon and the Planets move?  We will see that these are really the same question.  We begin, with my teacher Galileo, to take up the first question in this introductory chapter.  The second question will, as we said, lead us to the Calculus itself. 

We begin this introduction to the ideas of the Calculus with the study of freely falling objects.  The topic of this part is Freely Falling Objects, and in it you will discover an interesting fact about objects that fall freely.  We must abstract away of course the resistance of objects to the air and wind.  We are really talking about gravity itself, and about a simple fact.

Uniform Acceleration

Gravity acts on the speed of a falling object directly, and not directly on its position.  Gravity does not care how heavy an object is, or how it is shaped.  Once an object begins to fall, gravity causes its downward speed to grow in a uniform way.  Over equal intervals of time, the speed changes by equal amounts (at least when this happens close to the Earth).  This is called uniform acceleration, and that acceleration can be measured.  We will show in this exploration a natural way to measure it. 

Galileo's Formula

If we choose a unit of time T (say 1 second) then there is a distance D associated with it by the Earth itself.  That distance D is the distance any object would fall from rest in the interval T of time.  Near the earth, it is roughly 4.9 meters, or a little over 16 feet.  It is a natural unit to use, because, according to Galileo, the Distance (measured in units of D) that an object will fall in a time interval of length t (measured in units of T) is then

This is an experiment in which you may discover the law of "freely falling bodies". You choose an interval of time that you will call a "unit" time. This is, of course, an arbitrary choice. So you may choose it freely at the start of the experiment.The length of each duration will be determined by the number you set with the scroll bar:

As it happens, the unit of time does not matter. It could be seconds, or minutes or hours! Now Galileo reasoned (in a way that we will understand clearly only later) that if, when an object falls freely from rest, its velocity grows by equal amounts over equal intervals of time, the the distance it will fall through in an interval of time T is the distance it would fall if it fell at a constant velocity (the average of the values of its velocity at the beginning of the interval and at the end of the interval) over the same time interval T. This reasoning is based on an old Greek style of thinking that is due to Archimedes. It will be justified for us later when we study the technique of integration in the Calculus.

Galileo went on to show that if we choose as unit if distance, the distance D that the object fell in the first time interval T at the beginning of which it was dropped, then, while in the first interval, it fell distance D, in the next interval it would fall , and in the next, it would fall and so on. And if you ask what the total distance it falls over n intervals of length T, from the time it was dropped, you simply add up these odd numbers and discover that it must fall through a distance equal to the square of t units of distance D. That is what this experiment shows.

Thus, Galileo discovered a wonderful thing. As a body falls from rest, the intervals of distances it traverses in equal intervals of time have the ratios with the first interval of the successive odd integers: 1, 3, 5, 7, 9 and so on. It doesn't matter what unit we choose to measure time, or what unit we choose to measure distance. This is always true!

Now suppose that D is the distance the object fell in the first interval of time. Then in the second interval, it would fall and in the next interval it would fall and so on. We see that the distance the object falls by the end of the Kth interval is

Question 1: Show that the sum of the first K odd integers is the square of K.

When you press the button: the blue ball will begin to fall from the top of the screen on the left. It will fall very slowly at first because the simulation needs to make some accurate calculations in order to give meaningful results. It would fall slowly at first in any case because that is what freely falling bodies do!

In the blue graphics window, you will notice a series of rectangles being drawn. The heights of these rectangles represent the distances fallen over successive equal durations of time. As we said, Galileo noticed that the lengths of those intervals were not equal. This meant that freely falling bodies did not move "equably". They did not traverse equal distances in equal times.

But he did notice an interesting pattern in the lengths of successive intervals fallen. They got longer, to be sure, but they did it in the way just described. In order to help you see what Galileo saw, we placed a MathEdit on the page. It has three columns.

In the first column, is step number, the number of the time interval over which distance is measured. And in the second column is the set of distances traversed, over the first interval, then over the second, then over the third, et cetera, as Galileo might have measured them. In the third column, you see the ratios of distances fallen over successive intervals of equal duration tothe distance fallen over the first interval. This simply measures the distances in terms of the latter unit distance.

The numbers, themselves, are of course rather arbitrary. This is because they depend both on our clock and our ruler! The point of this lab is that there is something that does not depend on these things, and that is what Galileo found. With careful observation, you may find it too, if you do the following exercises. In any case, on the next page, we will give some hints that may help make things clearer.

Question 2: What does the fact that the intervals are getting longer mean for the motion of the ball?

Question 3: What pattern do you observe in the distances that are traversed in equal intervals of time? The numbers in the MathEdit column are quite accurate, but they are not exact, since the falling body was simulated with a differential equation. Try to guess the exact pattern.

Question 4: Vary the duration of the interval with the scroll bar. Try values 0.2 , 0.3, and 0.4. What remains the same in the MathEdit, and what does not? Can you formulate a law of falling bodies from these observations that is independent of units? It would say that the total distance fallen after the "nth" interval of time is proportional to some expression involving "n". An example of such a law is Galileo's law of freely falling bodies. They are rare!

Question 5: We neglected to mention the weight of the object. That is because it does not matter! This isn't obvious from this single experiment. But this is why Galileo is said to have performed the "Tower of Pisa" experiment, where he dropped two objects of different weights, but with the same shape off of the Tower. Actually, he probably never performed that experiment, because there are other less tiresome ways (There were no elevators in those days) to come to the same conclusion. In fact, he probably did not make measurements on objects falling freely, but he "slowed them down." See the coming section on Inclined Planes. Suppose now that the unit of time is seconds, and the distance is measured in meters. Estimate the speed with which the object is falling after precisely 1 second, 2 seconds, and 3 seconds. So the duration of each interval varies as you move the scroll bar between 0.1 and 0.5 sec. How can you make your estimates more precise using Galileo's law?