If a nail and a toothpick are simultaneously dropped from the same
height, they do not reach the ground at exactly the same instant. (Try
it with these or similar objects.)

In Galileo's attack on the Aristotelian cosmology, few details were actually new. However, his approach and his findings together provided the first coherent presentation of the science of motion. Galileo realized that, out of all the observable motions in nature, free-fall motion is the key to the understanding of all motions of all bodies. To decide which is the key phenomenon to study is the real gift of genius. But Galileo is also in many ways typical of scientists in general. His approach to the problem of motion makes a good "case" to be used in the following sections as an opportunity to discuss strategies of inquiry that are still used in science.

These are some of the reasons why we study in detail Galileo's attack on the problem of free fall. Galileo himself recognized another reason-that the study of motion which he proposed was only the starting phase of a mighty field of discovery:

My purpose is to set forth a very new science dealing with a very ancient subject. There is, in nature, perhaps nothing older than motion, concerning which the books written by philosophers are neither few nor small; nevertheless, I have discovered some properties of it that are worth knowing that have not hitherto been either observed or demonstrated. Some superficial observations have been made, as for instance, that the natural motion of a heavy falling body is continuously accelerated; but to just what extent this acceleration occurs has not yet been announced .... Other facts, not few in number or less worth knowing I have succeeded in proving; and, what I consider more important, there have been opened up to this vast and most excellent science, of which my work is merely the beginning, ways and means by which other minds more acute than mine will explore its remote corners.

Two New Sciences deals directly with the motion of freely falling bodies. In studying the following paragraphs from it, we must be alert to Galileo's overall plan. First, he discusses the mathematics of a possible, simple type of motion (which we now call uniform acceleration or constant acceleration). Then he proposes that heavy bodies actually fall in just that way. Next, on the basis of this proposal, he derives a prediction about balls rolling down an incline. Finally, he shows that experiments bear out these predictions. By Aristotelian cosmology is meant the whole interlocking set of ideas about the structure of the physical universe and the behavior of all the objects in it.

In fact, more than mere "superficial observations" had been made long before Galileo set to work. For example, Nicolas Oresme and others at the University of Paris had by 1330 discovered the same distance time relationship for falling bodies that Galileo was to announce in the Two New Sciences. It will help you to have a plan clearly in mind as you progress through the rest of this chapter. As you study each succeeding section, ask yourself whether Galileo is

-presenting a definition

-stating an assumption (or hypothesis)

-deducing predictions from his hypothesis

-experimentally testing the predictions

The first part of Galileo's presentation is a thorough discussion of motion with uniform speed, ... This is sometimes known as the Rule of Parsimony: unless forced to do otherwise, assume the simplest possible hypothesis to explain natural events.

Rephrasing Gallieo and using our symbols: for motion with uniform speed v, the ratio Dd/Dt is constant. Similarly, recall that for accelerated motion, we defined uniform acceleration as DD

a=Dv/Dt=constant

Other ways of expressing this relationship are discussed. That leads to the second part, where we find Salviati saying:

We pass now to . . . naturally accelerated motion, such as that generally experienced by heavy falling bodies . . . . in the investigation of naturally accelerated motion we were led, by hand as it were, in following the habit and custom of nature herself, in all her various other processes, to employ only those means which are most common, simple and easy . . . When, therefore, I observe a stone initially at rest falling from an elevated position and continually acquiring new increments of speed, why should I not believe that such increases take place in a manner which is exceedingly simple and rather obvious to everybody? If now we examine the matter carefully we find no addition or increment more simple than that which repeats itself always in the same manner. This we readily understand when we consider the intimate relationship between time and motion; for just as uniformity of motion is defined by and conceived through equal times and equal spaces (thus we call a motion uniform when equal distances are traversed during equal time-intervals), so also we may, in a similar manner, through equal time intervals, conceive additions of speed as taking place without complication ....Hence the definition of motion which we are about to discuss may be stated as follows:

Sagredo: Although I can offer no rational objection to this or indeed to any other definition devised by any author whosoever, since all definitions are arbitrary, I may nevertheless without defense be allowed to doubt whether such a definition as the foregoing, established in an abstract manner, corresponds to and describes that kind of accelerated motion which we meet in nature in the case of freely falling bodies . . . .Here Sagredo questions whether Galileo's arbitrary definition of acceleration actually corresponds to the way real objects fall. Is acceleration, as defined, really useful in describing their observed change of motion? Sagredo wonders about a further point, so far not raised by Galileo:

From these considerations perhaps we can obtain an answer to a question that has been argued by philosophers, namely, what is the cause of the acceleration of the natural motion of heavy bodies . . . .But Salviati, the spokesman of Galileo, rejects the ancient tendency to investigate phenomena by looking first for their causes. It is premature, he declares, to ask about the cause of any motion until an accurate description of it exists:

Salivati: The present does not seem to be the proper time to investigate the cause of the acceleration of natural motion concerning which various opinions have been expressed by philosophers, some explaining it by attraction to the center, others by repulsion between the very small parts of the body, while still others attribute it to a certain stress in the surrounding medium which closes in behind the falling body and drives it from one of its positions to another. Now, all these fantasies, and others, too, ought to be examined; but it is not really worth while. At present it is the purpose of our Author merely to investigate and to demonstrate some of the properties of accelerated motion, whatever the cause of this acceleration may be.Galileo has now introduced two distinct propositions:

(1) "uniform" acceleration means equal speed increments, Dv, in equal time intervals, Dt; and

(2) things actually fall that way.

Let us first look more closely at Galileo's proposed definition.

Furthermore, both definitions seem to match our common sense idea of acceleration about equally well. When we say that a body is "accelerating," we seem to imply "the farther it goes, the faster it goes," and also "the longer time it goes, the faster it goes." How should we choose between these two ways of putting it? Which definition will be more useful in the description of nature? This is where experimentation becomes important. Galileo chose to define uniform acceleration as the motion in which the change of speed v is proportional to elapsed time Dt, and then demonstrate that this matches the behavior of real moving bodies, in laboratory situations as well as in ordinary, "un-arranged," experience. As you will see later, he made the right choice. But he was not able to prove his case by direct or obvious means, as you shall also see.

After Galileo defined uniform acceleration so that it would match the way he believed freely falling objects behaved, his next task was to devise a way of showing that the definition for uniform acceleration was useful for describing observed motions.

Suppose we drop a heavy object from several different heights say, from windows on different floors of a building. We want to check whether the final speed increases in proportion to the time it takes to fall-that is, whether Dv is "proportional to" Dt, or what amounts to the same thing, whether Dv/Dt is constant. In each trial we must observe the time of fall and the speed just before the object strikes the ground.

But there's the rub. Practically, even today, it would be very difficult to make a direct measurement of the speed reached by an object just before striking the ground. Furthermore, the entire time intervals of fall (less than 3 seconds even from the top of a 10-story building) are shorter than Galileo could have measured accurately with the clocks available to him. So a direct test of whether Dv/Dt is constant was not possible for Galileo.

Which of these are valid reasons why Galileo could not test
directly whether the final speed reached by a freely falling object is
proportional to the time of fall?

(a) His definition was wrong.

(b) He could not measure the speed attained by an object just before it
hit the ground.

(c) There existed no instruments for measuring time.

(d) He could not measure ordinary distances accurately enough.

(e) Experimentation was not permitted in Italy.

Galileo's inability to make direct measurements to test his hypothesis-that Dv/Dt is constant in free fall-did not stop him. He turned to mathematics to derive from this hypothesis some other relationship that could be checked by measurement with equipment available to him. We shall see that in a few steps he came much closer to a relation„ship he could use to check his hypothesis.

Large distances of fall and large time intervals for fall are, of
course, easier to measure than the small values of Dd
and Dt that would be necessary to find the
final speed just before the falling body hits. So Galileo tried to
find, by reasoning, how total fall distance ought to increase with
total fall time if objects did fall with uniform acceleration. You
already know how to find total distance from total time for motion at
constant speed. Now we will derive a new equation that relates total
fall distance to total time of fall for motion at constant
acceleration. In this we shall not be following Galileo's own
derivation exactly, but the results will be the same. First, we recall
the definition of average speed as the distance traversed Dd divided by the elapsed time Dt:

This is a general definition and can be used to compute the average speed from measurement of Dd and Dt, no matter whether Dd and Dt are small or large. We can rewrite the equation as

This equation, still being really a definition of v

The answer involves just a bit of algebra and some plausible
assumptions. Galileo reasoned (as others had before) that for any
quantity that changes uniformly, *the average value is just halfway
between the beginning value and the final value*. For uniformly
accelerated motion starting from rest (where v_{initial} = 0
and ending
at a speed v_{final} this rule tells us that the average speed
is halfway. More generally the average speed would be between 0 and v_{final}
- that is,

v

(More generally, the average velocity would be

If this reasoning is correct, it follows that

Now we look at Galileo's definition of uniform acceleration: a = Dv/Dt. We can rewrite this relationship in the form

Dv = a x Dt. The value of Dv is just v

v

v

= 1/2 (a x Dt) x Dt

Or, regrouping terms.

This is the kind of relation Galileo was seeking-it relates total distance Dd to total time Dt, without involving any speed term.

Before finishing, though, we will simplify the symbols in the
equation to make it easier to use. If we measure distance and time from
the position and the instant that the motion starts (d_{initial}=
0 and t_{initial} = 0), then the intervals Dd
and Dt have the values given by d_{final}
and t_{final}. Because we will use the expression d_{final}/
t^{2}_{final} , many times, it is simpler to write it
as
d/t^{2}

-it is understood that d and t mean total distance and time
interval of motion, starting from rest. The equation above can
therefore be written more simply as

Remember that this is a very specialized equation-it gives the total distance fallen as a function of total time of fall but only if the motion starts from rest (v

Galileo reached the same conclusion, though he did not use algebraic forms to express it. Since we are dealing only with the special situation in which acceleration a is constant, the quantity 2a is constant also, and we can cast the conclusion in the form of a proportion: in uniform acceleration from rest, the distance traveled is proportional to the square of the time elapsed, or

For example, if a uniformly accelerating car starting from rest moves 10 m in the first second, in twice the time it would move four times as far, or 40 m in the first two seconds. In the first 3 seconds it would move 9 times as far-or 90 m. Another way to express this relation is to say that the ratio d

Why was the equation d = 1/2at^{2} more promising for
Galileo than a = Dv/Dt
in testing his hypothesis?

If you simply combined the two equations Dd
= v x Dt and Dv
= a x Dt it looks as if one might get the
result Dd = a xDt^{2}.
What is wrong with doing this?

Realizing that a direct quantitative test with a rapidly and freely falling body would not be accurate, Galileo proposed to make the test on an object that was moving less rapidly. He proposed a new hypothesis:

Thus Galileo claimed that if d/t^{2} is constant for a
body falling freely from rest, this ratio will also be constant,
although smaller, for a ball released from rest and rolling different
distances down a straight inclined plane.

Here is how Salviati described Galileo's own experimental test in *Two
New Sciences*:

A piece of wooden moulding or scantling, about 12 cubits long, half a cubit wide, and three finger-breadths thick, was taken; on its edge was cut a channel a little more than one finger in breadth; having made this groove very straight, smooth, and polished, and having lined it with parchment, also as smooth and polished as possible, we rolled along it a hard, smooth, and very round bronze ball. Having placed this board in a sloping position, by lifting one end some one or two cubits above the other, we rolled the ball, as I was just saying, along the channel, noting, in a manner presently to be described, the time required to make the descent. We repeated this experiment more than once in order to measure the time with an accuracy such that the deviation between two observations never exceeded one-tenth of a pulse beat. Having performed this operation and having assured ourselves of its reliability, we now rolled the ball only one-quarter of the length of the channel; and having measured the time of its descent, we found it precisely one-half of the former. Next we tried other distances, comparing the time for the whole length with that for the half, or with that for two-thirds, or three-fourths, or indeed for any fraction; in such experiments, repeated a full hundred times, we always found that the spaces traversed were to each other as the squares of the times, and this was true for all inclinations of the . . . channel along which we rolled the ball…

This picture painted in 1841 by G. Bezzuoli, attempts to reconstruct an
experiment Galileo is alleged to have made during his time as lecturer
at Pisa. Off to the left and right are men of ill will: the
blasé Prince Giovanni de Medici (Galileo had shown a
dredging-machine invented by the prince to be unusable) and Galileo's
scientific opponents. These were leading men of the universities; they
are shown here bending over a book of Aristotle, where it is written in
black and white that bodies of unequal weight fall with different
speeds. Galileo, the tallest figure left of center in the picture, is
surrounded by a group of students and followers.

Angle of Incline

For each angle, the acceleration is found to be a constant.
Spheres rolling down planes of increasingly steep inclination. At
90° the inclined plane situation matches free fall. (Actually, the
ball will start slipping instead of rolling long before the angle has
become that large.)

Galileo has packed a great deal of information into these lines. He describes his procedures and apparatus clearly enough to allow other investigators to repeat the experiment for themselves if they wished. Also, he gives an indication that consistent measurements can be made, and he restates the two chief experimental results which he believes support his free-fall hypothesis. Let us examine the results carefully.

(a) First, he found that when a ball rolled down an incline at a fixed angle to the horizontal, the ratio of the distance covered to the square of the corresponding time was always the same. For example, if d

In general, for each angle of incline, the value of d / t_{1}^{2}
was constant. Galileo did not present his experimental data in the full
detail which has become the custom since. However, his experiment has
been repeated by others, and they have obtained results which parallel
his. This is an experiment which you can perform yourself with the help
of one or two other students.

(b) Galileo's second experimental finding relates to what happens when
the angle of inclination of the plane is changed. He found that
whenever the angle changed, the ratio d / t^{2} took on a new
value, although for any one angle it remained constant regardless of
distance of roll. Galileo confirmed this by repeating the experiment *"a
full hundred times"* for each of many different angles. After
finding that the ratio d / t^{2} was constant for each angle of
inclination for which measurements of t could be carried out
conveniently, Galileo was willing to extrapolate. He concluded that the
ratio d_{l} / t^{2} is a constant even for larger
angles, where the motion of the ball is too fast for accurate
measurements of t to be made. Finally, Galileo reasoned that in the
particular case when the angle of inclination became 90°, the ball
would move straight down-and so becomes the case of a falling object.
By his reasoning, d/t^{2} would still be some constant in that
extreme case (even though he couldn't say what the numerical value
was.)