Free Fall
S. Gavin and S. P. Karrer
Physics Department, Wayne State University, Detroit, MI, 48201
[email protected]
I. Introduction
Our modern concept of free fall as motion under constant acceleration was the result of
roughly 1500 years' work. The solution of this problem by Galileo and his predecessors is a
particularly clear historical illustration of the utility of the scientific method. In this case study
we will look at what Galileo actually did in order to see how it fits with the modern conception
of the scientific method. But first, we will consider a few of Galileo’s intellectual ancestors to
see how pieces of the scientific method emerged (or did not emerge) in their work.
When thinking about the problem of free fall it is useful to keep the following points in mind:
1. The first people to consider this problem were only beginning to understand the
concepts of “velocity” and “acceleration”. They did not necessarily use these key
words as we now use them. In fact, some of them used “velocity” and “acceleration”
interchangeably, as in common speech. This was really confusing for them. The lack
of a standard jargon also confuses historians in their efforts to figure out what these
people were thinking.
2. Galileo did not have a formula sheet! Most his predecessors did not know any algebra
at all. Nobody in those days had considered representing physics quantities as letters
in algebraic equations. You might view equations as something that makes physics
complex – many students do! Yet, with a few key formulas and the definitions of v and
a, you were able to do the homework for this chapter in an hour.
3. Science was not a organized field of study when this problem was addressed. Back
then, they used to call science “natural philosophy” and, indeed, many of the people
who uncovered aspects of the physics of free fall were far more interested in
philosophical questions. A lot of the results prior to Galileo came up as side issues in
lengthy texts on other topics.
Finally, a disclaimer: Most treatments of science history – particularly those in physics classes
– simplify and rearrange the historical facts in order to clarify the physics principles. This
tends to launder the inherently murky processes of discovery and hide the scientific method.
We will also be guilty of this to some extent, although we will try to avoid it.
The scientific method in general is discussed in The Trouble with Science by Robin Dunbar,
(Harvard University Press, Cambridge, Mass.,1995). Chapters 1 – 3 provide a sufficient
background for this course.
Our discussion here follows the History of Free Fall by Stillman Drake, (Wall and Thompson,
Toronto, 1989). This book outlines the major contributors in the physics of falling from
Aristotle to Galileo.
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II. History
Aristotle formulated the first theory to explain the reason objects fall. He proposed that the
“natural state” of any object is to be at rest on the Earth. Objects that were forced to move
were given moving power by something else, such as the contact of a hand or foot. After
leaving the hand, the motion of a projectile is somehow sustained by the air, which keeps it
moving for some finite time. One might imagine that air rushes in to fill the void behind the
moving object, pushing it further along. But no matter the explanation, all things progressed
downward toward the center of the universe, which was also the center of the Earth.
Aristotle considered the different speeds of falling objects, arguing that a particular object will
fall more swiftly in a less dense medium. This would explain, for example, the slower rate of
descent of a rock falling through water as compared to air.
A common interpretation of Aristotle’s reasoning is that heavier objects would fall faster in a
given medium. Thus, a 5 kg iron ball would fall faster than a 1 kg ball. Whether or not Aristotle
actually said this is debatable. After all, Aristotle did not write all of his ideas down, and a lot
of what he did write may have been lost. But for centuries, Aristotle’s followers would cite him
as saying that the velocity of a falling object should be proportional to its weight (or mass).
Interestingly, Aristotle did not perform any experiments to test his ideas on free fall. However,
he did understand the importance of observations and experiments in his studies of other
aspects of nature. For example, he performed dissections to study the nature of animals.
Q1. To the philosophers of the time, Aristotle’s theory was considered a complete
explanation. How would it fit into the “scientific method” as we understand it today? [This
question has no “right” answer – its meant to generate discussion. Is it a hypothesis?
There are not many predictions that are concrete enough to test by experiment. The
dependence of fall velocity on medium is essentially correct. The theory that things are
attracted to the center of the Earth really couldn’t be tested by the ancients.]
Q2. How could we test whether air it responsible for continued motion? [One answer is to
conduct an experiment in a vacuum, but that was out of reach for Aristotle (who
thought a vacuum impossible anyway). Students may propose an experiment on the
motion of differently shaped bodies. There was such an experiment done on spears
with different tail ends (blunt versus tapered).]
Q3. What other aspects of Aristotle’s theory can or cannot be experimentally tested? [You
can drop the aforementioned iron balls to disprove that velocity depends on mass. If
you drop a feather and a hammer in air, however, this will seemingly support Aristotle.
Jupiter’s moons show that Earth isn’t the center of the universe, but that needs a
telescope. You also need Newtonian theory to relate orbits to gravity.]
Thomas Bradwardine developed a theory of the speed of falling bodies in the 14th century.
This theory is now called “Bradwardine's function,” even though he never expressed it
mathematically himself. In essence, he said that the speed of an object was equal to the ratio
of the motive power to resistance. Using modern algebra, we would express this statement as
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v=
F
,
R
(1)
where F is the “motive power” and R is the “resistance”. If gravity somehow provides the F,
then this formula would explain free fall.
He never expressed this idea in the form of (1) because equations had not yet been invented.
He wrote:
The ratio of speeds in motions follows the ratio of the motive powers to
resistances, and conversely. Or, to put the same thing in a another way,
the ratios of motive powers to resistances, and the speeds in motion,
exist in the same order of proportion.
He meant to explain why applying force to some objects would not make them move as much
(a large stone, for example). It seems that he did not imagine that his statement would imply a
formula for calculation or prediction of motion, like our modern mechanical relationships.
Nevertheless, it is significant that he is thinking in terms of proportionalities.
Q4. How is Bradwardine's theory a jump forward? How is it still fundamentally flawed?
[Bradwardine used math in physics and tried to create an objective, natural
explanation. It is flawed, for example, because it implies velocities are reached
instantaneously; see, however, Q5.]
Q5. How would we define v, F, and R today? [This is the formula for the terminal velocity
of an object falling in a dense medium like water. Think of F as the force of gravity and
Rv is the force of friction. The terminal velocity is reached when F=Rv, which gives (1).]
Q6. How is Bradwardine’s formula (1) consistent with Aristotle’s theory? Hint: think of your
answer to Q4. [The resistance R is larger in a higher density medium, like water.
Equation (1) implies that the object in a more dense medium would fall slower, as
Aristotle argued.]
Jean Buridan broke away from past philosophers with the idea of impetus. Previously, it was
thought that the projecting force was “unnatural” and, consequently, would naturally diminish
with time. In particular, Aristotle thought that the air around a projectile maintained its motion,
rather than resisting its motion via friction.
Buridan proposed instead that moving objects have a property called impetus that determines
how fast it moves. He claimed that impetus must encounter opposition to be reduced. This
impetus was then a property of the body, independent of the surrounding air. In free fall, he
argued that the weight of an object acted to add further impetus to that object. In each instant,
impetus would increase. This increase would produce an increase in speed.
Q7. What is its closest modern equivalent to impetus? [Impetus is closely equivalent to
Copyright 2008 Wayne State University. For educational use only. For permission to use: [email protected]
momentum, although Buridan never said that impetus = mv, nor did he emphasize its
vector nature. Nevertheless, Burdain’s observations are ancestors to Newton’s first
law. Both are seen as natural states and are properties of motion. Impetus is different
because it is described in a tangible way, as if impetus were an ethereal substance.]
G.B. Benedetti picked up the idea of impetus in the mid 1500’s. By that time, the theory had
been all but forgotten. He used this idea to argue that a falling body accelerates because its
impetus increases as it gets closer to the earth. Importantly, he also argued that objects of
different weights would fall at the same rate, in contrast to the theory of Aristotle. He argued
as follows: consider two equal weights connected by a weightless string. They must fall at the
same rate as a single body with their combined weight, since they are joined. If we now cut
the string, the two equal bodies must continue to fall at the same speed as before.
The conclusion that the rate of fall is independent of weight is often attributed to Galileo. In
fact a similar “thought experiment” appears in Galileo’s writings. Historians argue over
whether Galileo had been aware of Benedetti’s work.
Simon Stevin was the first to experimentally test the ideas of Buridan and Benedetti and
documented his results in 1586. It is astounding that this may have been the first free-fall
experiment ever. Stevin's experiment involved dropping two lead spheres of vastly different
weight from the same height. It was found that they reached the ground at nearly the same
instant.
In general, the scientific community before this was more interested in logic and reasoning
than proof. Stevin's experiment was briefly noted and he moved on to the mathematical and
philosophical questions that were his primary interest.
Q8. How do the joint efforts of Benedetti and Stevin fit into the steps of the scientific method?
How is it similar to the way modern science works? [Benedetti’s hypothesis leads to a firm
prediction of equal rates of fall. This is tested by Stevin’s experiment. It is similar as
theorists today will have their ideas tested by experimentalists. It is different because
this was hardly a collaboration. The theory had been presented as true before being
tested.]
Q9. Discuss the value of Benedetti’s contribution. Why is this the key to applying the scientific
method to the free-fall problem? [It is the first hypothesis that could easily be
experimentally tested with the equipment of the day. All the other guys were too
vague.]
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III. Galileo
In the early 1600’s, Galileo realized that one could characterize the motion of a freely falling
object by studying how far the body fell in a given time interval. Through step-by-step
reasoning and careful experimentation, he found that the distance an object falls is
proportional to the square of the time. To discover this law, he needed to be able to
accurately measure distance and time and, therefore, had to invent several techniques to do
so. Clocks of the day did not have a second hand. Galileo started out measuring time by
humming a tune and using the rhythm as a metronome. A true renaissance man would be a
musician, after all! He later developed a water clock, which was essentially an hourglass that
used water.
But the problem is that falling objects fall very fast, making it hard to note the distance
traveled at each “beat.” Galileo slowed the motion down by rolling balls down a shallow
incline. In an intuitive leap, he realized that making the incline steeper and would make the
motion closer and closer to a sheer drop. Today we would view this as a limiting process.
Galileo realized that he could use the slow roll of a heavy ball down a shallow incline as a first
step to understanding the free-fall problem.
Galileo’s basic apparatus is as follows: an inclined ramp with a groove suitable for a certain
sized ball. The ball was rolled down from various heights.
Though this method did not give time in terms of seconds or any other regular unit, this did
not matter. Galileo worked in arbitrary units, proportions, and ratios, which later canceled out.
He was able to see that if the time elapsed in the motion was doubled, the distance covered
was quadrupled.
Though he did not have the mathematical language to express it himself, Galileo had
determined the simple relationship regarding gravity: the acceleration a = Δx/Δt is constant.
This implies that in any given amount of time, an object will gain the same amount of velocity.
Assuming this, we can develop other relationships of motion. Consider distance traveled
during a certain time with a certain average velocity. By definition, we have vav = Δx/Δt. If we
are accelerating constantly, vav = 1/2(vinitial+vfinal) and from rest: vinitial = 0 and vav = vfinal/2. The
amount of velocity gained during constant acceleration from rest is vfinal=aΔt. This means that
vav = aΔt/2 or that
!x = 12 a!t 2 .
(2)
The point is that Galileo’s observations had to be accurate enough to really demonstrate that
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Δx is proportional to the square of Δt. His later water clock measurement was very accurate;
he even accounted for water which stuck to the side of any container used. Galileo developed
a simple valve he could open and close quickly. The author points to sketches and a high
level of accuracy as evidence he did not simply plug the tube with his finger. However, a
modern experimentalist was able to obtain results more accurate than Galileo's with a simple
hand-release system.
Galileo next turned his attention to freely falling objects, with the hypothesis that the
acceleration would also be constant for that motion. His experiment involved dropping a fairly
heavy weight from a rather tall height. Indeed, Galileo's experiment backed up this equation.
Observe that it would have been very difficult to establish equation (2) from direct
measurement given the inaccuracy of Galileo’s techniques. Before Galileo had started, he
had no idea that Δx would be proportional to the square of Δt. However, once he had
established that result using the inclined plane, he could check that it was also consistent with
freely falling objects by dropping weights from a few different heights.
Q9. Galileo focused on measuring the dependence of Δx on Δt. What if he had instead chosen
to study a) the change of velocity Δv in time Δt, or b) the change of velocity Δv over the
distance Δx? How could he have done this experimentally? Would the result have been
equivalent in theory?
Q10. How does Galileo’s method fit into the steps of the scientific method?
Q11. Examine each step in Galileo’s chain of reasoning. How does each help or hinder
experimental progress? For example, could the water clock be relied upon for consistent
timing?
Discussion Questions: In what sense is Aristotle’s theory more complete than Galileo’s? In
what sense is Galileo’s result “more important” than those of his predecessors?
Other sources:
http://www.iit.edu/~smart/martcar/lesson2/lesson2.htm
This site starts by saying Galileo came out and talked about constant acceleration. This is
somewhat contrary to the book. The mathematics and symbols used in the description were
supposedly not available to Galileo at the time.
Copyright 2008 Wayne State University. For educational use only. For permission to use: [email protected]