4.1 Galileo's Important Experiment
Galileo was born in Pisa in 1564, where he grew up in a musical family. His father, Vincenzio, was an esteemed lutist who also made controversial experiments with intonations, intervals, and tuning that challenged ancient authorities [1]. His brother became a professional musician and Galileo played the lute, just as his father. He entered the University of Pisa to study medicine, but soon switched to mathematics. Vincenzio's zest for experimentation seems to have rubbed off on Galileo, because when he got a position at the university in 1589, he began to study the motion of falling bodies.
According to Aristotle, bodies fell at a constant speed that was proportional to their weight. Since free fall typically occurs very fast it is difficult to form an idea of it from direct observation. To be able to study it at all, Aristotle had evidently slowed objects down by letting them fall in water [1]. He had concluded that, in the absence of a resisting medium such as water or air, the speed of fall would be infinite. In Aristotle's view of the world, things fell because they sought their natural place at the center of the universe. He thereby made a difference between the “natural”, downward motion of falling objects and “violent”, upward motion that he considered contrary to nature.
Galileo showed by thought experiment that Aristotle was wrong in thinking that the speed of fall was proportional to the weight of the falling object. If we suppose that a heavier stone falls faster than a light one, he reasoned, tying the stones together would result in a paradox. The lighter one would then retard the fall of the heavier stone, and the heavier stone would speed up the fall of the lighter one. Despite the fact that they are heavier when tied together they would fall slower than the large stone does on its own [2].
According to tradition, he also demonstrated in a legendary experiment at the Leaning Tower of Pisa that objects of different weight fall at the same speed. The only known account of this experiment arose shortly before Galileo's death and historians of science often doubt that it ever took place. On the other hand, the experiment is so easily performed that it is difficult to imagine that Galileo would have settled with just the thought experiment. Whether he conducted it at the Leaning Tower or not, he probably tried it in one variant or other. In any case, the experiment was not unique to Galileo. Others had performed it before him and shown that Aristotle's ideas about free fall did not hold water. Even though flaws were known in Aristotle's physics, they were generally not considered serious. Galileo's contribution was to demonstrate that the errors were not superficial. Aristotle's system could not be easily fixed – it had to be replaced.
So, Galileo knew that objects of different weight fell through a given distance in the same time. The question was how they acquired their speed. Did it occur instantly at the point of release? Or did they get a series of small and rapid spurts of uniform speed, which had been the common picture during the Middle Ages? The concept of uniform acceleration was not yet known. Galileo realized that the speed acquired by an object in free fall increased with the distance it fell. A pile driver, driving a stake into the ground, doubled its effect if it was dropped from a doubled height. He observed that the weight of the driver stayed the same, so the effect must be due to a change in its speed. In his last book, Dialogues Concerning Two New Sciences, he claimed that the falling body acquired its speed through continuous, uniform acceleration. This meant that the object acquired equal increments of speed in equal intervals of time. Since any time interval may be divided into an infinite number of instants, he said, the acceleration must be continuous [2]. In the same book he describes an ingenious experiment from which he obtains a precise mathematical law, describing the motion in free fall. The law relates distances fallen from rest to the times required to cover those distances. Two distances fallen by an object in free fall, s1 and s2, are to each other as the squares of the times required to cover those distances, t1 and t2, or:
(4.1) 
We would say that the distance is proportional to the square of the time.
The impact of the law was tremendous. Before it, any event in Nature had to be ascertained by observation. Now, an aspect of the world's behavior could be computed with a pencil and paper. It allowed scientists to predict future events and retrodict past events.
Galileo was convinced that slowing a falling object down with water, as Aristotle had done, would disturb the motion more than it would illuminate its nature. Instead of impeding the fall of an object in a resisting medium he extended the time of fall by letting a ball roll down an inclined plane. The greater the angle of inclination, he reasoned, the closer the motion would be to free fall. Decreasing the angle of inclination, the fall would take place over longer times, making it possible to study. By varying the inclination he hoped to find the general characteristics of the motion.
He prepared a wooden board for the experiment by cutting a channel along its edge. It was straight, smooth, and polished. The channel was lined with parchment, which was also as smooth and polished as possible. “After placing the board in a sloping position”, he said, “a hard, smooth, and very round bronze ball was rolled along the channel”. He noted the time required to make the descent; the measurement was repeated more than once until the deviation between two observations would not exceed one tenth of a pulse beat [2].
After this he measured the times needed to cover other distances on the board. He compared these with the time he had measured for the whole descent. Rolling one quarter of the length, for example, took exactly half the time of the whole length, as illustrated in Figure 4.1. “[I]n 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 plane” [2]. He had obtained Equation 4.1.
Figure 4.1 Schematic description of the experiment on the inclined plane. At the end of the slope is a wooden block. The two balls represent two starting points for the descent, and the distance s is one fourth of the total length. Galileo found that it took the ball exactly half the time to cover this distance as compared to the whole length.
To obtain an accurate mathematical law he had to measure time with sufficient precision. Since there were no mechanical clocks in Galileo's day, this was a crucial part of the development of the experiment. He describes a simple timekeeper in his book, built from a large vessel of water. A small pipe was soldered to its bottom to produce a thin jet of water that could be collected in a small glass during the time of each descent. After the descent he weighed the water on a “very accurate balance”. The ratios of these weights, he said, gave the ratios of the times, “with such accuracy that there was no appreciable discrepancy in the results, although the operation was repeated many, many times” [2].
This experiment was groundbreaking in more than one way. Scientists had tried to understand velocity in terms of distances, but Galileo realized that it was more productive to study it in terms of time. This made it possible for him to introduce the concept of acceleration. The experiment also marks the beginning of modern experimental science by being the first systematically planned experiment that resulted in a precise mathematical law [1]. In going from passive observation of the heavens to active experimentation, science had moved from being a spectator to taking the director's seat. Instead of sitting back and enjoying the show, it set the stage for Nature's performances and took command of the events on it.
The law of free fall made it possible for Galileo to establish that projectiles follow parabolic trajectories. As described in the previous chapter, this was the trajectory that Newton identified with the orbit of the moon when he merged the celestial and terrestrial mechanics. Galileo simply divided the motion of a projectile into a horizontal part and a vertical part. The horizontal motion was unimpeded and uniform, while the vertical motion obeyed the law of free fall. The result was a parabola. This meant that there was no difference between Aristotle's violent, upward motion and the natural, downward motion. They were one and the same motion, described by a single law.
Galileo's experiment on the inclined plane has passed into history as a turning point in science but questions have been raised about it. Was it really an experiment? Some have claimed that it cannot have been more than a physical demonstration of the law of free fall, which he somehow must have obtained by other means. Did Galileo really have the precise experimental means required to determine his accurate law?