Galileo, perhaps more than any other single person, was responsible for the birth of modern science – Steven Hawking 
Galileo was fascinated by motion and continually experimented with pendulums, cannons, and rolling balls to understand why bodies move the way they do. The arguable culmination of these efforts occurred in 1604 when he discovered what became known as “The Law of Fall.” The vertical distance travelled from rest (h) during free fall increases with the square of time (t).
h 𝜶 t2 Galileo’s Law of Fall
Galileo went on to assert that given the Law of Fall and given that the distance fallen (h) equals average speed (v) multiplied by time, then speed itself must be proportional to time.
v 𝜶 t
Combining the two relationships, Galileo arrived at one of the most significant discoveries in history.
h 𝜶 v2
Simply put, these findings were momentous, being the first to 1) identify the importance of v2 in science, and 2) discover the trade-off between what would become known as kinetic energy (1/2 mv2) and potential energy (mgh) as formalized in the conservation of mechanical energy (m = mass, g = gravitational acceleration) that was established around 1750.
1/2 mv2 + mgh = constant Conservation of Mechanical Energy
With this background, let’s now look at how Galileo accomplished this great feat.
The Law of Fall determined by an ingenious experiment
Hold a ball straight out in front of you. Then drop it. Watch how fast it accelerates and hits the ground. How could you possibly quantify this event, especially if you lived back in the 1600s absent of any kind of timing device? The fact that Galileo figured this out fascinates me. Here’s how he did it.
Galileo first focused on what he could directly measure: time and distance. But how did he measure these? When objects fall, they fall fast. So he slowed things down. He let balls roll down an inclined plane, which decreased the force in the direction of motion (look to your right). In this way he was able to mark on the plane distance from rest at fixed time increments.
But wait a minute! Fixed time increments? Yes! How do we know? Because we have the original data! One would have thought all of Galileo’s papers would have been analyzed by the 20 century, but in 1973, Stillman Drake, a leading Galileo expert, discovered otherwise . He was going through Galileo’s own notebooks and surprisingly unearthed the experimental data supporting the Law of Fall (look to your left). Galileo published the result but not the data leading to the result.
But wait another minute! How did Galileo measure those fixed time increments, especially in an era when the necessary timing devices didn’t even exist? Ah! This is where things get interesting, because Galileo didn’t say. Into this void stepped Drake. Drake suggested that since Galileo was raised in a musical world, then he likely had a deep respect for the strong internal rhythm innate to human beings. He proposed that Galileo made use of this by singing a song or reciting a poem and using the cadence to mark time with rubber “frets” along the incline during the experiments to create audible bumps when the ball passed by. By adjusting or tuning these frets, Galileo was able to accurately synch the bump sounds with his internal cadence, thus providing a means to achieve equal divisions of small time increments. This proposed approach is strongly supported by the fixed time increments in the data. To Drake, the only method that would result in accurate fixed time increments would be a fixed cadence. “But wait!,” you say, yet again. “How could this possibly provide the necessary accuracy?” Well, just observe yourself listening to live music when the drummer is but a fraction of a second off-beat. You cringe, right? This is because your innate rhythm is that strong.
Now let’s take a step back and consider the larger impact that Galileo had on science.
Galileo’s discoveries, including The Law of Fall, led to the rise of modern science. Here’re some reasons why.
The dawn of a new variable to science – time
Galileo was one of the first to use the concept of time as a dimension in a mathematical relationship. As noted by science historian Charles Gillispie , “Time eluded science until Galileo.” Linking time with another dimension, distance, opened the door to developing more complex relationships involving speed and acceleration.
Galileo’s brought mathematics into physics
Historically, physicists and mathematicians didn’t interact. Physicists resisted the use of math since professors in this area were invariably philosophers and not mathematicians. Galileo joined the two fields together by using a mathematical approach to describe and quantify the physical world and so test the hypotheses he formed. Moreover, he believed that, “[The universe] is written in the language of mathematics.”  and thus that mathematics could be used to describe all natural phenomena and conversely that all natural phenomena must follow mathematical behavior. In his search for the Law of Fall, for example, he believed that a simple equation existed and then found the equation.
The scientific method
Although we may not recognize it, we work today in a world largely created by Galileo. We make observations of some process or phenomenon and make a hypothesis, e.g., a mathematical model, to explain it. We then design experiments to generate data to test the hypothesis. Is it right or wrong? This approach is built on Galileo’s approach that favors data over preconceived ideas.
Galileo and the launch of the scientific revolution
[T] the study of nature entered on the secure methods of a science, after having for many centuries done nothing but grope in the dark. – Kant in reference to Galileo and others using experiments to understand nature. 
The scientific revolution arguably started, if one had to pick a year, in 1453 when Copernicus put the sun back at the center of the solar system where it belonged. But while Copernicus may have started the revolution, Galileo clearly fanned its flames. His conviction that practical and controlled experimental observation, enhanced by mathematical theory and quantification, was the key to a more satisfying understanding of nature became an inspiration to those who followed.
So what did Galileo truly do that made the difference?
Between Galileo and Aristotle there were just a lot of guys with theories they never bothered to test – Helen Monaco’s character in Philip Kerr’s Prayer: A Novel 
So what did Galileo truly do that made the difference? Data! It’s fascinating to know that while everyone from Aristotle to those immediately preceding Galileo thought about all sorts of things, many of the same things that Galileo was to think about, none of them took any measurements. Galileo measured while others thought. We see this around us today. Much thinking, proposing, and speculating. But without measurements, it really doesn’t mean anything. As a former boss of mine once wisely said, “One data point is worth one thousand opinions.” Rarely has this been better put.
Thank you for reading my post. I go into much greater detail about the life and accomplishments of Galileo Galilei (1564-1642) in my book, Block by Block – The Historical and Theoretical Foundations of Thermodynamics. It was Galileo’s work that helped eventually led to the 1st Law of Thermodynamics based on energy and its conservation.
The above illustrations are from my book. My thanks to Carly Sanker for bringing her great skill to creating them from my ideas. She is an excellent artist.
 Hawking, Stephen W., 1988, A Brief History of Time: From the Big Bang to Black Holes. A Bantam Book. Toronto: Bantam Books, p. 179.
 Drake, Stillman. 1973. “Galileo’s Discovery of the Law of Free Fall.” Scientific American 228 (5): pp. 84–92; 1975. “The Role of Music in Galileo’s Experiments.” Scientific American 232 (6): pp. 98–104.
 Gillispie, Charles Coulston. 1990. The Edge of Objectivity: An Essay in the History of Scientific Ideas. 10. paperback printing and first printing with the new preface. Princeton, NJ: Princeton Univ. Press. p. 42.
 Popkin, Richard Henry, ed. 1966. The Philosophy of the Sixteenth and Seventeenth Centuries. New York: The Free Press. p. 65.
 Kant, Immanuel. 1896. Immanuel Kant’s Critique of Pure Reason: In Commemoration of the Centenary of Its First Publication. Macmillan. p. 692.
 Kerr, Philip. 2015. Prayer: A Novel. G.P. Putnam’s Sons. p. 73.