Galileo's most important legacy comes from his studies on mechanics. It is in Pisa, around the age of 25, that he started to interest himself on the laws of motion and maybe even making some experiments, even though his most important work concerning this study was done between 1604 and 1609, in Padua.
Nonetheless, in Pisa he wrote a series of notes (that were meant to be published in a book, De Motu, that was never ultimated) disserting on motion. His approach is mainly philosophical, but became more mathematical as his work progresses. He discusses many topics, for example he says that the falling speed of different materials is related to the ratio between the density of the material and that of the medium.
Apparently his methods were not much appreciated in Pisa so that he left in 1592 to Padua, where he found a more favorable intellectual environment. In 1596 he published another book Le Meccaniche, in which he disserts about bodies falling on an inclined plane: the force acting on a descending body is to his weight as the height of the plane is to its length. In 1604, in a letter to his friend Paolo Sarpi, we learned about the development of his theories as he describes the time-squared law, according to which the distances of falling bodies increase proportionally with the square of time, that is d = A t^2 . He defined then an uniformly accelerated motion as one in which the speed at any point X is proportional with its distance OX from the origin, and he wrote, mistakingly, that from this definition the time-squared law could be mathematically derived. Such a demonstration implies that the average speed is proportional to the square of the falling distance, which means that it is proportional to the fourth power of time! Galileo will, later in his life, reject this definition and reformulate this law.
In the next years he studied thoroughly the fall of bodies on an inclined plane and formulated another law that says the speed of a body is, at any time, proportional to the time passed since the beginning of acceleration. His attention was driven to the study of the inclined plane because he developed a great interest in the motion of projectiles: rather than shooting real ones in the air he studied their falling motion, concluding that their trajectory of motion was a parable.
He abandoned definitively his first definition of accelerated motion, and reformulated it as the motion that, abandoning a steady position, in equal intervals of time, sums equal “moments of speed”. Another concept he stated is that the speed of a body falling without friction on an inclined plane is independent of the inclination angle.
Galileo’s theories were not mere speculation, they were supported by accurate experiments he makes. A description of his experimental apparatus can be found in his book, Discourses on Two New Sciences (1638): he used a wooden bar about 7 m long and 30 cm wide, polished as good as possible, on which he dropped a bronze sphere. For measuring time he hung a tank of water in a high position and on its bottom was a tiny pipe from which water would come out: this water was collected and weighted, and the ratios between weights would give the times ratios.
In this book he also sayed that no change of motion can happen without a cause, and only steadiness or uniform motion can be causeless, a concept that will be later reformulated to become the first law of Newton. What he never came to explain is how this motion arises. Also he could not solve the puzzle of why bodies of different magnitudes would behave identically when falling but, when subject to a sudden motion (that is a sudden force), the dimensions of the body became important. This enigma was later solved by Newton, introducing the concept of mass.
On the whole he left an important work. With the definition of accelerated motion and with the formulation of the law of falling bodies the basic structure for developing of mechanics was ready, even if he did not face the problem systematically and did not have a completely homogeneous vision of the laws of motion, as he could never overcome some inconsistences that were be later fixed by Newton and others.