Final answer:
Galileo found laws of falling objects by careful experimentation and observation, concluding that objects accelerate uniformly regardless of mass and using mathematical formulations to predict motion. His findings greatly influenced the shift towards experimental science and paved the way for Newtonian physics.
Step-by-step explanation:
Galileo's approach to finding the laws of falling objects was to conduct experiments and make precise observations of how objects accelerate, including experiments where objects fell freely or rolled down a ramp. He discovered that objects accelerate uniformly, meaning that in equal intervals of time, they gain equal increments in speed, regardless of their mass. Galileo's experiments, such as supposedly dropping two objects of differing mass from the Tower of Pisa and observing their fall times, helped to establish the principle that in the absence of air resistance, all objects fall at the same rate. His use of mathematical formulation to express these laws enabled others to predict how far and how fast objects would move in various lengths of time.
Furthermore, Galileo was crucial in shifting the scientific perspective towards observation and experimentation as the definitive methods for determining the truth, over abstract logic or dogma. His astronomical observations with the telescope supported his experimental findings and challenged the prevailing Aristotelian physics, which held that more massive objects should fall faster than lighter ones. His work laid the foundational principles for what would later be refined and encapsulated in Isaac Newton's Philosophiae Naturalis Principia Mathematica.