 If you had to assign a year to the birth of modern physics, a good choice would be 1638, in which appeared Galileo's book Discourses and Mathematical Demonstrations relating to two new sciences, more specifically Galileo's treatment of naturally accelerated motion. Now the actual text doesn't look very much like a modern physics book. There are geometric constructions and theorems related to accelerated motion, but you won't find a single equation. However, what you will find are experimental data and inductive reasoning leading to a mathematical theory of motion. On the surface, a modern physics text looks quite different because it has lots of equations. However, this is really more a matter of notation than of substance. The modern notation, to be sure, is more useful in applications, but the basic science hasn't changed. Here is the fundamental question. If we have two objects of different weight, say these two lead balls, how will they fall under the influence of gravity? Quoting Galileo, Aristotle says that an iron ball of 100 pounds falling from a height of 100 cubits reaches the ground before a one-pound ball has fallen a single cubit. I say that they arrive at the same time. Now, although Galileo presents logical arguments for his position as Aristotle did, his ultimate appeal is to experiment. So let's perform the experiment. Now, unlike Galileo, we have gadgets like video cameras and spreadsheets to make our job much easier. Here we'll see the experiment slowed down by a factor of 7. The red circles indicate where the balls start falling from rest. It indeed appears that they fall together. Let's watch again. Back to Galileo. You find on making the experiment that the larger outstrips the smaller by two finger breaths. Note that Galileo does not claim that the experiment quote-unquote proves his statement. But continuing on, he says, you would not hide behind these two fingers the 99 cubits of Aristotle. Nor would you mention my small error and at the same time pass over in silence his very large one. This is a modern scientific point of view. No measurement can ever be performed exactly, so no physical theory can ever be quote proven. Best we can do is to seek out a theory that most accurately explains our observations. Galileo also understands that his same time statement is an idealization of the real world that neglects air resistance. Well, if this was the end of it, people like Einstein probably would not have referred to Galileo as the father of modern physics. After all, he wasn't the first person in history to perform this experiment. But Galileo was just getting started. For me, the following statement of his sums up what he did next. In those sciences where mathematical demonstrations are applied to natural phenomena, the principles once established by well-chosen experiments become the foundation of the entire superstructure. I think that's a great description of fundamental physics. Galileo designed experiments that allowed him to abstract from his limited set of measurements fundamental physical principles that he then cast into a mathematical theory. And from there, the techniques of mathematics could be used to predict as yet unobserved phenomena. Galileo didn't have the technology to follow objects in free fall, so he slowed the process down by rolling balls down inclined planes. In one version, a pendulum caused a bell to ring every time it passed through the vertical, thereby serving as a clock. And then bells were placed at various positions on the ramp, and they were rung in succession as the ball rolled down the ramp. So by synchronizing the bell rings, it was possible to relate distance travel to elapsed time. Fortunately, technology makes the experiment much easier for us to perform. Here we'll take every tenth frame of our free fall video and track the position of one of the lead balls. We can determine the distance the ball has fallen by determining the image pixel of its center and relating pixel size to the size of measured features on the concrete wall. We also know the elapsed time between video frames. And so we end up with a plot of position in meters versus time in seconds. If we call the change in distance in time between two frames, respectively delta x and delta t, then by definition the average velocity of the ball is change in distance over change in time, meters per second. If we plot average velocity versus time based on our data, we get something that's close to being aligned. It's not exactly aligned, but we might idealize it as a line and then write velocity increases proportional to time. The constant of proportionality we'll call g is the gravitational acceleration. If we go and find the line that best fits our data, we arrive at a value of g equal 9.89 meters per second. The fit's not perfect, but it's good enough that we might be led to propose the following theory. For any freely falling object starting from rest and neglecting air resistance, the velocity after time t is v equals gt, where the gravitational acceleration g is a constant to be determined by measurement. Let's go back to our original data. It follows some curve and Galileo's description of this curve is, the spaces described by a body falling from rest with a uniformly accelerated motion are to each other as the squares of the time intervals employed in traversing these distances. We express this more compactly as the equation x equals 1 half gt squared. This distance t is time. It's the same physics, just a more convenient expression. The relation between our velocity expression, delta x over delta t, and this position expression can be obtained if we calculate what delta x would be. If we start out at time t and we move into the future to t plus delta t, our change in position is, well, we end up at 1 half gt plus delta t squared. We started at 1 half gt squared. Subtract those. Expand out the square, the t squared at the beginning and at the end, cancel out. We end up with the expression that our displacement is gt delta t plus 1 half g delta t squared. Then divide change in distance by change in time delta t. The factor delta t in the first term cancels. The second term is left with 1 half g delta t. Imagine now that you make this measurement over an extremely short period of time. In fact, a period that approaches zero and the delta t goes to zero, goes away, leaves you with gt. We then write dx dt, which is the instantaneous velocity is equal to gt. That was our original line that we developed a little while ago. In other words, by this mathematical manipulation, we show the equivalence between x equals 1 half gt squared and v is equal to gt. Indeed, we can make any of the following three statements and they give us essentially the same physics. Position x is 1 half gt squared. Velocity v is gt or acceleration is a constant. Uniform acceleration is just equal to the gravitational constant g. And indeed, as people followed Galileo's example of developing mathematical theories for the physical world, they were motivated to develop new methods of mathematical analysis. Indeed, that little delta t goes to zero thing we just did is called differential calculus. So let's go back to our original position versus time data. The best way to get the gravitational acceleration is to directly fit the x equals 1 half gt squared curve to this data. If we do that, we find g is 9.75. So notice we're getting slightly different values of g depending on which approach we take. That's the nature of experimental data. There's some uncertainty in the values you get. For comparison, the standard value is just slightly more than 9.8. So for two specific lead balls, we show that they fell at essentially the same rate. We determined the gravitational acceleration constant and came up with a mathematical description of position, velocity, and acceleration. Is this true for every object? We can't prove it, but we assume it is and then continue to test that. For example, is it true for a ball and a feather? Well, in air it's not, but in a vacuum it is.