## The Mathematics of Motion

Philosophy is written in this grand book—I mean the universe—which stands open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth. (Galileo, The Assayer)

Gravity is experienced by the everyday observer because it changes the motion of objects. If you jump up from the surface of the Earth with some initial velocity, say, 1 meter/ second, then before you get very far, gravity will quickly adjust your velocity to be back towards the surface of the Earth, and you will be pulled back to the ground. Therefore, gravity is something that changes velocities. The strength of gravity is measured by how much it can change the velocity of an object. Before we can obtain a quantitative understanding of gravity, we must develop one simple mathematical tool that shows us how to calculate the change in the velocity of a moving body. This simple mathematical tool, the slope of a curve, turns out to be fundamental in every branch of science. If you have never calculated the slope of a curve before, it's important and worthwhile to take enough time to carefully check every step of the calculations in this chapter. The only mathematical prerequisite for reading this chapter is a familiarity with algebra and the concept of a function.