Paths, An Introduction

Introduction

As an airplane passes high overhead, it often leaves a contrail behind. This is a visible representation of the plane's "path" through the sky. The moon also traces an elliptical path around the earth, although there is no visible trail. Anything that moves through space, during a fixed interval of time (start to finish), defines a path.

Sometimes it is convenient to consider the components of the path. If the path is defined by a function p(t), the component functions might be x(t), y(t), and z(t). These functions give the x y and z coordinates at time t. Return to the path of the moon around the earth, i.e. its orbit. If we select our coordinate system carefully, z(t) = 0. The moon orbits in a fixed plane, and never leaves that plane. The other components, x(t) and y(t), cycle around, looking a bit like cos(t) and sin(t), although the moon's orbit is not a perfect circle.

As we shall see, calculus can be used to find the length of the path, the instantaneous velocity and direction of the moving particle at any given time, the force required to keep the object turning in its orbit, and so on.