4.2 A Satellite in Orbit
Figure 4.3 represents the Earth and a satellite traveling in a circular path around the earth. Radial vectors r1 and r2 represent the distance from the center of the satellites circular path to its position at times t1 and t2. Vector v1 represents the satellites velocity at time t1, and h is the distance the satellite has fallen by time t2 from where it would be in the absence of the Earths gravity. Vectors r1 and are r2 have the same magnitude; both are equal to the radius of the satellites circular path, r.
As can be seen from Figure 4.3, h is much smaller than the radius vectors.
Now, we will study the right triangle formed by vectors r1, r2 + h, and vt, where t is the difference between t2 and t1 (t = t2 t1).
According to the Pathagorean Theorem,
Since h is already small, h2 is very small, and we will ignore it. The r2 cancels out on each side of the equation. Thus, solving for h, we get
Substituting ac for v2/r (Equation 4.3), we find that
The distance that an orbiting object falls due to the force of gravity is equal to one half of the product of its centripetal acceleration and time.