Satellites orbit the earth, the earth orbits the sun, and our entire solar system orbits the center of the Milky Way galaxy. Not all orbits are circular, but in this section we’ll limit our analysis to circular orbits.

How does a satellite orbit the earth? What forces act on it? To answer these important questions, let’s return, for a moment, to projectile motion. Projectile motion occurs when the only force on an object is gravity. Our analysis of projectiles assumed that the earth is flat and that the acceleration due to gravity is everywhere straight down. This is an acceptable approximation for projectiles of limited range, such as baseballs or cannon balls, but there comes a point where we can no longer ignore the curvature of the earth.

FIGURE 8.10 shows a perfectly smooth, spherical, airless planet with one tower of height h. A projectile is launched from this tower parallel to the ground $(\theta =0\text{°})$ with speed $v_0$. If $v_0$ is very small, as in trajectory A, the “flat-earth approximation” is valid and the problem is identical to Example 4.4 in which a car drove off a cliff. The projectile simply falls to the ground along a parabolic trajectory.

FIGURE 8.10 Projectiles being launched at increasing speeds from height h on a smooth, airless planet.

As the initial speed $v_0$ is increased, the projectile begins to notice that the ground is curving out from beneath it. It is falling the entire time, always getting closer to the ground, but the distance that the projectile travels before finally reaching the ground—that is, its range—increases because the projectile must “catch up” with the ground that is curving away from it. Trajectories B and C are of this type. The actual calculation of these trajectories is beyond the scope of this textbook, but you should be able to understand the factors that influence the trajectory.

If the launch speed $v_0$ is sufficiently large, there comes a point where the curve of the trajectory and the curve of the earth are parallel. In this case, the projectile “falls” but it never gets any closer to the ground! This is the situation for trajectory D. A closed trajectory around a planet or star, such as trajectory D, is called an orbit.

The most important point of this qualitative analysis is that an orbiting projectile is in free fall. This is, admittedly, a strange idea, but one worth careful thought. An orbiting projectile is really no different from a thrown baseball or a car driving off a cliff. The only force acting on it is gravity, but its tangential velocity is so large that the curvature of its trajectory matches the curvature of the earth. When this happens, the projectile “falls” under the influence of gravity but never gets any closer to the surface.

In the flat-earth approximation, shown in FIGURE 8.11a, the gravitational force acting on an object of mass m is

But since stars and planets are actually spherical (or very close to it), the “real” force of gravity acting on an object is directed toward the center of the planet, as shown in FIGURE 8.11b. In this case the gravitational force is

That is, gravity is a central force causing the centripetal acceleration of uniform circular motion. Thus the gravitational force causes the object in Figure 8.11b to have acceleration

An object moving in a circle of radius r at speed $v_{\text{orbit}}$ will have this centripetal acceleration if

That is, if an object moves parallel to the surface with the speed

then the free-fall acceleration is exactly the centripetal acceleration needed for a circular orbit of radius r. An object with any other speed will not follow a circular orbit.

FIGURE 8.11 The “real” gravitational force is always directed toward the center of the planet.

The earth’s radius is $r=R_{\text{e}}=6.37\times {10}^6\text{m}$. (A table of useful astronomical data is inside the back cover of this book.) The orbital speed of a projectile just skimming the surface of an airless, bald earth is

Even if there were no trees and mountains, a real projectile moving at this speed would burn up from the friction of air resistance.