Not too long ago, perhaps in the heyday of your grandparents. it was a fanciful and far-fetched idea that humans would soon be in comfortable spaceships high above the atmosphere orbiting the planet earth. As recently as 1969, when the first edition of this book was being written, humans first reached the moon. Today the news you see on TV from around the world is beamed to and from satellites. As you're reading this, astronauts and cosmonauts may be circling you. We are preparing for space exploration that will take us who knows where. Our spacefaring accomplishments, amazing by present-day standards yet perhaps quaint and antiquated by tomorrow's, had their beginnings back in 1665 on a farm in Woolsthorpe, England. It was there that Isaac Newton thought of universal gravitation and its role in the motions of the moon, planets, and satellites.
Perhaps the sight of a falling apple triggered Newton's idea that the pull of the earth on the apple extends also to the moon. The young Newton - still in his early 20s -wanted to explain the fact that the moon does not follow a straight-line path, but instead circles about the earth. He had the concept of inertia developed earlier by Galileo: without an outside force, a moving object continues its motion at constant speed in a straight line. He knew that if an object undergoes a change in speed or direction, then a force is responsible. Newton had the insight to see that the force that pulled the apple may extend to the moon and pull it into a circular path about the earth.
In the previous chapter we saw that the moon falls beneath the straight-line path it would follow if no force acted on it. Newton hypothesized that the moon is simply a projectile circling the earth under the attraction of gravity. This concept is illustrated in a drawing by Newton, shown in Figure 9.1. He compared the motion of the moon to that of a cannonball fired from the top of a high mountain. He imagined that the mountain top was above the earth's atmosphere, so air resistance would not slow the motion of the ball. If a cannonball were fired with a small horizontal speed, it would follow a parabolic path and soon hit the earth below. If it were fired faster, its path would be less curved and it would hit the earth farther away. If the cannonball were fired fast enough, Newton reasoned, the parabolic path would become a circle and the cannonball would circle indefinitely. It would be in orbit. So Newton visualized satellites nearly 300 years before they became reality.
Both cannonball and moon have "sideways" velocity, or tangential velocity - the velocity parallel to the earth's surface - sufficient to ensure motion around the earth rather than into it. Without air resistance to reduce its speed, the moon "falls" around and around the earth indefinitely.
For the idea to advance from hypothesis to theory, it would have to be tested. Newton's test was to see if the moon's fall beneath its otherwise straight-line path was in correct proportion to the fall of an apple or any object near the earth's surface. He reasoned that the mass of the moon should not affect how it falls, just as mass has no effect on the acceleration of freely falling objects on earth. How far the moon falls and how far fore suppose the velocity an apple near the earth's surface falls should relate only to their respective distances from the earth's center. If the distance of fall for the moon and the apple are in correct proportion, then the hypothesis that earth gravity reaches to the moon must be taken seriously.
The distance from the center of the earth to the center of the moon was known to be 60 times the distance from the center of the earth to the center of the apple close to the earth's surface. The apple will fall nearly 5 meters (4.9 meters) in its first second of fall. Newton reasoned from Kepler's laws that gravitational attraction to the earth must be weakened by the inverse square of distance. So if the moon is 60 times as far away, its fall toward earth should be 1/(60)2 the fall of an apple near the earth's surface. In one second the moon should fall 1/(60)2 of 4.9 meters, which is 1.4 millimeters.
Newton confirmed his hypothesis. Every second the moon falls 1.4 millimeters beneath the tangent line it would follow without gravity. Newton concluded that the earth and planets orbit the sun in the same way that the moon orbits the earth. The planets continually fall around the sun in closed paths. Why don't the planets crash into the sun? They don't because of their tangential velocities. What would happen if their tangential velocities were reduced to zero? The answer is simple enough: their motion I would be straight toward the sun and they would indeed crash into it. Any objects in the solar system with insufficient tangential velocities have long ago crashed into the sun. What remains is the harmony we observe.
Recall from Chapter 3 that the tangential speed a projectile needs to orbit the earth is 8 kilometers per second. In each second the projectile goes 8 kilometers horizontally (tangential), and falls 4.9 meters vertically (radial) - the same distance the earth curves for each 8-kilometer tangent. So an 8-kilometer-per-second cannonball fired horizontally , Tangential velocity, from Newton's mountain would follow the earth's curvature and coast around the earth again and again (provided the cannoneer and the cannon got out of the way). Fired slower, the cannonball would strike the earth's surface; fired faster it would overshoot a circular orbit as we will discuss shortly. Newton calculated the speed for circular orbit, and since such a cannon-muzzle velocity was clearly impossible, he did not foresee people launching satellites (he did not foresee multistage rockets).
Note that in circular orbit the speed of a satellite is not changed by gravity; only the direction changes. Compare a satellite in circular orbit with a bowling ball rolling along a bowling alley. Why doesn't the gravitational force acting on the bowling ball change its speed? Because gravity is not pulling forward or backward; gravity pulls straight downward. There is no component of gravitational force in the direction of motion (Figure 9.5).
The same is true for a satellite in circular orbit; it is ways moving perpendicularly to the earth's gravitational field. It does not move in the direction of the field, which would
*Or, working backwards, (,0014 m)(602) = 4.9 m.