Circling the Earth
Let's take the analogy of the baseball pitcher a step farther. When a baseball is thrown in a straight line, we already said that the ball would fall to Earth because of gravity and atmospheric drag. Let's pretend again that there is no atmosphere, so there is no drag to slow the baseball down. Now, let's assume that the person throwing the ball throws it so fast that as the ball falls towards the Earth, it also travels so far, before falling even a little, that the Earth's surface curves away from the ball's path.
In other words, the baseball falls as it did before, but
the ball is moving so fast that the curvature of the Earth becomes a
factor and the Earth "falls away" from the ball. So, theoretically,
if a pitcher on a 100 foot (30.48 m) high hill threw a ball straight
and fast enough,the ball would circle the Earth at exactly 100 feet
and hit the pitcher in the back of the head once it circled the globe!
The bad news for the person throwing the ball is that the ball will
be traveling at the same speed as when they threw it, which is about
8 km/s or several times faster than a rifle bullet. This would be very
bad news if it came back and hit the pitcher, but we'll get to that
in a minute.
How fast does the baseball need to travel to never hit the ground? The answer is called the circular velocity, and at the surface of Earth (give or take 100 m) the answer is about 8 kilometers per second (km/s). To compare this with speeds you are used to: 1 km/s = 3600 km/hour = about 2300 miles per hour!
You can see why the shuttle rocket tanks have to be so big; they have to push the shuttle to at least 8 km/s to get it into orbit around Earth!
If we threw the baseball even harder, it would go into a bigger and bigger orbit, with the closest approach to Earth staying the same. If we could throw it hard enough, in fact, we could put it into an infinitely big orbit. If we did that, it would never come back, and we can say it escaped. This escape speed for Earth is about 11 km/s, or about 25,000 miles per hour. This is one of the steps that a rocket to Mars or Venus or Jupiter must achieve.
Newton realized that the question of what happens to a baseball (or bullet) and what happens when the Moon goes around the Earth were really the same question. Any time you throw a ball or fire a bullet you are putting something into an orbit around the center of the Earth, although in those cases nearly all of the orbit is underground! The same math and logic may be used to figure out what will happen in the case of the ball and the case of the Moon, as this picture illustrates: