Terrestrial Gravity: Galileo Analyzes a Cannonball Trajectory
From the earliest times, gravity meant the tendency of most bodies to fall to earth. In contrast, things that leaped upwards, like flames of fire,
were said to have “levity”. Aristotle was the first writer to attempt a quantitative description of falling motion: he wrote that an object fell
at a constant speed, attained shortly after being released, and heavier things fell faster in proportion to their mass. Of course this is nonsense,
but in his defense, falling motion is pretty fast—it’s hard to see the speed variation when you drop something to the ground. Aristotle most
likely observed the slower motion of things falling through water, where buoyancy and fluid resistance dominate, and assumed that to be a slowed-down
version of falling through air—which it isn’t.
Galileo was the first to get it right. (True, others had improved on Aristotle, but Galileo was the first to get the big picture.) He realized that
a falling body picked up speed at a constant rate—in other words, it had constant acceleration (as he termed it, the word means “addition of
speed” in Italian). He also made the crucial observation that, if air resistance and buoyancy can be neglected, all bodies fall with the same
acceleration, bodies of different weights dropped together reach the ground at the same time. This was a revolutionary idea—as was his assertion
that it should be checked by experiment rather than by the traditional method of trying to decipher what ancient authorities might have meant.
Galileo also noted that if a ball rolls without interference on a smooth horizontal surface, and friction and air resistance can be neglected, it will
move with constant speed in a fixed direction—in modern language, its velocity remains constant.
He considered the motion of an object when not subject to interference as its “natural” motion.
Using his terminology, then, natural horizontal motion is motion at constant velocity, and natural vertical motion is falling at constant
acceleration.
But he didn’t stop there—he took an important further step, which made him the first in history to derive useful quantitative results about
motion, useful that is to his boss, a duke with military interests. The crucial step was the realization that for a cannonball in flight, the
horizontal and vertical motions can be analyzed independently. Here’s his picture of the path of a horizontally fired cannonball:
The vertical drop of the cannonball at the end of successive seconds, the lengths of the vertical lines ci, df, eh are the same vertical distances
fallen by something dropped from rest. If you drop a cannonball over a cliff it will fall 5 meters in the first second, if you fire it exactly
horizontally at 100 meters per second, it will still fall 5 meters below a horizontal line in the first second. Meanwhile, its horizontal motion will
be at a steady speed (again neglecting air resistance), it will go 100 meters in the first second, another 100 meters in the next second, and so on.
Vertically, it falls 5 meters in the first second, 20 meters total in two seconds, then 45 and so on.
Galileo drew the graph above of the cannonball’s position as a function of time, and proved the curve was parabolic. He went on to work out the
range for given muzzle velocity and any angle of firing, much to the gratification of his employer.
Newton’s Universal Law of Gravitation
Newton then boldly extrapolated from the earth, the apple and the moon to everything, asserting his Universal Law of Gravitation:
Every body in the universe attracts every other body with a gravitational force that decreases with distance as 1/r2.