Motion in Two Dimensions

Michael Fowler


Here we put together Galileo's picture of natural vertical motion (something falling) with his idea of natural horizontal motion (neglect friction) to see how he was able to analyze, for the first time, the motion of projectiles, then as now a subject of great interest to the military, among others.

Summary of Galileo's "Natural" Vertical Motion

Question: In what respect was Galileo's view of falling motion the same as the Greeks' but different from our own?

Answer: He thought falling motion was "natural" as opposed to "violent" motion, he did not view a falling body as having an external force acting on it, except for frictional drag (air resistance). We view falling motion as motion of a body acted on by the force of gravity.

Note: The idea of gravity as a force-at-a-distance, that could even act through a vacuum, was introduced by Newton, and derided as occult and superstitious. The idea of a force before that time was basically a contact push or pull, and other observed forces, such as magnets, were explained in terms of the magnet emitting small particles which interacted with iron, etc. (Of course, this has some similarities with the modern quantum view!).

Galileo did think the acceleration he described might be accounted for by some "cause", but didn't think he had anything useful to say on that subject. To quote from Two New Sciences, page 166 (end of this excerpt), The present does not seem to be the proper time to investigate the cause of the acceleration of natural motion concerning which various opinions have been expressed by various philosophers, some explaining it by attraction to the center, others to repulsion between the very small parts of the body, while still others attribute it to a certain stress in the surrounding medium which closes in behind the falling body and drives it from one of its positions to another. Now, all these fantasies, and others too, ought to be examined; but it is not really worth while. At present it is the purpose of our Author merely to investigate and to demonstrate some of the properties of accelerated motion (whatever the cause of this acceleration may be)-meaning thereby a motion, such that the momentum of its velocity goes on increasing after departure from rest, in simple proportionality to the time, which is the same as saying that in equal time-intervals the bodv receives equal increments of velocity… .

Question: What was the essential advance Galileo made over the Greek view in describing falling?

Answer: He separated out the effect of the medium fallen through, and defined the natural motion as that with no medium effect at all. In contrast to the Greeks, he made careful quantitative experiments, and established that natural falling motion is constant acceleration.

"Natural Horizontal Motion"

Galileo also introduced the concept of "natural horizontal motion".

Question: What do you think he could mean by that? (Hint: what about friction?)

Answer: to quote Galileo (page 244): Imagine any particle projected along a horizontal plane without friction … that this particle will move along this same plane with a motion which is uniform and perpetual, provided the plane has no limits. Think of lobbing a piece of ice almost horizontally onto a smooth frozen lake surface. Or, just think of a ball rolling on a smooth, hard horizontal surface, such as a tabletop. Of course, these things eventually come to rest, because there is some friction, but it's clear that, for a long time, things keep moving without being pushed, contrary to Aristotle's view.

Perhaps the Greeks didn't have frozen lakes, or smooth enough marble floors and balls to observe very low friction motion? We might add, though, that of course the Greeks did observe a thrown ball continue to move through the air after leaving the hand. Since they believed that any motion needed a continuously applied force to maintain it, they thought that the air itself parted in front of the ball then slipped around to the back and pushed on the ball from behind, as it rushed in to fill the space left by the moving ball. After all, they doubtless thought, if the air isn't pushing the ball, what is? Galileo's insight was that for something to continue moving at a steady velocity doesn't need a push at all--it just needs the frictional forces to be eliminated.

Describing Two-Dimensional Motion: Projectiles

Galileo's great contribution to the understanding of motion came from analyzing the following

Question: Suppose a ball is rolling across the top of a smooth table, friction being negligible, so it has constant horizontal velocity. Suppose it rolls off the edge of the table, shooting towards the floor. How would you describe its motion after it leaves the table edge? (Hint: Galileo thought about its vertical motion and its horizontal motion separately!)

 Galileo's Answer: But if the plane is limited and elevated, then the moving particle, which we imagine to be a heavy one, will on passing over the edge of the plane acquire, in addition to its previous uniform and perpetual motion, a downward propensity due to its own weight; so that the resulting motion which I call projection is compounded of one which is uniform and horizontal and of another which is vertical and naturally accelerated.

Compound Motion

The conceptual breakthrough here is the new idea that the motion of a projectile can be separated into two components, vertical and horizontal, and they can be analyzed separately. Nobody had thought like that before! It proved to be an extremely fruitful approach—it was the first move towards describing velocity as a vector having components.

But is it true? It has to be experimentally established that the horizontal and vertical velocity components (or just "motions") do in fact behave as claimed! One good demo shoots a ball horizontally at the same time another ball is dropped. The balls can be heard to strike the floor simultaneously, whatever the height of the apparatus. (Care must be taken to ensure that the ball does go off horizontally.) It is possible to use the video camera here too, for example for a ball rolling off a table, or even a ball thrown, to check that in that case too the horizontal velocity is constant.

Graphing the Motion

It is important for the student to use this information to construct a graph of the path of a ball after it leaves a smooth horizontal table top and flies through the air. (Galileo's own version is here.) The student must understand how this combination of motions produces the parabola.

Of course, at high speeds or in dense media, friction alters projectile trajectories. We shall return to this point after discussing Newton's Laws.