Galileo contra Aristotle

It is safe to say that Aristotle, renowned and influential for many centuries, definitely got on Galileo’s nerves when it came to physics.

This is also true concerning the lazy-thinking followers of Aristotle in Galileo’s day. Galileo made sure that, in his classes, conclusions based on observations, particularly observations of measurable quantities, had the power to lock down a physical concept. No amount of philosophical hot air could defeat a set of measurements from the great book of nature. Authority resides not in the famous books of Aristotle but in the great book of nature, of which one makes observation and takes measurement.

Galileo simplified his task by sticking to measurable motions.

Aristotle’s book, Physics, concerned motion, but he also included a diversity of motions that we and our professor, Galileo, would not include in a physics text. For instance, under the topic of motion, Aristotle considered this question, “Why did the Spartans go to war?” That is a motion of a kind, but not measurable with meter sticks and a pendulum. Aristotle also considered the change of bronze into a bronze statue. He considered aging, learning, doctoring, rolling, building and leaping as motions. A projectile, like an arrow shot from a bow, is propelled along its path by the air behind it through which it has already moved, according to Aristotle. To read Aristotle on this topic is to learn patience — it is very complicated.
Galileo simplified his task. He restricted his study of motion to positions and distances, times and intervals of time, from Event A in space and time, to event B in space and time, events between which a time interval and a distance interval could be calculated. In a previous chapter, we considered a short history of events on the Florida Turnpike, viz.

The distance interval is 7.8 miles, 13 km. The time interval is two hours, 7200 seconds. From these kinds of measurements, Galileo has something to say. For instance, he might say that the driver’s average speed is 3.9 mph.
However, from this short table of information, neither we nor Galileo can derive any accelerations of the car along the turnpike.

What is the difference between an accelerated state of motion and one without acceleration?

Galileo took a freely falling body as his exemplar of accelerated motion. One could study all other types of accelerated motion using free-fall acceleration concepts.

Of critical importance is the idea of “continually acquiring new increments of speed… which repeats itself always in the same manner.” This is what we mean about the acceleration of an object in free fall, that for every second of free fall in the gravitational field at the surface of Earth, the downward speed increases 9.8 m/sec, always in the same manner, until it impacts the ground. We customarily call this acceleration g,

This is the acceleration everyone on Earth feels every day. Astronauts up in the International Space Station, ISS, of course, feel a bit less acceleration, being a significant distance up in space above the surface of Earth. For everyone on Earth, though, from the top of Mount Everest to the deeps of the Mariana Trench in the Pacific Ocean, g is our gravitational acceleration, courtesy of all the kilograms of Earth’s mass.

Is there a state in between accelerated states?

Galileo’s first conclusion concerning free fall and vertical acceleration was that two inclined planes of equal heights but different tilt angles would still accelerate a ball to the same speed at the bottom of the ramp. For instance, two inclined planes of height 0.10 m, i.e., 10 cm, and bases 1.0 m and 0.5 m, have tilt angles 5.7º and 11.3º respectively.

If released from rest at the top of the ramp, each ball acquires the same speed. In this diagram, the length of the arrows is the same, indicating equal speeds.  The acceleration of gravity, g, is responsible for accelerating each ball to its later speed, even though it is not straight down free fall.

Given an inclined plane of any angle, you can figure out the fraction of g that pulls an object down the slope.

Just as a bird flying northeast has some speed directly to the north and some speed perpendicular to that, to the east, so an object on a ramp, with tilt angle ,

will get a fraction of g down the ramp and a fraction of g perpendicular to that, into the ramp. To figure out the proportions, rotate the inclined plane triangle so that the inclined surface becomes horizontal, as in Fig. 2a. Then sketch out an accelerations triangle of the same shape, like the yellow triangle in Fig. 2a. Now flip it and rotate the yellow triangle so that its short side is along the ramp surface, as shown in Fig. 2b. Figure 2c shows the “exploded” version of the yellow accelerations triangle, with the longest side labeled g, the downslope fraction labeled D, and the fraction absorbed by the ramp labeled A. Galileo showed that this similar triangle gives you the fraction of g that accelerates a body down the inclined plane. The fraction into the ramp has no affect, if the ramp is rigid. But the fraction of g along the downslope gets the object moving.

Another way to interpret this result, using proportional triangles, is to say that the amount of gravitational pull down the ramp, D, is the same size pull, U, that you need to exert up the ramp, to keep the object at rest. Your pull U balances gravity’s pull D.

Galileo used inclined plane results like this to attack other problems. Eventually, he tackled the problem of a projectile moving through its arc. He proved that, ignoring the air resistance, the path of a projectile through the air is a parabola. He envisioned a horizontal plane along which an object moves at a constant speed, that the horizontal plane came to the end, and that the object keeps moving out past the edge of the plane, like a Ferrari driving off a cliff. His analysis of the object’s motion rested on two assumptions:

1. that the motion retained its horizontal component unperturbed, and
2. that the object acquired equal increments of downward speed in equal intervals of time, in the normal free fall manner.

Aristotle would never have accepted this splitting of the motion into independently analyzable parts, horizontal and vertical, nor would he have accepted that a state of motion could persist, unperturbed, at constant speed. Aristotle knew only a state of rest and a state of forced motion, a moveable object moved by a mover. Galileo insisted on this.

Galileo insisted on unperturbed motion horizontally at constant speed, adding a third state to the roster of possible states of motion.

He asserted this with confidence because he had experimented with his inclined planes. He knew that, on a horizontal plane, an object would be indifferent to being at rest or to moving along a horizontal line at constant speed.

How Galileo taught the third state of motion

Galileo taught this idea to his students at Pisa in the following manner. Consider for a moment that you were comparing motions of an object on an ideal, frictionless inclined plane to its motion on an ideal, frictionless horizontal plane.

The object on the ramp gets a small fraction of the full free fall force of gravity, as usual.  In addition, you maintain, contra Professor Galileo, that the object on the horizontal plane needs a certain minimum amount of OOMPH to get it going along the plane, denoted by the small red push force vector. We will call that amount of push . According to you, the object will move along the horizontal plane only if it experiences this amount of push. Your push force might be 0.02 metric units of force.

At this point, Aristotle would be smiling at you, saying, “Οε τρίγωνο αγόρι! Είναι τα γενέθλιά σας!”^[2]

In addition, say that the object’s vertical weight force is W; thus, the value of W might be 0.10 metric units of force. To support that object hanging on a string, you need an equally sized force Y, 0.10 metric units of force, pointed upward.

The ratio of the two forces, your conjectured push force on the object and the ordinary weight force of the object, p:W, is 1:5 in this example.  That ratio, 1:5, between your conjectured push force and the object’s ordinary weight force, now becomes the key, viz.

This state of motion, at constant speed and constant direction, is in between Aristotle’s two states, rest and forced motion. Perhaps we can give Aristotle his due. He worked hard with what he had. We have also improved our view of the elements of nature: Aristotle numbered four elements, fire, air, water, earth.  However, in the past century or so, we have extended and refined the list of elements to over 100 on the periodic table. This concept, the inertial state of motion, is crucial to our entire scientific enterprise, where making predictions and designs requires accurate computation that truly models nature itself.

The parabolic shape of the path of a projectile develops out of the two separated equations of motion,

1. of an object in an inertial state of motion going horizontally off the edge of a horizontal plane at speed , and
2. of the same object starting suddenly to drop.

If one starts the clock at the instant the object goes over the side, and if one takes that edge of the plane to be the origin of the x-y coordinate system, then the positions become functions of time, and , pronounced “x of t” and “y of t,” viz.

Now, rearrange the first equation so that

We can now put the quotient, in the y equation wherever is.  Then the formula for y becomes a function of x,

in the classic form that parabolas take, with a factor of x², just as you might remember it from geometry class in high school.

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For a projectile, the shape is an up side down parabola, because of free fall.  That is, the acceleration of gravity is downward.  We should therefore use in the drop distance formula. Here is a projectile in motion off a frictionless, horizontal plane, with initial speed , and with the edge of the plane at .  It is an up side down semi-parabola, with impact about 36 m below the plane, , and about 72 meters to the right,

A full trajectory of a projectile, like a baseball heading for the outfield, would be a full parabola, up side down.

The diagram above is a real “moon shot” home run at Tropicana Field, where the fence in deep center is 404 feet from home plate. or, 123 meters.  The initial speed components, and , are .

“Tropicana Field”by Kwong Yee Cheng is licensed under CC BY-NC-SA 2.0

We have mentioned the downward force of gravity and fractions of it along the surface of an inclined plane. In the next chapter, we will make a systematic survey of the concept of force, as we study Sir Isaac Newton’s three laws of motion.