Galileo and the Mathematics of Motion

Part I: The Inclined Plane Experiments

Behind me is a replica of Galileo's workroom -- as recreated at the Deutsches Museum in Munich, Germany. As you can see, it contains many of the mathematical instruments and experimental equipement Galileo used in his studies -- the most prominent of which is the inclined plane. This was certainly constructed to help with his study of motion -- accelerated motion in particular.

Earlier we discussed the kind of demonstration Galileo may well have performed himself from the leaning tower of Pisa -- which disproved Aristotle's theory of motion by showing that objects of different weight fall with the same accelerating speed, hitting the ground at virtually the same time.

Galileo wanted to study gravity -- and how it affected acceleration -- in great detail, but falling objects accelerated too quickly, and the time was too short, to make accurate observations.

Was there any way he could try to slow down the effect of gravity -- to observe the rate of acceleration in slow motion? This is exactly what the inclined plane allowed Galileo to do.

The animation linked to this picture shows that at an angle of 60 degrees,

the rate of descent along an inclined plane is not much slower than the case of a freely falling object, but at 30 degrees,

you can see that it is now possible to begin measuring ratios of time and distance with reasonable accuracy.

The force of gravity continues to act on the billiard ball as it begins at time t-zero to roll down the plane, and it goes faster and faster, until at the bottom it rolls onto the horizontal plane.

At this point, as Galileo reasoned, gravity is no longer pulling on the ball to continue accelerating its motion; instead, the effect of gravity is now uniform, or constant, and ideally the billiard ball will now continue to move in a straight line, with a constant uniform motion. This, in fact, was one of Galileo's important insights about motion, and is a version of the law of inertia.

If we now lower the plane even more, the force of gravity will be diluted even further, so that for example, if we now roll the billiard ball down the inclined plane from rest at t-zero, the intervals of time and distance can be measured even more accurately.

But how? Galileo had no stop watch -- not even a pendulum clock. Actually, he used a klepsydra, a version of the ancient water clock, which provided a relative measure of distances in terms of amounts of water collected in a jar as the billiard ball rolled down the inclined plane. But there was another way to measure time intervals that Galileo also used.

Equal intervals of time are easily measured by musical intervals. This brings us back to the renaissance music we heard at the beginning of this program -- for it is not difficult to measure equal intervals of time quite accurately by ear -- and Galileo found something quite surprising when he tried this.

Let's begin with the plane at an inclination of 20 degrees. In the video clip attached to the picture below

we start the ball rolling at time t-zero and count equal intervals of time as it rolls down the plane. The t's appear on the computer simulation, showing how far the ball rolls in each interval of time. But now look at what happens if we take the distance covered in the first time interval as a unit of measure.

In the first second the ball has covered a distance of one unit. In the next second it covers three times this distance, and in the next second, because acceleration continues to make the ball move faster, it covers a distance five times the initial distance. As Galileo discovered, from one second to the next as the ball rolls down the inclined plane, the ratios of the distances covered increase by odd numbers, by intervals of 1, 3, 5, 7, 9, etc.

Trying this again with the plane at different angles always produces the same progression of distances covered in equal time intervals. This is always in proportion to the sequence of odd numbers: 1, 3, 5, ... and so on.

This progression of distances by odd numbers as Galileo observed for uniformally accelerating objects is quite remarkable, but there's more. Let's now analyze this a bit more carefully.

After 1 second the ball had covered a distance of one unit. In the next second, it covers three more units, so at the end of the first two seconds it has covered a total of four distance units. In the third second, it covers 5 more units, for a total distance after three seconds of 9 units. If the ball continues for another second, it will cover 7 more units for a total distance of 16 units after four seconds.

As Galileo realized after considerable effort, both experimental and mathematical, there is an even more remarkable connection bere between time and distance -- namely that acceleration acting uniformly on a falling object does so mathematically, in such a way that the distance covered is directly proportional to the square of the time, as we can see so clearly in the table below representing all the data of our experiment and Galileo's calculation of the square of the time - which equals the total distance covered!

Using the mathematical symbol representing the relation of being proportional we write this as:

Go on to the next section: Part II: Galileo's Analysis of Projectile Motion

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