From Impetus to Inertia

He was a major contributor to the Copernican Revolution, but a greater contribution of Galileo Galilei is his work on the motion of falling bodies.  He was born in Pisa, Italy in the same year as Shakespeare, on the day that Michelangelo died: 1564. He died in 1642, the year Newton was born. His work "Discourses on Two New Sciences" was published in 1638 while he was under house arrest for his support of the Copernican view of planetary motion. This work was much more that an explanation of the mechanics of how objects moved, it was the beginnings of modern science. For the first time a theory was valid only when verified by experiment, and the scientific method was established. Here is a brief description of his "new science" - the presentation is streamlined, but the ideas are Galileo's.

Uniform Motion: Wherein equal distances are covered in equal times.

Uniformly Accelerated Motion: Wherein the motion gains equal increments of speed in equal time intervals.

These relations he acquired through experimentation and reduction of data - what we call the experimental method began here.

The connections between distance, velocity and acceleration can be seen directly when presented in graphical form:

Here, a graph of acceleration with a constant value of 2 meters/sec2 is shown - it's just a straight line of value 2, which never changes. This is a particularly important case: First, it's simple. Second it is the same situation we find all the time when something falls under the influence of gravity. (If it were the case of Earth's gravity, the acceleration would be equal to 9.8 meters/sec2.)

OK: If you accelerate (from rest) at 2 meters/sec2 for say, 1 second, how fast are you going? You should be able to figure it out - the answer is 2 meters/sec. After 2 seconds your speed will be 4 meters/sec., after 3 seconds it will be 6 meters/sec., etc. A graph of this is shown below the acceleration curve. For completeness I have added a constant starting velocity to the motion. That is, rather than starting from rest I start at an initial speed of 1 m/s. Of course, this starting speed just pushes the curve upward by 1 m/s everywhere along the curve.

One more step: From the velocity we should be able to find the distance covered. This is shown in the third graph. In the first second we move at 1 m/s due to our initial velocity; 1 m/s for 1 second gets us to the 1 meter point. We must add to this the distance traveled due to the acceleration. The velocity changes during the first second, but our average velocity (due to the acceleration only - we already have counted our starting velocity) is 1 m/s (We start at 0 m/s and end at 2 m/s. The average is (2 + 0)/2 = 1). The distance covered after 1 second is then 2 meters (1 meter due to our initial velocity and 1 due to the acceleration). Keep doing this and you get the curve shown. A chart of values is shown below.

Distance covered due to a constant acceleration of 2 m/s2 and an initial velocity of 1 m/s.
t (seconds) 1 2 3 4
x (due to v0) 1 2 3 4
x (due to a) 1 4 9 16
x (total) 2 6 12 20

 Galileo found this out through experimentation - rolling balls down an inclined plane. Newton derived the same from basic principles - his laws of motion.

 

Before leaving this matter, there's one important observation to make. If you followed the calculations above you arrived at the results shown, but it took some careful reasoning and an understanding of how things move. Put that aside now and look at a purely geometrical problem related to the curves: Start with the acceleration curve and calculate the area under the curve from time zero until time 1 second. It's easy to see that the area is (height times width) 2 x 1 = 2. After 2 seconds the total area is 2 x 2 = 4. After 3 seconds the area is 6, then 8 then 10. These numbers are exactly the velocity achieved because of the acceleration! Of course, when you multiply length times height you must also multiply the units, which are m/second2 times seconds = m/second, the units for velocity. Conclusion: the area under the acceleration curve is a measure of the velocity. As an exercise you should calculate the area under the velocity curve. Guess what - it provides the values for x, the displacement. Geometry solves problems in physics! Finally, if you remember what the slope of a line is, you can see that the slope of the velocity curve is exactly 2, the value of acceleration (the units are also correct). The language of physics is mathematics.

Go to the page on basic ideas about motion