*©Fernando Caracena 5 August 2012*

A continuation of Grokking Galileo's Physics Part I.

*Visualization of mathematical patterns in Nature*

Visualization of mathematical patterns is an important tool in physics, which is a common path to demonstrating concepts and understanding them. See for example some videos: on Dark Energy and on plotting. Mathematics gives the physicist a tool for using these patterns as a basis for reasoning about the material universe and to make predictions about what will happen in nature as a result of various physical constraints.

*Galileo's Observations about accelerated motion*

Here is an example of how to use algebra to formulate patterns in observations of nature mathematically. Galileo did a variety of experiments involving the acceleration of gravity by rolling small marbles along straight grooves down inclined planes. As you can find out for yourself, marbles roll down incline planes slower the more the plane's slope approaches the horizontal and faster, as the incline approaches the vertical. Because of crude equipment, Galileo was not capable of observing directly the acceleration of objects falling straight down with any precision; however, he found that he could slow the accelerated motion down on inclined surfaces to the point that he could study the motion with enough precision to resolve its temporal patterns.

In his 1638 CE book “Dialogues Concerning Two New Sciences”, Galileo writes what we will call his first observation:

“...I discovered by experiment some properties of [motion] which are worth knowing and which have not been hitherto either observed or demonstrated. Some superficial observations have been made [in the past], as, for instance, that the free motion ... of a heavy falling body is continuously accelerated;..... so far as I know, no one has yet pointed out that the distances traversed, during equal intervals of time, by a body falling from rest, stand to one another in the same ratio as the odd numbers beginning with unity."

We call the following, Galileo's second observation:

It has been observed [by others] that missiles and projectiles describe a curved path of some sort; however no one has pointed out the fact that this path is a parabola ...”

Note that in the above quote, Galileo stresses the mathematical nature of his discoveries. They are numerical (ratios of odd integers) and geometrical (parabolic trajectories).

The following quotes shows that Galileo designed very practical experiments, from which he abstracted mathematical laws of motion:

“... I have attempted in the following manner to assure myself that the acceleration actually experienced by falling bodies is that above described. A piece of wooden moulding or scantling, about 12 cubits long, half a cubit wide, and three finger-breadths thick, was taken; on its edge was cut a channel a little more than one finger in breadth; having made this groove very straight, smooth, and polished, and having lined it with parchment, also as smooth and polished as possible, we rolled along it a hard, smooth, and very round bronze ball. Having placed this board in a sloping position, by lifting one end some one or two cubits above the other, we rolled the** **ball, as I was just saying, along the channel, noting, in a manner presently to be described, the time required to make the descent. We repeated this experiment more than once in order to measure the time with** **an accuracy such that the deviation between two observations never exceeded one-tenth of a pulse-beat.”

“....For the measurement of time, we employed a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent, whether for the whole length of the channel or for a part of its length; the water thus collected was weighed, after each descent, on a very accurate balance; the differences and ratios of these weights gave us the differences and ratios of the times, and this with such accuracy that although the operation was repeated many, many times, there was no appreciable discrepancy in the results.”

A lot of Galileo's reasoning was based on ratio's, that is, his thinking was geometrical. He measured time using a water clock, the units of which were proportional to the weight of water emptying out of a large water tank into a small vessel.These were not absolute units, but ratios, the ratio of times being the ratio of weights of water collected during various periods of time.

Now, let us use Galileo's first observation: “that the distance[ interval]s traversed, during equal intervals of time, by a body falling from rest, stand to one another in the same ratio as the [successive] odd numbers beginning with unity” to derive equations that express the relations between distance, speed and time of any object accelerated by the earth's gravity. Time is measured as a whole number of equal intervals of time (time/fixed interval of time = n). The first increment of distance traveled is used as the unit of distance. Each successive increment of time increments the distance traveled by the next odd integer times that unit of distance. The total distance traveled is the sum of the odd integers up to n time multiplied by the first increment of distance, or as follows: total distance traveled = n^{2 }times the first increment of distance.

Table 1 summarizes Galileo's first observation where the distance and time increment ratios are compared.

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Table 1

Ratios:

time(n) distance increment total distance traveled

1 1 1

2 3 4

3 5 9

4 7 16

...

n 2*n-1 n^{2 }

...

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Algebraic Summary

We have discussed Galileo's observations regarding an object beginning from rest at an initial time (t=0) and moving in a straight line under a constant acceleration (a=constant). http://www.ghyzmo.com/wp-admin/post.php?post=152&action=editFrom the previous section, we know that the speed at which the object moves is just the area of the rectangle swept out in time between the time axis and the acceleration line (parallel to the time axis). For a particular time (t), this idea translates into the following relation:

v(t)=a*t. (2)

A plot of v(t), given by (2) as a function of time, is a straight line that has a slope equal to the acceleration (a).

The total distance traveled under these conditions is just the area of the expanding triangle that is formed between the straight-line plot of v(t) and the time axis, which at any time is given by the following:

D(t)=1/2 * v(t)*t. (3a)

Further, (3a) can be reduced to an equation depending on acceleration and time by using (2),

D(t)=1/2 * a *t^{2}. (4)

Given the above algebraic background, we are now ready to show that Galileo's first observation verifies the above equations. Note that the distance interval for the first unit of time elapsed is

D_{1}=1/2 * a *Δt^{2 }, (5)

;and the one for the nth time (n*Δt) is

D_{n}=n^{2}*(1/2 * a *Δt^{2}). (6)

From (5) and (6) we can just solve for the ratio,

D_{n}/D_{1}=n^{2}, (7)

which is just the mathematical equivalent of Galileo's first observation.

Continue with Ballistics of Galileo.