## Info

Time

Figure 3.9. Graphical representation of distance versus time for Galileo's inclined-plane experiments. The data from Table 3.1 fit this graph.

Time

Figure 3.9. Graphical representation of distance versus time for Galileo's inclined-plane experiments. The data from Table 3.1 fit this graph.

which we saw in the previous section indicates uniformly accelerated motion. The graph in Fig. 3.9 has the same characteristic upward curving as that shown in Fig. 3.8b. Thus Galileo was able to determine experimentally how objects fall; namely, with uniformly accelerated motion. This is the kind of motion a falling object has when the effects of the resistive medium can be eliminated and is the kind of motion that characterizes the basic falling process.

Galileo certainly was aware that falling objects, in air or in liquids, do not keep falling faster and faster; that is, he knew that the resistance of the medium will eventually cause the acceleration to cease, and that the falling velocity will reach a constant value. This final, maximum velocity is now generally referred to as the terminal velocity for a falling object. It was precisely because of the resistive effects of the air (or a liquid) that Galileo decided he needed to study falling with pendulums and inclined planes, thereby keeping the speed low enough to minimize the effects of the medium. In order to understand the net result of the falling process, let us consider Fig. 3.10.

Figure 3.10. Graphical representation of motion of light and heavy falling objects.

The figure represents the velocity (or speed) of both a heavy and a light object, as a function of time, and thus should be compared with Figs. 3.7a and 3.8a. The diagonal line represents the motion of an object falling with a constant (uniform) acceleration, which is the motion an object would have when falling in a vacuum. In anything but a vacuum, resistance of the medium increases as the speed increases, until the resistive force equals the downward force of gravity, and the acceleration stops. The resulting terminal velocity for an object will depend not only on its weight but on its size and shape as well. The terminal velocity for a falling person (a sky diver, for example) may be as much as 130 mph, depending on body position, clothing, and so on. The horizontal lines in Fig. 3.10 indicate the kind of motion predicted by Aristotle for a light and heavy object. These lines represent constant (although different) speeds, with no indication of a time during which the object speeds up from zero speed to its final terminal velocity.

Although it is important to understand the actual motion of a falling object, as shown in Fig. 3.10, from a scientific viewpoint, it is more important to understand the true nature (the "true reality" in Plato's "Allegory of the Cave") of falling, disregarding the resistance of the medium. Galileo asserted that this true nature is that objects fall with uniformly accelerated motion. This was a very important step toward understanding the falling process. Once the how of falling motion is known, it is then necessary to know why they fall. The understanding of why was provided by Isaac Newton, who was born in the same year that Galileo died. Before discussing Newton's contributions, it is necessary to consider some other important contributions to the science of mechanics by Galileo and also the general relationship of mathematics to science.

### Optional Section 3.1 Motion of Falling Bodies

As discussed above, Galileo determined that objects fall with uniformly accelerated motion. The mathematical formulae that express this relationship are v t and