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Pounding the German lines opposite Baley court Woods, near Nixeville, Department of the Meuse, with French 340-mm.
Galileo tried it himself. In the presence of the professors and students of the University of Pisa he dropped balls of different sizes and materials from the top of the tower of Pisa (Fig. 76), 180 feet high, and found that they fell in practically the same time. He showed that even very light bodies like paper fell with velocities which approached more and more nearly those of heavy bodies the more compactly they were wadded together. From these experiments he inferred that all bodies, even the lightest, would fall at the same rate if it were not for the resistance of the air.
That the air resistance is indeed the chief factor in the slowness of fall of feathers and other light objects can be shown by pumping the air out of a tube containing a feather (or some small pieces of tissue paper) and a coin (Fig. 77). The more complete the exhaustion the more nearly do the feather and the coin fall side by side when the tube is inverted. The air pump, however, was not invented until sixty years after Galileo's time.
FIG 77. Feather and coin fall together in a vacuum
90. Exact proof of Galileo's conclusion. We can demonstrate the correctness of Galileo's conclusion in still another way, one which he himself used.
Let balls of iron and wood, for example, be started together down the inclined plane of Fig. 78. They will be found to keep together all the
FIG. 78. Spaces traversed and velocities acquired by falling bodies in one, two, three, etc. seconds
way down. (If they roll in a groove, they should have the same diameter; otherwise, size is immaterial.) The experiment differs from that
of the freely falling bodies only in that the resistance of the air is here more nearly negligible because the balls are moving more slowly. In order to make them move still more slowly and at the same time to eliminate completely all possible effects due to the friction of the plane, let us follow Galileo and suspend the different balls as the bobs of pendulums of exactly the same length, two meters long at least, and start them swinging through equal arcs. Since now the bobs, as they pass through any given position, are merely moving very slowly down identical inclined planes (Fig. 65), it is clear that this is only a refinement of the last experiment. We shall find that the times of fall, that is, the periods, of the pendulums are exactly the same.
From the above experiment we conclude with Galileo and with Newton, who performed it with the utmost care a hundred years later, that in a vacuum the velocity acquired per second by a freely falling body is exactly the same for all bodies.
91. Relation between distance and time of fall. Having found that, barring air resistance, all bodies fall in exactly the same way, we shall next try to find what relation exists between distance and time of fall; and since a freely falling body falls so rapidly as to make direct measurements upon it difficult, we shall adopt Galileo's plan of studying the laws of falling bodies through observing the motions of a ball rolling down an inclined plane.
Let a grooved board 17 or 18 ft. long be supported as in Fig. 78, one end being about a foot above the other. Let the side of the board be divided into feet, and let the block B be set just 16 ft. from the starting point of the ball A. Let a metronome or a clock beating seconds be started, and let the marble be released at the instant of one click of the metronome. If the marble does not hit the block so that the click produced by the impact of the ball coincides exactly with the fifth click of the metronome, alter the inclination until this is the case. (This adjustment may well be made by the teacher before class.) Now start the marble again at some click of the metronome, and note that it crosses the 1-ft. mark exactly at the end of the first second, the 4-ft. mark at the end of the second second, the 9-ft. mark at the end of the third second, and hits B at the 16-ft. mark at the end of the fourth second. This can be tested more accurately by placing B successively at the
9-ft., the 4-ft., and the 1-ft. mark and noting that the click produced by the impact coincides exactly with the proper click of the metronome.
We conclude, then, with Galileo, that the distance traversed by a falling body in any number of seconds is the distance traversed the first second times the square of the number of seconds; that is, if D represents the distance traversed the first second, S the total space, and t the number of seconds, S = Dť2.
92. Relation between velocity and time of fall. In the last paragraph we investigated the distances traversed in one, two, three, etc. seconds. Let us now investigate the velocities acquired on the same inclined plane in one, two, three, etc. seconds.
Let a second grooved board M be placed at the bottom of the incline, in the manner shown in Fig. 78. To eliminate friction it should be given a slight slant, just sufficient to cause the ball to roll along it with uniform velocity. Let the ball be started at a distance D up the incline, D being the distance which in the last experiment it was found to roll during the first second. It will then just reach the bottom of the incline at the instant of the second click. Here it will be freed from the influence of gravity, and will therefore move along the lower board with the velocity which it had at the end of the first second. It will be found that when the block is placed at a distance exactly equal to 2 D from the bottom of the incline, the ball will hit it at the exact instant of the third click of the metronome, that is, exactly two seconds after starting; hence the velocity acquired in one second is 2 D. If the ball is started at a distance 4 D up the incline, it will take it two seconds to reach the bottom, and it will roll a distance 4 D in the next second; that is, in two seconds it acquires a velocity 4 D. In three seconds it will be found to acquire a velocity 6 D, etc.
The experiment shows, first, that the gain in velocity each second is the same; second, that the amount of this gain is numerically equal to twice the distance traversed the first second. Motion, like the above, in which velocity is gained at a constant rate is called uniformly accelerated motion.
In uniformly accelerated motion the gain each second in the velocity is called the acceleration. It is numerically equal to twice the distance traversed the first second.
93. Formal statement of the laws of falling bodies. Putting together the results of the last two paragraphs, we obtain the following table, in which D represents the distance traversed the first second in any uniformly accelerated motion.
v = at,
Since D was shown, in § 92, to be equal to one half of the acceleration a, we have at once, by substituting a for D in the last line of the table,
These formulas are simply the algebraic statement of the facts brought out by our experiments, but the reasons for these facts may be seen as follows:
Since in uniformly accelerated motion the acceleration a is the velocity in centimeters per second gained each second, it follows at once that when a body starts from rest, the velocity which it has at the end of t seconds is given by v = at. This is formula (1).
To obtain formula (2) we have only to reflect that distance traversed is always equal to the average velocity multiplied by the time. When the initial velocity is zero, as in this case, and the final velocity is at, average velocity = (0 + at) ÷ 2 at. Hence
S = 1 at2.
This is formula (2).
These are the fundamental formulas of uniformly accelerated motion, but it is sometimes convenient to obtain the final velocity v directly from the total distance of fall S, or vice versa. This may of course be done by simply substituting in (2) the value of t obtained from (1), namely, This gives
v = √2 aS.