seconds and one-tenth the space fallen through is 15312 feet; and we know that the space fallen through in three seconds is 144 feet; subtract 144 feet from 1531 feet, and the remainder is 919 feet. This is therefore the space fallen through in one-tenth of a second occurring at the end of three seconds after the beginning of the motion. 92. Thus we can determine the space through which a body falls in any time which may be specified; we shall now determine the velocity of the falling body at any instant which may be specified; or in other words the rate at which the body is then moving. In the following Table the first column gives the number of seconds since the beginning of the motion, and the second column gives the velocity at that instant. At the end of 1 second. .32 feet per second; at the end of 2 seconds...... .2 x 32 feet per second; and so on. Thus we have an easy Rule for finding the velocity, expressed in feet per second, at the end of any number of seconds since the beginning of the motion: multiply the number of seconds by 32. For example, required the velocity at the end of seven seconds; 7 x 32=224; thus the falling body at the end of seven seconds is moving at the rate of 224 feet per second. Again, required the velocity at the end of two seconds and a half; we have × 32=80; thus the falling body at the end 2 = 5 2 ; and 5 2 of two seconds and a half is moving at the rate of 80 feet per second. 93. It will be seen that the number 32 which occurs in the Table and Rule of the preceding Article is double the number 16 which occurs in the Table and Rule of Art. 88. 94. The case of falling bodies offers a very simple example of variable motion; the velocity, that is the rate of motion, increases just as fast as the time increases, so that, for example, at the end of five seconds the velocity is five times as great as at the end of one second. 95. But here a very important point requires to be explained. As the velocity of a falling body is continually changing, how can we speak of the velocity at any specified instant? When we say that at the end of three seconds the velocity is 96 feet per second, we mean that if no change of motion took place afterwards the motion would be uniform, and at the rate of 96 feet per second. This is the exact meaning of the statement; and we may illustrate it by putting it in another form which is perhaps easier for a beginner, though not quite so exact. Suppose we ask what space is really fallen through in a very short time directly after the end of the first three seconds, say in onetenth of a second. If the motion were uniform and at the rate of 96 feet per second, the space fallen through in onetenth of a second would be × 96 feet, that is 9% feet, 1 10 that is 9 feet. We found in Art. 91 that the space through which the body really falls in this tenth of a second is 91 feet, which is somewhat greater than the result obtained on the supposition of uniform motion at the rate of 96 feet per second. In this way we are led to another answer to the question, what is meant by saying that the velocity of a falling body at the end of three seconds is 96 feet per second: we may say that for a very short time the body without sensible error may be considered to be moving uniformly at the rate of 96 feet per second. 96. The beginner should illustrate the important statement at the end of the preceding Article by other numerical examples. For instance, take a shorter interval of time than one-tenth of a second, say one-twentieth of a second, or one hundredth of a second; then calculate by the method of Art. 91 the space actually fallen through in this interval directly after the end of the first three seconds: it will be found to differ but very slightly from the space which would have been described in the interval by a body moving uniformly at the rate of 96 feet per second. The smaller we take the interval the more closely will the two results agree. Similarly we can illustrate the statement that at the end of four seconds the velocity of a falling body is 128 feet per second; and so on. 97. We have now stated the laws which hold with respect to the motion of falling bodies, but, as will often happen in the course of treatises on Natural Philosophy, we must next indicate some slight modifications and corrections of the general statements which have been made. It will be seen that the matters to which we proceed, though of considerable interest as to theory, do not sensibly impair the practical accuracy of what has gone before. 98. We have taken 16 feet as the space through which a body falls during the first second of its motion; but the number is really rather different for different places, and at London it is about 16 feet and 1 inch. It increases as the distance from the equator increases, and is about an inch greater at the poles than at the equator. So also the velocity at the end of the first second, which we denoted by 32 in Art. 93, is really different in different places, being for any place whatever double the number which denotes at that place the space fallen through during the first second. 99. Again, we have spoken of falling bodies, without any distinction, as if the motion were precisely the same for all bodies whatever; but strictly speaking this is not true. The air which surrounds the earth resists the motion of falling bodies, and the resistance is more influential for light bodies, like cork or paper, than for heavy bodies, like stone or lead: the reason of this will be explained hereafter. But it is found by experiment that, if the air be removed, what is called a heavy body and what is called a light body fall down equal spaces in the same time. This can be shewn by the aid of the air-pump, an instrument to be described hereafter; the experiment is very impressive, and has been ascribed to Newton. But even without an air-pump it is easy to shew that the difference in the motions of falling bodies is not due to the kind of substance of which the body is composed. For gold, which in the case of a sovereign falls as fast as anything which we have commonly in view, may be beaten out to a thin leaf which almost floats on the air; and on the other hand a sheet of paper when open falls very slowly, but when rolled up into a tight ball falls like wood or stone. It must be T. P. 3 observed that the resistance of the air increases very greatly as the velocity of the moving body increases; and thus the laws which we have given as those of falling bodies would require some practical correction if they were to be applied to cases of bodies falling during long times. 100. Finally, the laws which we have stated apply to falling bodies near the surface of the earth. If we had the power of ascending to a distance of hundreds of miles from the earth, and dropping a body from such a point, the motion would be of a different kind: this will be noticed hereafter. 101. As an example of the laws of falling bodies it is sometimes proposed to find the depth below the ground of the surface of the water in a well. Suppose, for instance, that a stone is dropped into a well, and that in two seconds it is heard to strike the water. Since in two seconds a body falls through 64 feet, we may take this for the required depth. This is really a trifle too great, because it takes some time, though very little, for the sound of the splash to reach the ear; and thus the real time of the motion of the stone to the water is somewhat less than two seconds. But it need scarcely be said that dropping stones into a well is a practice to be avoided for fear of choking the well. There can, however, be no objection to pouring back into the well some of the water drawn from it; and this is sometimes done for the amusement of visitors in the case of wells which are famous for their depth. But when the well is very deep the resistance of the air on the drops of falling water will be so great as to render the experiment worthless with regard to any numerical result. In the fortress of Königstein, in Saxony, there is a well which is known to be 640 feet deep, so that the splash, if there were no resistance of the air, would reach the ear in about seven seconds after the water is poured back; but practically the time is about fifteen seconds. 102. The direction in which a body falls is that which is called the vertical direction, which is perpendicular to a horizontal plane; and thus it is different at different places on the earth; but the difference is not sensible to common observation so long as we keep within a few miles of the same spot. See Art. 58. 103. We have hitherto spoken of falling bodies as phenomena which are observed, without referring to the cause of the phenomena. We will now briefly allude to this point. The earth in fact draws bodies to itself, somewhat in the same way as a magnet draws a piece of iron towards itself. This attraction of the earth, as it is called, gives rise to the weight of a body which is supported, and makes a body fall which is unsupported. The effect of the attraction is slightly diminished by the rotation of the earth on its axis; so that, for instance, if there were no rotation a body at the equator would fall in the first second through about two-thirds of an inch more than it actually does. The word gravity is used to denote this power which the earth possesses, as shewn in the weight and the fall of bodies; the effects are said to be produced by the force of granity, or simply by gravity. V. RELATIVE MOTION. COMPOUND MOTION. 104. We know from Astronomy that the earth is not at rest, but really possesses two different motions at the same time; it moves nearly in a circle about the sun once in a year, and it turns round its axis once in a day. Speaking roughly, we may say that in consequence of the motion about the sun the earth moves through somewhat more than one and a half millions of miles in a day; or we may say that it moves through a space about equal to two hundred times its own diameter. In the former mode of statement the velocity seems almost inconceivably great; in the latter it seems more moderate. In consequence of the earth's turning round its axis a place on the equator describes in one day a circle of which the circumference is about twenty-five thousand miles. Now all people and things on the earth have these two motions, and hence |