Page:Encyclopædia Britannica, Ninth Edition, v. 11.djvu/77

Rh Lay a small flat feather upon the top of a penny piece held horizontally. Then let the penny fall ; it will be followed with equal rapidity by the feather, which will be found to remain in contact with the penny throughout the entire descent. In this case the penny piece displaces the air, and thus to a great extent shields the feather from the resistance to which it would be exposed without such pro tection ; it is thus found that the two objects fall to the ground from the same height at the same time. The various experiments to which we have referred suffice to establish the very important result that the time occupied by a body in falling to the surface of the earth, if dropped from a point above it, is independent of the mass of the body as well as of the materials of which the body is composed. There are, no doubt, certain apparent exceptions to the generality of this statement. The law, as we have stated it, does surely not apply to the case of a balloon or a live bird. In each of these cases the air is made, directly or indirectly, to supply a force which overcomes the force of gravity and neutralizes its effects; but if there were no air, then the balloon and the bird would fall to the ground in precisely the same time as a 56 B) weight would do when dropped from the same height. It will not be necessary for us to introduce any further reference to the resistance of the air, and we shall discuss the phenomena presented by falling bodies as they would occur in a space from which the air has been removed. We have by these considerations cleared the way for a very important quantitative determination. Taking a given interval of time, for example, one second, we see that the height through which a heavy body will fall in one second depends neither upon the mass of the body nor on the materials of which it is composed. This is therefore a constant at any given place on the earth s surface for every description of body, and it is of fundamental importance to determine that quantity accurately. By an indirect method, founded on pendulum observations, it is possible to deter mine this quantity with far greater accuracy than would be attainable by actually making the experiment. The value as thus found is slightly different at different parts of the earth though constant at each one. At any part of the United Kingdom it may be taken as 16 - 1 feet. When the distance which the falling body moves over in the first second has been ascertained, it is possible to find the distance which will be accomplished in two seconds, or indeed in any number. The difficulty of the question arises from the circumstance that, as the velocity of the falling body is gradually increasing, the distance moved over in the second second is greater than it was in the first, and generally that the distance in any second is greater than the distance accomplished in any previous second. Imagine the &quot;lift&quot; in a hotel to be a room 16 1 feet high ; then when the lift is at rest, a stone will take one second to fall from the top of the room to the floor. But now suppose the experiment to be repeated, when the lift is either ascending or descending. It will be found that no matter what be the velocity of the lift, provided it remains uniform for a second, and no matter whether the lift be ascending or descending, the stone will still take exactly one second to fall from the ceiling to the floor. To illustrate the important conclusions which can be drawn from this experiment, let us make some suppositions with reference to the velocity of the lift. Suppose that the lift is descending with a velocity of 5 feet per second. Then since it is found that the stone will reach the floor in one second, it is manifest that during that second the stone must actually have fallen through a distance equal to the height of the room augmented by the 5 feet through which the floor of the room has descended. The total dis tance traversed by the stone is therefore 16-1 + 5 = 21-1 feet. It is, however, to be observed that at starting the stone must necessarily have had the same velocity as the lift, i.e., 5 feet per second. The observed facts can therefore be explained by supposing that the stone retained its initial velocity of 5 feet per second, and that gravity acted upon the stone so as to draw it 16-1 feet nearer the earth than it would have been had gravity not acted. On the other hand, suppose that at the time when the experiment was made the lift was ascending with an uniform velocity of 5 feet per second. Then the actual distance travelled by the stone in falling will be less than the height of the ceiling by the distance through which the floor has been raised, i.e., 16 1 -5 = ll g l feet. Observation nevertheless shows that the time occupied in falling from the ceiling to the floor is still one second. The observed facts can be explained by remembering that at the moment of starting, the stone must actually have had the same velocity as the lift, i.e., an upward velocity of 5 feet per second. If there fore gravity had not acted, the stone would in one second have ascended through a vertical distance of 5 feet. The observations are therefore explained by supposing that gravity in this case also draws the body 16 1 feet nearer the earth in one second than the body would have been had gravity not acted. By suitable contrivances it is possible to ascertain that a body dropped from rest will in a time of two seconds move over a space of 6 4 4 feet. We have already seen that during the first second the body will fall 16 I feet. It follows that in the second second the space described by a body falling freely from rest is 64 4 - 16 1 = 48 3. It is thus obvious that the space described in the second second is three times as great as the space described in the first second. To what is this difference to be ascribed 1 At the commencement of the first second the body was at rest; at the conclusion of the first second the body had attained a certain velocity, and with this velocity the body com menced its motion during the second second. The total distance of 48 3 feet accomplished during the second second is partly due to the velocity possessed by the body at the commencement, and partly to the action of gravity during that second. By the principle just ex plained, we are able to discriminate the amounts due to each cause. It appears, from the experiments already re ferred to, that during the second second as during the first the effect of gravity is simply to make the body 16 1 feet nearer the earth than it could otherwise have been. But the body moves altogether 48-3 feet in the second second, and as the action of gravity during that second will only account for 16 - l feet, it follows that the residue, amounting to 48 3 - 16 1 == 32 - 2 feet, must be attributed to the velocity accumulated during the first second. We are therefore led to the very important result that a body falling freely from rest in the United Kingdom will have acquired a velocity of 32 &quot;2 feet per second when one second has elapsed. It need not be a matter for surprise that, though at the close of the first second the velocity acquired is 32 2, the distance moved over during that second is only 16 1. It will be remembered that the body starts from rest, and that while in the act of falling its velocity is gradually increasing. The body, therefore, moves much further in the last half of the second than it did in the first half, and consequently the total distance travelled must be less than the distance which would have been accomplished had the body been moving during the whole second with the velocity acquired at its termination. It might not be easy to arrange a direct experiment to show how far the body will fall during the third second ; we can, however, deduce the result by reasoning from what we have already learned, Let us suppose that the lift 