FALLING BODIES. Owing to gravity (q.v.), all terrestrial bodies, if unsupported, fall, or move towards the earth's center. When a falling body is absolutely without support, it is said to fall freely, as distinguished from oue descending an inclined plane or curved surface. We shall here consider the two cases of free descent and of descent on inclined planes.
1. Bodies falling first fact of observation regarding falling bodies is that they fall with a variable velocity; from this we infer that they are acted upon by some force. Again, on observing how the velocity varies, we find that its increments in equal times are equal; from this we conclude that gravity is a uniform force, which it is, at least sensibly, for small distances above the earth's surface. We have next to find a measure for this force. By experiment it is found that a body in 1" falls through 16.1 ft., and that at the end of 1" it moves with such a velocity, that if it continued to move uniformly after the 1" expired, it would pass over 32.2 ft. in the next second. Hence 32.2 ft. is the measure of the velocity which has been generated in 1", and is therefore the measure of the accelerating force of gravity; for the measure of accelerating force is the velocity which it will produce in a body in a second of time. The quantity 32.2 ft. is usually denoted by the letter g; and it is proper to mention here that this quantity measures the accelerating force of the earth's attraction on all bodies. Experiment shows that under the exhausted receiver of an air-pump all bodies fall with equal rapidity, and that the difference of velocities of falling bodies in air is due entirely to the action of air upon them.
As the accelerating force is uniform, it follows that the velocity generated in any time, t, will be given by the formula v=gt. Since the force is uniform, it must generate an equal velocity every second. In t", therefore, it must generate a velocity yt, since it produces g in 1'. In 2", a falling body will be moving with a velocity of 64.4 ft.—i.e., were the velocity to become constant for the third second, it would in that second move through 64.4 feet.
We are now in a position to inquire mote particularly,liow bodies. fall, and to answer such questions aijint time will it body falling keelS'ttakil tAalFillrough a given space? Second: What velocity will it gain in falling through a given space? Third: How high will a body ascent when projected straight up with a given velocity? etc. Let A be the point from which a body falls, and B its position at the end of the time t; and let AB = S. Then we know that at B the body has the velocity gt. Suppose, now, the body to be projected upwards from B towards A with this velocity gt—gravity acting against it, and tending to retard its motion. We know that at the end of a time t it will be again at A, having exactly retraced its course, and lost all the velocity with which it started from B, because gravity will just take the same time to destroy the velocity gt which it took to produce it.
From this consideration we may obtain an expression for the space AB or S in terms of the time t. In the time t, the body rising from B with a velocity =0 would ascend, if not retarded, a height (g:,`,. t, or gt". But in the time t, gravity, we know, carried it through S; it will therefore, in the same time, by retarding it, prevent it going to the height ye by a space = S. The space through which it actually ascends is then represented by the difference but this space we know to be AB or S. Therefore S = or 2S = or S = We may give this equation another form. vl For 12 being the velocity acquired in the time t, gt, .• t = Then S = ig.—=— Hence = 2gS. From these formula;, we see that when a body falls from rest under the action of gravity, its velocity at any time varies as the time, and the square of its velocity as the space described.
If the body, instead of starting from rest, has an initial velocity V; and if v, as before, be the velocity at the time t, then evidently v is = the original velocity+ that which is generated by gravity, or v = V +gt; and the space will be that which would have been described by the body moving uniformly with a velocity V + that which it would describe under gravity alone, or S = • With regard to the last two formulae, it is easy to see that they may be made to suit the case of a body projected upwards with a velocity V, by a change of signs; thus, v =V —ft, and S — gravity here acting to destroy velocity, and diminish the height attained. From the general formula; in the case of an initial velocity, whether the body be projected upwards or downwards, we may express v in terms of S, as we did in the case of motion from rest. For e = (V =V' ±2g(Vt+ = These are all the formulas applicable to the case of falling bodies, and by their means all problems in this branch of dynamics may be solved. It also appears that the formula; above investigated apply to all cases of rectilinear motion of bodies considered as particles under the action of any uniform force. In all such cases, if f measure the accelerating force S = = 2fS, for the case of motion from rest; and S = Vt t and for the case of an initial velocity.
The reader can easily frame examples illustrative of the formula; for himself. We subjoin one: A stone falls down a well, and in 2" the sound of its striking the bottom is heard. How deep is the well? Neglecting the time occupied in the transmission of sound, the formula S = applies, or S = depth = t being 2"; . • . depth = 2g, or 64.4 feet.
2. Bodies descending inclined planes. In this case the formulas already investigated apply with a slight change. In the figure, if P be a body on the inclined plane AB, descending under gravity, we observe that only that resolved part of gravity parallel to AB is effective to make it descend, the other part at right angles to AB merely pro ducing pressure on the plane. The angle of inclination of the plane being a, we know (see COMPOSITION AND