ACCELERATION, in Mechanicks, the Increase of Velocity in a moving Body. See VELOCITY, and ACCELERATED MOTION.
Deceleration stands directly opposed to Retardation, which denotes a Diminution of Velocity. See RETARDATION.
Acceleration is chiefly used in Physics, in respect of falling Bodies, i.e. of heavy Bodies tending towards the Centre of the Earth by the Force of Gravity. See GRAVITY, and CENTRE.
That Natural Bodies are accelerated in their Descent is evident from various Considerations, both a priori and posteriori. Thus, we actually find, that the greater Height a Body falls from, the greater Impression it makes, and the more vehemently does it strike the subject Plane, or other Obstacle.
Cause of the Acceleration of Falling Bodies.
Various are the Systems and Opinions which Philosophers have produced to account for this Acceleration. Some attribute it to the Pressure of the Air: The farther, say they, a Body falls, the greater Load of Atmosphere is of consequence incumbent on it: and the Pressure of a Fluid is in proportion to the perpendicular Altitude of the Column thereof —Add, that the whole Body of the Fluid pressing in innumerable right Lines, which all meet in a Point, viz. the Centre; that Point, by the meeting of those Lines, sustains, as it were, the Pressure of the whole Mass: Consequently, the nearer a Body approaches thereto, the Effect or Pressure of more united Lines must it sustain. See AIR, and ATMOSPHERE.
But what overturns this account is that as the pressure of the air downwards increases, so, by the known laws of statics, does the resistance or the force with which the same fluid tends to repel or drive the body upwards again. See FLUID.
Others insist that the incumbent air is the grosser and more vaporous, the nearer the earth; and filled with more heterogeneous particles, which are not true elastic air: and hence, say they, a descending body, meeting continually with less resistance from the elasticity of the air, and having the same force of gravity still acting on it, will necessarily be accelerated. See ELASTICITY.
But what overturns all accounts where the air or atmosphere is concerned is that the acceleration holds in vacuo, and even more regularly than in air. See VACUUM.
The Peripatetic account is worse than this: the motion of heavy bodies downwards, say they, arises from an intrinsic principle, which makes them tend to the centre, as their proper sphere or element, where they would be at rest: hence, add they, the nearer bodies approach thereto, the more is their motion intended. See ELEMENT, QUALITY, etc.
The Galenists, on the other hand, hold that the earth emits a sort of attractive effluvia, innumerable threads whereof continually ascend and descend; which threads, proceeding like radii from a common centre, divaricate the more, the further they go: so that the nearer a heavy body is to the centre, the more of these magnetic threads it receives; and hence the more is its motion accelerated. See EFFLUVIA, and MAGNETISM.
But this is rebutted by an easy experiment: for if a ball be let fall out of the lowest window of a high tower and also out of the highest, the acceleration will be the same in both cases, notwithstanding the greater vicinity to the centre in the one than in the other case.
The Cartesians account for the acceleration from the repeated pulses of a subtle ethereal matter, which is continually acting on the falling body and impelling it downwards. See CARTESIANISM, ETHER, ELEMENT, MATTER, SUBTILE, etc.
After all, the cause of acceleration is nothing mysterious; the principle of gravitation, which determines the body to descend, determining it to be accelerated by a necessary consequence. See GRAVITATION.
For, suppose a body let fall from on high: the primary cause of its beginning to descend is, doubtless, the power of gravity; but once the descent is commenced, that state becomes in some measure natural to the body; so that, if left to itself, it would persevere in it forever, even though the first cause should cease; as we see in a stone cast with the hand, which continues to move after it is left by the cause that gave it motion. See LAW OF NATURE.
But, besides the propensity to descend impressed by the first cause, and which of itself were sufficient to continue the same degree of motion once begun, in infinitum; there is a constant accession of subsequent efforts of the same principle, gravity, which continues to act on the body already in motion, in the same manner as if it were at rest.
Here, then, being a double cause of motion, and both acting in the same direction, viz. directly towards the center of the earth; the motion they jointly produce must necessarily be greater than that of one of them. And the velocity thus increased, having the same cause of increase still persisting, the descent must necessarily be continually accelerated.
For, supposing gravity, whatever it be, to act uniformly on all bodies, at equal distances from the earth's center; and that the time in which a heavy body falls to the earth be divided into equal parts infinitely small: let this gravity incline the body towards the earth's center, while it moves, in the first infinitely small part of the time of its descent; if after this, the action of gravity be supposed to cease, the body would proceed uniformly on towards the earth's center, with a velocity equal to the force of the first impression.
But, now, since the action of gravity is here supposed still to continue; in the second moment of time, the body will receive a new impulse downwards, equal to what it received at first; and thus its velocity will be double of what it was in the first moment: in the third moment it will be triple; in the fourth quadruple, and so on continually. For the impression made in one moment is not at all altered by what is made in another; but the two are, as it were, aggregated, or brought into one sum.
Wherefore, since the particles of time are supposed infinitely small, and all equal to one another; the impetus acquired by the falling body will be everywhere as the times from the beginning of the descent. And hence, since the quantity of matter in the body given continues the same; the velocity will be as the time in which it is acquired.
Further, the space passed over by a moving body in a given time, and with a given velocity, may be considered as a rectangle made by the time and the velocity.
Suppose A (Tab. Mechanics, Fig. 61) a heavy body descending, and let AB represent the time of its descent; which line suppose divided into any number of equal parts, AC, CE, EG, etc. representative of the intervals, or moments of the given time. Let the body descend through the first of those divisions, AC, with a certain equable velocity arising from the proposed degree of gravity: this velocity will be represented by AD; and the space passed over, by the rectangle CAD.
Now, as the action of gravity in the first moment produced the velocity AD in the body before at rest, in the second moment, the same will produce in the body so moving a double velocity, CF; in the third moment, to the velocity CF will be added a further degree, which, together therewith, will make the velocity EH, which is triple of the first and so on for the rest. So that in the whole time AB, the body will have acquired a velocity BK. Again, taking the divisions of the line, e.g. AC, CE, etc. for the times, the spaces gone through will be the areas or rectangles CD, EF, etc. So that in the whole time AB, the space described by the movable will be equal to all the rectangles, i.e. to the dented figure ABK.
Such would be the Case, if the Accessions of Velocity only happened in certain given Points of Time, e.g. in C, in E, etc., so that the Degree of Motion should continue the same until the next Period of Acceleration comes up.
If the Divisions or Intervals of Time were supposed less, e.g. by halves; then the Dentures of the Figure would be proportionably smaller; and it would approach so much the nearer to a Triangle. If they were infinitely small, i.e. if the Accessions of Velocity were supposed to be made continually, and in every Point of Time, as is really the Case, the Rectangles thus successively produced will make a full Triangle, e.g. ABE, (Fig. 63.) Here, the whole of AB consists of the little Portions of Time A i, A 2, etc., and the Area of the Triangle ABE, of the Sum of all the little triangular Surfaces answering to the Divisions of the Time: The whole Area of Triangle expresses the Space moved through in the whole Time AB, and the little Triangles A i, etc. the Spaces gone through in the Divisions of Time.
But these Triangles being similar, their Areas are to one another, as the Squares of their homologous Sides AB, A i, etc., and consequently, the Spaces moved, are to each other as the Squares of the Times.
Laws of Acceleration.
Hence we easily infer the great Law of acceleration, viz. "That a descending body uniformly accelerated, describes, in the whole time of its descent, a space which is just half of what it would have described in the same time, with the accelerated velocity it has acquired at the end of its fall."For, the whole space the falling body has moved through in the time AB, we have already shown, will be represented by the triangle ABE; and the space the same body would move through in the same time, with the velocity BE, will be represented by the rectangle ABEF. But the triangle is known to be equal to just half the rectangle. Therefore, the space moved is just half of what the body would have moved with the velocity acquired at the end of the fall.
Corollary. Hence, (i) we gather that the space moved with the last acquired velocity BE, in half the time AB, is equal to that really moved by the falling body in the whole time AB.
(ii) If a falling body describes any given length in a given time, in double that time it will describe four times that length; in thrice the time, nine times, etc. and universally, if the times be in arithmetical proportion, 1, 2, 3, 4, 5, etc., the spaces described will be 1, 4, 9, 16, etc.
(iii) The spaces described by a falling body, in a series of equal moments or intervals of time, will be as the unequal numbers 1, 3, 5, 7, 9, etc. And since the velocities acquired in falling are as the times, the spaces will also be as the squares of the velocities; and both times and velocities in a sub-duplicate ratio of the spaces.
The motion of a body ascending or impelled upwards is diminished or retarded from the same principle of gravity acting in a contrary direction, in the same manner as a falling body is accelerated. See RETARDATION.
A body thus projected upwards rises until it has lost all its motion, which it does in the same time that a body falling would have acquired a velocity equal to that with which the body was thrown up.
And hence, the same body thrown up will rise to the same height from which, falling, it would have acquired the velocity with which it was thrown up.
And hence, the height which bodies thrown up with different velocities do ascend to, are to one another as the squares of those velocities.
Acceleration of Bodies on inclined Planes. — The same general Law obtains here as in Bodies salling perpendicularly: The Effect of the Plane is to make the Motion slower; but the Inclination being everywhere equal, the Retardation rising therefrom will proceed equally in all Parts, at the Beginning and the Ending of the Motion. The particular Laws can be found under the Article Inclined Plane.
Acceleration of the Motion of Pendulums. — The Motion of pendulous Bodies is accelerated in their Descent;but in a less Ratio than that of Bodies salling perpendicularly. See the Laws thereof under the Article PENDULUM.
Acceleration of the Motion of Projectiles. See PROJECTILE.
Acceleration of the Motion of compressed Bodies, in expanding or restoring themselves. See COMPRESSION, DILATATION, TENSION, FIBRE, &c.
That the Motion of compressed Air, expanding itself by its Elasticity to its former Dimensions, is accelerated, is evident from various Considerations. See AIR, ELASTICITY, &c.
Acceleration is also applied in ancient Astronomy in respect of the Fixed Stars. This Acceleration was the Difference between the Revolution of the Primum Mobile and the Solar Revolution; which was computed at 3 Minutes and 56 Seconds. See STAR, PRIMUM MOBILE, &c.
