Acceleration

“ Energy of Acceleration and Work of Retardation for a Shifting Body.—Let w be the weight of a body which has a motion of translation in any path, and in the course of the interval of time Δt let its velocity be increased at a uniform rate of acceleration from v1 to v2. ”

“ The rate at which a velocity is increasing is called the acceleration. If a moving body is, at any particular instant, gaining an additional velocity in an element of time , then the acceleration at that instant may be written
;
but is itself . Hence we may put ; and this is usually written ;
or the acceleration is the second differential coefficient of the distance, with respect to time. Acceleration is expressed as a change of velocity in unit time, for instance, as being so many feet per second per second ”

“ Hence it is clear that equal spaces will be traversed in equal times by two bodies, one of which, starting from rest, moves with a uniform acceleration, while the momentum of the other, moving with uniform speed, is one-half its maximum momentum under accelerated motion. ”

“ Acceleration will take place from the highest to the lowest point of flight, and the energy stored as energy of position will be completely returned in its original form of energy of motion. The effect of the working of the reversible cycle, then, on the rotatory system, under the given conditions, is simply to produce alternately a retardation and a corresponding acceleration. ”

“ On the other hand, a body thrown upward has uniformly retarded motion, that is, its acceleration is downward. The velocity acquired by a falling body in unit time is called its acceleration, or the acceleration due to gravity, and is equal to 32.16 ft. (980 cm.) per second, downward, each second of time. In one second, therefore, a falling body gains a velocity of 32.16 ft. ”

“ That which we call acceleration pertains wholly to deflection, or in very small part to speed, as loss of speed by one and gain by another.
There is still another set of relations to be considered. A body is composed of particles; in order that they should remain within the sphere of the body their paths must be composite, and in order that their paths may be composite there must be a sufficient number of internal collisions to deflect them and retain them within that sphere. If the body itself is moved the paths of the several particles in the average must thus be rendered less composite ”

“ In a common case, such as when a stone is tied to a string and whirled round, this acceleration is produced by the string which pulls the stone; in a spinning-top the acceleration of the outer parts is produced by the forces binding them on to the inner part, and so on. ”

“ Furthermore we may remark that any velocity once imparted to a moving body will be rigidly maintained as long as the external causes of acceleration or retardation are removed, a condition which is found only on horizontal planes; for in the case of planes which slope downwards there is already present a cause of acceleration, while on planes sloping upward there is retardation; from this it follows that motion along a horizontal plane is perpetual; for, if the velocity be uniform, it cannot be diminished or slackened, much less destroyed. ”

“ The conception of 'force' as the product of 'mass' and 'acceleration' is based on the fact - easily experienced by anyone who cycles along a level road - that it is not velocity itself which requires the exertion of force, but the change of velocity - that is, acceleration or retardation ('negative acceleration' in the sense of mathematical physics) ; also that in the case of equal accelerations, the force depends upon the mass of the accelerated object. The more massive the object, the greater will be the force necessary for accelerating it. ”

“ To the engineers the task of keeping their mighty engines in such tune as to maintain constantly the peak acceleration of three Earth gravities; to the computers that of so directing their ever-changing course as to win every possible centimeter of precious tangential velocity. ”

“ As to the wave of acceleration, this is a perturbation altogether analogous to light waves, which starts from the electron at the instant when it undergoes an acceleration, and which is then propagated by successive spherical waves with the velocity of light. Whence follows: in a straight and uniform motion, the energy is wholly conserved; but, when there is an acceleration, there is loss of energy, which is dissipated under the form of luminous waves and goes out to infinity across the ether. ”

“ We therefore have hold on the aether through accelerations, and acceleration has an absolute sense as determining the production of waves from matter that has undergone a change in velocity, and the aether manifests its reality as the vehicle, as the carrier of energy transported by these waves. ”

“ Let us admit it then, and consider a body under the action of a force. The relative motion of this body with respect to an observer moving with a uniform velocity equal to the initial velocity of the body, should be identical with what would be its absolute motion if it started from rest. We conclude that its acceleration must not depend upon its absolute velocity, and from that we attempt to deduce the complete law of acceleration. ”

“ Energy Stored and Restored by Deviations of Velocity.—Thus a body alternately accelerated and retarded, so as to be brought back to its original speed, performs work during its retardation exactly equal in amount to the energy exerted upon it during its acceleration ”

“ In ordinary physical investigations dealing with velocities, as for instance with the course of a projectile, we pass at once from the study of acceleration to that of momentum and so to that of force; for change of momentum, which is proportional to force, is the product of the mass of a body into its acceleration or change of velocity. But we can take no such easy road of kinematical in­ves­ti­ga­tion in this case. The “velocity” of growth is a very different thing from the “velocity” of the projectile. The forces at work in our case are not susceptible of direct and easy treatment ”

“ We may obtain several such properties by considering how the accelerations of a system, at a given instant, will be modified by changes of the forces or of the constraints to which the system is subject. Let us suppose that the forces of a system receive the increments , in consequence of which, and of certain additional constraints, which do not produce any discontinuity in the velocities, the components of acceleration receive the increments . ”

“ How much does the velocity change in a small interval of time? That’s called the acceleration, and when it is multiplied by the coefficient called the mass of an object, or its inertia coefficient, then that together is the force. One can measure this. For instance, if one has a stone on the end of a string and swings it in a circle over the head, one finds one has to pull, the reason is that although the speed is not changing as it goes round in a circle, it is changing its direction; there must be a perpetually in-pulling force, and this is proportional to the mass. ”

“ This fact is that if the actual changes of motion which the different particles of bodies experience are each resolved in its appropriate way, each component acceleration is precisely such as is prescribed by a certain law of Nature, according to which bodies in the relative positions which the bodies in question actually have at the moment, [1] always receive certain accelerations, which, being compounded by geometrical addition, give the acceleration which the body actually experiences. ”

“ If we confine ourselves to small velocities, then we have the same expression for the kinetic energy, that is stated by mechanics for the living force. The magnitude of acceleration, however, cannot be derived from that without further ado.
Acceleration presupposes a variability of velocity. The expressions for electromagnetic energy, however, are only derived under the assumption of a time-independent value of velocity. ”

“ We find under these conditions that the force of inertia is proportional to the acceleration with a coefficient of proportionality analogous to mass, but which is here a function of the velocity, and increases indefinitely, like the kinetic energy, as the velocity tends to approach that of light. ”

“ As to the acceleration...."
"That word means picking up speed, doesn't it?" interrupted Dorothy.
"The rate of picking up speed," corrected Seaton. "That is, if you were going forty miles per hour one minute, and fifty the next minute, your acceleration would be ten miles per hour per minute. See? It's acceleration that makes you feel funny when you start up or down in an elevator." ”

“ Accordingly we have here the superposition of two different states, namely, the velocity acquired during the preceding fall which if acting alone would carry the body at a uniform rate to infinity, and the velocity which results from a natural acceleration downwards common to all bodies. ”

“ This mass, in turn, reveals itself in the resistance a particular object offers to any change of its state of motion. Where different accelerations and the same mass are considered, the factor m in the above formula remains constant, and force and acceleration are directly proportional to each other. Thus in the acceleration is discovered a measure for the magnitude of the force which thereby acts. ”

“ If we multiply these monthly differences, or mean monthly velocities, by 12, to bring them into a form comparable with the {74} annual velocities already represented on our acceleration-curves, we shall see that the one series of observations joins on very well with the other; and in short we see at once that our acceleration-curve rises steadily and rapidly as we pass back towards the date of birth. ”

“ It is true that the effects of the wave of acceleration constantly accumulate, but this accumulation itself is so slow that thousands of years of observation would be necessary for it to become sensible. ”

“ Here a body, or system of bodies, originally at rest, has been set in motion and has received acceleration up to a certain velocity, the work which has been done in that acceleration being stored up as “actual energy” in the body itself. ”

“ From observation we now know the greatest acceleration and maximum velocity of an earth particle likely to be encountered; and these are measures of the destructivity. The engineer is therefore dealing with known forces, and he has to bear in mind that these are chiefly applied in a horizontal direction. A formula connecting the acceleration requisite to overturn bodies of different dimensions has been given. The acceleration which will fracture or shatter a column firmly fixed at its foundation to the moving earth may be expressed as follows:—
where
a = the acceleration per sec. ”

“ So that if the disparity of velocities is so great as to overcome all that resistance which commonly arises either from the attrition of contiguous bodies as they slide by one another, or from the cohesion of continuous bodies that are to be separated, or from the weights of bodies to be raised, the excess of the force remaining, after all those resistances are overcome, will produce acceleration of motion proportional thereto, as well in the parts of the machine as in the resisting body. ”

“ By experiment it has been found that a body falling freely from rest in a vacuum near the earth's surface follows approximately the law , where s = space (height) in feet, t = time in seconds. Find the velocity and acceleration (a) at any instant; (b) at end of the first second; (c) at end of the fifth second.
which tells us that the acceleration of a falling body is constant; in other words, the velocity increases 32.2 ft. per sec. every second it keeps on falling. ”

“ This opposes itself, less or more, accordingly as it is to open more slowly or hastily to make way for the moveable, which being by its own nature, as I have said, continually accelerated, consequently encounters a continually increasing resistance in the medium, until at last the velocity reaches that degree, and the resistance that power, that they balance each other; all further acceleration is prevented, and the moveable continues ever after with an uniform and equable motion. ”

“ The development of coal-power in the nineteenth century furnished the first means of assigning closer values to the elements; and the appearance of supersensual forces towards 1900 made this calculation a pressing necessity; since the next step became infinitely serious.
A law of acceleration, definite and constant as any law of mechanics, cannot be supposed to relax its energy to suit the convenience of man. ”

“ The greater the velocity, the more energy is required to impart a given acceleration. To increase the velocity from ten feet per second to twenty feet per second, the applied force must continue through one second of time, and more energy is required to follow a rapidly moving body, and continue to apply to it a given force for one second than would be required to follow and maintain the application of the same force to a body moving more slowly—the distance traveled is greater in one case than in the other. ”