Velocity

“ It is plain that the three quantities, s, v, t, are the three intrinsic principles of movement. In fact, the velocity, v, is the act, or the form, of movement; whence the epithet of uniform applied to all movement of constant velocity; the amount, s, of space measured is the term actuated by the said velocity; the time, t, is the duration of the movement, that is, its actuality; for as movement is essentially successive, its actuality also is successive, and constitutes a length of time. Here, then, we have most distinctly the three principles of movement. ”

“ If a body is resisted in the ratio of its velocity, and moves, by its vis insita only, through a similar medium, and the times be taken equal, the velocities in the beginning of each of the times are in a geometrical progression, and the spaces described in each of the times are as the velocities. Case 1. Let the time be divided into equal particles; and if at the very beginning of each particle we suppose the resistance to act with one single impulse which is as the velocity, the decrement of the velocity in each of the particles of time will be as the same velocity. ”

“ The velocity with which a denser fluid flows into a rarer one is equal to the velocity a heavy body acquires in falling through a height equal to the difference of altitude of two columns of the heavier fluid of such heights as will produce the respective pressures; and, therefore, when the difference of pressure or amount of rarefaction in the chimney is known, it is easy to tell the velocity of motion which ought to be produced by it. ”

“ The concept of velocity, as ordinarily defined, involves the two concepts of space and time. The operations by which we measure the velocity of an object are these: we first observe the time at which the object is at one position, and then later observe the time at which it is at a second position, divide the distance between the two positions by the time interval, and if necessary, when the velocity is variable, take the limit. ”

“ We shall investigate the velocity of such plane waves by a method which is only a slight modification of a method given by W. J. M. Rankine (Phil. Trans., 1870, p. 277) .
Whatever the form of a wave, we could always force it to travel on with that form unchanged, and with any velocity we chose, if we could apply any “external” force we liked to each particle, in addition to the “internal” force called into play by the compressions or extensions. For instance, if we have a wave with displacement curve of form ABC (fig. 8) , and we require it to travel
Fig. 8. ”

“ Velocity is measured in centimeters per second, and since the second is longer in the moving system, while the centimeter in the direction which we are considering is the same in both systems, the observer A, always using the units of his own system, concludes that the change in velocity of the ball b is smaller in the ratio than the change in velocity of the ball a. The change in velocity of each ball multiplied by its mass gives its change in momentum. ”

“ For the exposition of velocity, which, by the definition thereof, is a motion which, in a certain time, passeth over a certain space, it is requisite, not only that time be exposed, but that there be also exposed that space which is transmitted by the body, whose velocity we would determine; and that a body be understood to be moved in that space also ”

“ If that were the nature of its velocity, it would serve to illustrate our first and erroneous conception of Becoming, but that is not the nature of the velocity. It is continually changing. The velocity, therefore, of the descending stone never is any one velocity; it is always becoming another velocity. Its velocity, therefore, has no Being, because Being implies some continuance or permanence. It is properly called a Becoming. Such is one illustration by means of which you may be aided in familiarising your minds to the conception of Becoming, or process as distinguished from that of Being. ”

“ THE THEOREM OF THE ADDITION OF VELOCITIES EMPLOYED IN CLASSICAL MECHANICS
Let us suppose our old friend the railway carriage to be travelling along the rails with a constant velocity v, and that a man traverses the length of the carriage in the direction of travel with a velocity w. How quickly or, in other words, with what velocity W does the man advance relative to the embankment during the process? ”

“ Natural philosophers can tell you how long it will take to reach the earth. By artificial contrivances they can calculate the ratio at which its velocity becomes increased. But no natural philosophy can calculate or can tell you what the particular velocity of the falling body is at any given moment, however short. The truth is, that the stone never has any particular, that is, any definite and constant velocity. Its velocity is always changing. ”

“ The greater velocity with which your little brother moves, renders his momentum equal to yours. You have the most gravity, he the greatest velocity; so that, upon the whole, your momenta are equal: for you, no doubt, remember that momentum is weight multiplied into velocity. [33] You have here then a striking instance of mechanical advantage gained by opposing motion to matter, or velocity to weight ”

“ Thus any motion with finite velocity is a mean between these two extremes, and as such the result of a particular co-ordination of levity and gravity. This makes it evident that to speak of a velocity taking its course in space, whether with reference to light or to a physical body in motion, is something entirely unreal. ”

“ Beginning then with the earth's velocity of rotation at its lowest, we may suppose it gradually and continually increased, and, as we have already seen, the change in the energy of the system is to be expressed by multiplying the relative rotation into the change of the earth's angular velocity. ”

“ Consider a body at rest, impart to it an impulse, that is to say make a given force act upon it for a given time; the body moves, acquires a certain velocity; the body being impelled by this velocity, apply to it again the same force for the same time, the velocity will be doubled; if we still continue, the velocity will be tripled when we shall have given a third time the same impulse. So beginning again a sufficient number of times, the body will end by acquiring a very great velocity which can exceed all limit, an infinite velocity. ”

“ To have even a faint idea of this enormous velocity, let us make a little comparison. A body falling from the summit of a steeple a hundred and fifty feet high, dashes against the pavement with a velocity of fifty five miles an hour. Falling from the summit of St. Peter's, it strikes the earth at the rate of 300 miles an hour, or five times quicker than the rapidest express train. Falling from the neutral point, the Projectile should strike the Earth with a velocity of more than 25,000 miles an hour! ”

“ If it were possible to determine the velocity of light by observing the time it takes to travel between one station and another on the earth's surface, we might, by comparing the observed velocities in opposite directions, determine the velocity of the aether with respect to these terrestrial stations. ”

“ It is enough for our author at present that we understand his object to be the investigation and examination of some phenomena of a motion so accelerated, (no matter what may be the cause,) that the momenta of velocity, from the beginning to move from rest, increase in the simple proportion in which the time increases, which is as much as to say, that in equal times are equal additions of velocity. ”

“ It can, indeed, be demonstrated that there is a critical velocity for every globe, such that if a body be projected upwards with that critical velocity or any greater one, it will be permitted to escape altogether. ”

“ Therefore the greatest velocity that the body can acquire by falling, is to the velocity acquired in any given time, as the given force of gravity which perpetually acts upon it, to the resisting force which opposes it at the end of that time. ”

“ In a moment, Leonard; then I want you and Jim to ask me any questions you like. All this that I have said, is prefatory. The question is one solely of velocity. The rest is automatic. The natural laws governing the attributes of matter in relation to its velocity are these:
“First: A substance whose velocity is increased loses density proportionately. For instance, our vehicle, I consider it as a whole, you understand. ”

“ Fig. 105 shows the vertical and horizontal velocity curves and the contours of equal velocity in a rectangular channel, from one of Bazin’s gaugings.
§ 103. Experimental Observations on the Vertical Velocity Curve.—A preliminary difficulty arises in observing the velocity at a given point in a stream because the velocity rapidly varies, the motion not being strictly steady. If an average of several velocities at the same point is taken, or the average velocity for a sensible period of time, this average is found to be constant. ”

“ 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. ”

“ The velocity of the pulses is equal to that which heavy bodies acquire by falling with an equally accelerated motion, and in their fall describing half the altitude A. For the pulse will, in the time of this fall, supposing it to move with the velocity acquired by that fall, run over a space that will be equal to the whole altitude A ”

“ But the truth to which they were at last led, was, that such causes would account for all the diminution of velocity which bodies experience when apparently left to themselves and that without such causes, the motion of all bodies would go on forever, in a straight line and with a uniform velocity. ”

“ XVI, it is manifest that the velocity of a body, moving in the line RPB about the centre S, in any place P, is to the velocity of a body describing a circle about the same centre, at the distance SP, in the subduplicate ratio of the rectangle 1⁄2L SP to SY2. ”

“ Accordingly, that part of the Third Law of Motion which we are now considering, that the velocity generated is as the force, was obtained, in fact, by a separate train of research. The first exemplification of this 251 law which was studied by mathematicians, was the motion of bodies upon inclined planes: for the force which urges a body down an inclined plane is known by statics, and hence the velocity of its descent was to be determined. ”

“ But as the velocity increases uniformly with the time, we shall obtain the average speed, by finding that which the body had in the middle of the interval which elapsed between the beginning and end of the fall, and thus the space through which the body has actually fallen is that through which it would move in the same time with this average velocity uniformly continued. ”

“ We may therefore suppose the velocity to be so increased, that it would describe an arc of 1, 2, 5, 10, 100, 1000 miles before it arrived at the earth, till at last, exceeding the limits of the earth, it should pass quite by without touching it. ”