Relative velocity

“ The only meaning which is given to the word "velocity" in scientific discussion, which can be stated without assuming the truth of a scientific theory, is , where and are Distances and Times bearing relation A to distances and times. Other quantities, like Absolute Velocity, which are called velocities because they are related in a certain way to Relative Velocities, can only be defined by stating the relation in the form of an equation, which is, in fact, the expression of a scientific theory. ”

“ The velocity with which a particle of matter revolves, whether on the surface, or in the interior of the globe, is proportional to its distance from the axis of rotation; and therefore, when a body comes nearer to the axis, it loses a part of the motion which it had before; which part, of consequence, is communicated to the whole mass of the earth, and therefore tends to increase the velocity with which it revolves. ”

“ The speed with which the surface waves move outwards in all directions, varies not only with the force of the wave, but also with the kind of material through which they pass. This velocity may be in the neighborhood of twenty miles per second, while in others the velocity is as great as 140 miles per second. ”

“ It is therefore evident that any falling body is accelerated in its motion. This shows that the effect of gravitation is not to create a sudden velocity, but to add velocity to velocity, and continually to increase velocity. ”

“ Now if we apply to the perturbation the maximum value which the observations allow, when they have been corrected for the known acceleration due to the variation of the eccentricity of the terrestrial orbit, we find the velocity of the attractive force to amount to fifty millions of times the velocity of light. ”

“ But by the same argument it may be alledged, that a body arriving at a certain place, and there slopping, has no ultimate velocity: because the velocity, before the body comes to the place, is not its ultimate velocity ”

“ Then if V represent the force arising from the gravity, proportional to the arc CD to be described, by which the body is acted upon in D, and R be put for the resistance, V - R will be the whole force with which the body is urged in D. Therefore the increment of the velocity is as V - R and the particle of time in which it is generated conjunctly. But the velocity itself is as the contemporaneous increment of the space described directly and the same particle of time inversely. ”

“ Whence since the weight of a globe of water, in the time in which the globe with a velocity uniformly continued describes a length of 30,556 inches, will generate all that velocity in the falling globe, it is manifest that the force of resistance uniformly continued in the same time will take away a velocity, which will be less than the other in the ratio of 1 to 376, that is, the part of the whole velocity. And therefore in the time that the globe, with the same velocity uniformly continued, would describe the length of its semi-diameter, or 3 inches, it would lose the part of its motion. ”

“ Now we want to investigate whether this definition is not in sharp contrast to the ordinary illustration of velocities. The distances proportional to the relevant velocities are used in ordinary mechanics as representing the velocities of uniform motions. Formula (2) leads to the same result at the limits of our ordinary experience. Only at velocities nearly comparable to the velocity of light, a notable difference occurs that quickly leads to infinite distortion. ”

“ 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. ”

“ It will be observed that, according to the first hypothesis, the velocity with which light traverses a body must vary with the motion of that body. If the motions of the body and the ray are like-directed, the velocity of light ought to be increased by the whole velocity of the body. ”

“ That the resistance to the same surface with different velocities, is in slow motions nearly as the squares of the velocity, but gradually increasing more and more in proportion as the velocities increase. ”

“ When estimated straight across from surface to surface, that is, in the direction of the liveliest change of pressure, it is the maximum pressure gradient. Along this normal direction the air tends to flow with an acceleration proportional to the gradient. The velocity thus acquired by any portion of air in being pushed along the line of falling pressure, combined with its velocity due to other causes, gives its true velocity. ”

“ The slowness with which powders subside, depends partly on the specific gravity of the substance, and partly on the magnitude of the particles themselves. Bodies, in falling through a resisting medium, after a certain time acquire a uniform velocity, which is called their terminal velocity, with which they continue to descend: when the particles are very small, and the medium dense, as water, this terminal velocity is soon arrived at. ”

“ We may in the first place construct a concrete standard for velocity, as, for example, by stretching a string between two pegs on a board with a fixed weight. If we strike the string, a disturbance travels along the string which we can follow with the eye, and we define unit velocity as the velocity of this disturbance. An object has greater than unit velocity if it outruns the disturbance, and less if it lags behind. ”

“ What is required in addition is to adopt some moving entity as giving our definition of uniform velocity. Considering our known universe it is self-evident that we should choose in our definition of uniform velocity the velocity of light, since this selection could be made by an observer anywhere in our universe. ”

“ Or there is still another most interesting way of defining velocity, in which the analysis into space and time is not made at all, but velocity is directly measured by building up the given velocity by physical addition of a unit velocity selected arbitrarily. ”

“ If we communicate velocity to a body by the hand or by a spring, the effect we produce in a second of time is lessened, when the body has already a velocity which withdraws it from the pressure of the agent. But it appears that this is not so in the case of gravity; the velocity added in one second is the same, whatever downward motion the body already possesses. A body falling from rest acquires a velocity, in one second, of thirty-two feet ”

“ If a solid body fails to impart all its rotational velocity to the medium there, then it will certainly fail to impart its full rotational velocity to the enveloping medium 100 miles away, and fail still more at a distance of 1000, and still more at a distance of 100,000,000, and so on proportionate to the distance. ”

“ Why does the velocity of its "violent" motion decrease as it rises, while that of its "natural" fall increases as it descends. If we mark closely the relation between these two facts, the problem will solve itself. We shall see, as Galileo did, that the decrease of velocity in rising and the increase of velocity in falling are one and the same phenomenon, viz., an increase of velocity towards the earth. Accordingly, it is not a place that is assigned to the body, but an increase of velocity towards the earth. ”

“ 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. ”

“ But if the matter repelled from the head was thus driven over a distance of ninety million miles in twenty-four hours, at the outside, the average velocity of its motion was about a thousand miles per second, or nearly three times as great as the greatest velocity which the sun can communicate by his attractive energy to matter approaching him from without, even though such matter come to him from an almost infinite distance, and in a perfectly straight line—the conditions most favorable for giving a high rate of final velocity. ”

“ There are, of course, an infinite number of different movements possible, for there is every degree of velocity and difference of direction to be considered. On the other hand there is only one type of rest, or absolute quiescence, and it would be just as likely that a body should possess any stated velocity, say ten miles, ten yards, and three quarters of an inch per second, as that it should possess that particular characteristic implied in the absence of all movement whatsoever. ”

“ 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 ”

“ We know by experiment what the effect of increased velocity has upon any whirling body; it tends to enlarge the body at those parts where the velocity is the greatest, the consequence being that the bulging out of the atmosphere would be greatest at the equator. ”

“ The important points for us to notice at present are that the definition of velocity actually used involves the concept of extended time, and that it would be possible to define velocity in different ways, which would give quite a different complexion to phenomena at high velocities, but which would leave untouched our ordinary experience. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”