Trajectory

“ Is the investigation of the path, or trajectory as it is called, of a body which is projected into space. A body thus projected is acted upon by two forces, the force of projection, which, if acting alone, would carry the body onwards forever in the same direction and at the same rate; and the force of gravity, which tends to draw the body downwards towards the earth. The force of projection acts only at the commencement of the body’s motion ”

“ The science of the mathematics of motion deals with the speed of one and its trajectory, the speed of another and its trajectory, and of their collisions, and for this purpose it has to deal with the measure of their relations, and forever relation is considered and thus an illusion is sometimes produced, when motion itself seems to be wholly relation. ”

“ Speed is the substrate, foundation, or ground of path, for every speed produces a path, or in other terms, every path is dependent on a speed. Every unit having extension and speed has persistent duration; but as these units change in position and also change in trajectory, they could not change if there were not something that persisted through change. Persistence, therefore, is the substrate, foundation or ground of change. Consciousness is the substrate or ground of choice, for if there is no consciousness there can be no choice. ”

“ The inclination of the trajectory decreases from the origin to the summit, where it is nothing, it increases in the descending branch from the summit to its termination, and if the ground did not interpose an obstacle, it would become vertical at an infinite distance. An element of the trajectory in the descending branch has a greater inclination than the corresponding element of the ascending branch. ”

“ The trajectory of Earth deviates as little from a circle, that, when we are dealing with sun rays, we can neglect the velocity component , on which the variation of the oscillation period depends (§ 37) . ”

“ No bullet or shell travels in a straight line. As soon as it leaves the muzzle of the gun, it begins to fall, and its course through the air is a vertical curve that brings it eventually down to the ground. This curve is called the "trajectory." No gun is pointed directly at a target, but above it, so as to allow for the pull of gravity. The faster the bullet travels, the flatter is this curve or trajectory, because there is less time for it to fall before it reaches its target. Modern rifles fire their missiles at so high a speed that the bullets have a very flat trajectory. ”

“ Caroline. But, granting this, the crew of a vessel have a proof of their motion, which the inhabitants of the earth cannot have,—the apparent motion of the objects on shore, or their having passed from one place to another.
Mrs. B. Have we not a similar proof of the earth's motion, in the apparent motion of the sun and stars? Imagine the earth to be sailing round its axis, and successively passing by every star, which, like the objects on land, we suppose to be moving instead of ourselves. ”

“ The experience of space arises from free motion, especially locomotion; speed is a direct sensation; and time is the other factor of speed. Time is measured by any uniformly moving body—that is by space and speed together. Muscular action impeded, the sense of force or resistance, is another primary sensation; and by inference from this arises our notion of “matter,” which is sometimes spoken of as a permanent possibility of sensation. Hardness and softness, roughness and smoothness, are all inferences from varieties of touch. ”

“ If, on the path of a body falling vertically from rest, one lays off a portion which is traversed in any time you please and whose upper terminus coincides with the point where the motion begins, and if this fall is followed by a motion deflected along any inclined plane, then the space traversed along the inclined plane, during a time-interval equal to that occupied in the previous vertical fall, will be greater than twice, and less than three times, the length of the vertical fall. ”

“ Thus 'tis a Law of Motion, discover'd by Experience, that the Moment or Force of any Body in Motion is in the compound Ratio or Proportion of its solid Contents and its Velocity; and consequently, that a small Force may remove the greatest Obstacle or raise the greatest Weight, if by any Contrivance or Machinery we can encrease the Velocity of that Force, so as to make it an Overmatch for its Antagonist. ”

“ Although the doctrine of motion in connexion with force, which is the subject of mechanics, is by far the most important form in which the consideration of motion enters into the formation of our sciences, the Pure Doctrine of Motion, which treats of space, time, and velocity, might be followed out so as to give rise to a very considerable and curious body of science. ”

“ We must here remember that motion as speed is inherent and constant in the particle and that motion as path is always determined from without, but the particles within the body are all external to one another, and therefore the direction of motion which is internal to the body is in obedience to the laws of heredity, and the direction of motion which comes from without the body is heat, gravity and strain. ”

“ The explanation of this phenomenon appears to me to be as follows: a wing, or any surface whatever, which moves against the air, meets, at the beginning of its motion, a considerable resistance, on account of the inertia of the air, which resists every displacement; but, little by little, the air yields, and the velocity of its motion gradually increases, until it may equal that of the moving surface which displaces it; this phase of motion having been reached, the resistance diminishes. ”

“ According to our rule, we must begin by asking what is the immediate use of thinking about force; and the answer is, that we thus account for changes of motion. If bodies were left to themselves, without the intervention of forces, every motion would continue unchanged both in velocity and in direction. ”

“ We say that Being is that which is, and we explain the extension of time by referring to movement, while we also explain movement by referring to time and velocity, and again we explain velocity by referring to the extension of time and movement. This is no vicious circle; for every one knows these entities before hearing their formal definition. Now, the same is true with respect to space and ubication; for the notion of space is intuitive, and before we hear its philosophical definition, we know already that it is the region of all possible ubications and movements. ”

“ When our motion in this direction ceases, however, gravitation, imperishable, continues to exert its equalizing influence, the result being a start in the opposite direction, and we will then reverse our course, and retrace our path, crossing again the central band of attraction, to retreat and fly to the opposite side of the power of greater attraction, into the expanse from which we came, and that is now above us. ”

“ The causes, as they exist at any moment, having produced a certain motion, that motion, becoming itself a cause, reacts upon the causes, and produces a change in them. By altering the distance and direction of the central body relatively to the planet, and the direction and quantity of the force in the direction of the tangent, it alters the elements which determine the motion at the next succeeding instant. ”

“ Simple modes of motion. To slide, roll, tumble, walk, creep, run, dance, leap, skip, and abundance of others that might be named, are words which are no sooner heard but every one who understands English has presently in his mind distinct ideas, which are all but the different modifications of motion. Modes of motion answer those of extension ; swift and slow are two different ideas of motion, the measures whereof are made of the distances of time and space put together ; so they are complex ideas, comprehending time and space with motion. ”

“ Though Space, Time, Matter, and Motion, are apparently all necessary data of intelligence, yet a psychological analysis (here indicated only in rude outline) shows us that these are either built up of, or abstracted from, experiences of Force. Matter and Motion, as we know them, are differently conditioned manifestations of Force. Space and Time, as we know them, are disclosed along with these different manifestations of Force as the conditions under which they are presented. Matter and Motion are concretes built up from the contents of various mental relations ”

“ In this argument he assumes that the transference of motion is the transfer of speed, but we have demonstrated that the transference of motion is only the transfer of direction by change in the paths of each, which is simple and can be understood by a boy. ”

“ The line of a falling body, called also the line of direction, is interesting as being that direction in space at any point of the earth’s surface with reference to which all other directions are named, and by which they are to be determined. ”

“ The motion of the earth's axis would reconcile all the facts known, and it must be regarded as a geological desideratum to determine its amount and direction, and to assign the cause of such a movement. ”

“ Or rather, out of this primitive conception of Motion, the adult conception of it is developed simultaneously with the development of the conceptions of Space and Time: all three being evolved from the more multiplied and varied impressions of muscular tension and objective resistance. Motion, as we know it, is thus traceable, in common with the other ultimate scientific ideas, to experiences of force. ”

“ Again, let the nature investigated be the Spontaneous Motion of Rotation; and in particular, whether the Diurnal Motion, whereby to our eyes the sun and stars rise and set, be a real motion of rotation in the heavenly bodies, or a motion apparent in the heavenly bodies, and real in the earth. ”

“ In the physical world we are congnizant of but four elementary or primordial ideas, beside the inevitable Time and Space. They are Matter, Force, Position, and Motion. Of these, motion is simply change of position; and force is recognized as the agent in every change of motion. ”

“ In considering the power of traction or impulsion necessary to move a body, whether on a canal or on a road. I must carefully distinguish that force which is requisite to put the body from a state of rest into a state of motion, from that which is requisite to sustain the motion when once imparted to it. ”

“ Speed is the essential of motion, but the same particle in motion traverses a path. Motion, therefore, is speed and path, and can be measured in terms of space. Speed and path constitute motion.
The essential of time is persistence, but the relation of time is change; change may be measured in terms of motion. ”

“ The navigator who has measured the earth’s circuit by his hourly progress, or the astronomer who has paced a degree of the meridian, can alone form a clear idea of velocity, when we tell him that light moves through a space equal to the circumference of the earth in the eighth part of a second—in the twinkling of an eye. ”

“ Motion. A continued and successive change of place, either of a whole body, or of the particles of [Pg 211] which a body is composed; the earth in revolving upon its axis only, would not change its place as a body, but all the particles of which it is composed, would revolve round a common axis of motion. In revolving in its orbit, its whole mass is constantly occupying a new portion of space. ”

“ But a line can be traversed in either direction, while conceptual time is supposed to permit but one way of passing from one instant to another in its course. An uniform flow, or other motion, has, like time, a fixed direction, but might be conceived as returning into itself without detriment to its uniformity. ”

“ The varieties of force themselves have been instructively reduced to a single basis—that of motion; electricity, gravity, light, and all the rest, are at present referred to the movement in particular orbits and planes of the ultimate particles which build up all masses. ”

“ We can only learn the laws of movement by observing movement—the swing and poise of the figure, the relation of the lines of limbs and drapery to the direction of force and centre of gravity, so important in composition. ”

“ All these millions, these thousands of millions, of suns hasten through boundless space with a speed which attains a velocity of two, three, and even four thousand metres a second. Motion maintains the equilibrium of the universe, and constitutes its organization, energy, and life. ”

“ Every body cast into the air moves under the influence of two distinct forces. By the one it is carried forward with an equal motion, and describes equal spaces in equal times, in the direction in which it was projected; and by the other, which we call gravity, is drawn downwards in lines perpendicular to the surface of the earth, with a motion continually accelerated, or whose velocity is always increasing. ”

“ Those that occur in one and the same line are either contrary to one another in direction or maintain the same direction. As all these motions are contemplated as taking place alone, there results immediately from the relation of the lines, that is, of the spaces of motion described in equal time, the relation of velocity. ”

“ In addition to all this, our whole sidereal system is carried along in space by a joint movement. And thus everything moves, everything circulates, everything rushes, with breathless speed, towards a goal which is unknown and which is never reached. ”

“ Moment of Force.—In the study of motion we found that the quantity of motion is called momentum and is measured by the product of the mass times the velocity. In the study of parallel forces, especially such as tend to produce rotation, we consider a similar quantity. It is called a moment of force, which is the term applied to the effectiveness of a force in producing change of rotation. ”