Conservation of momentum

“ The third law of motion tells us that action and reaction are equal and opposite, so that when two bodies come into collision the forces at work generate equal and opposite quantities of momentum. We shall best see the meaning of this law by a numerical example, bearing in mind that momentum means the product of mass into velocity. ”

“ In the case of a connected system in motion, the configuration is continually changing at a rate depending on the velocities so that we can no longer assume that the momentum is the time-integral of the force which acts on it. ”

“ But in this case, each impulse produces its effect instantaneously, and the time is employed, not in the transition from any cause to its effect, but in the intervals between the action of the several causes, during which intervals the body goes on with the velocity already communicated to it. In each impulse, force produces motion: and the motion goes on till a new change takes place, by the same kind of action. The force may be said, in the language employed by the Critic, to be transformed into momentum ”

“ 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. ”

“ The smaller we suppose the increment of time δt, the greater must be the impressed forces, but the time-integral, or impulse, of each force will remain finite. The limiting value of the impulse, when the time is diminished and ultimately vanishes, is defined as the instantaneous impulse, and the momentum p, corresponding to any variable q, is defined as the impulse corresponding to that variable, when the system is brought instantaneously from a state of rest into the given state of motion. ”

“ It must be recollected that a volume of air, when once put in motion, will move on, like any other body, by the mere force of its own momentum, till that motion is destroyed by its friction against the substances along or through which it is impelled. ”

“ To get an idea of the significance of these tensor triples, we first consider the mechanical momentum and its corresponding mechanical momentum flux. The total momentum flux into the interior of a room, i.e. the increase in momentum in the interior per unit time, is equal to the resulting mechanical force which acts on the entire mass located within that space. ”

“ But the pendulum produces its mechanical effect upon the escapement, at the moment of its contact, and each wheel upon the next, at the moment of its contact. As has 192 been said in a Review of this work, ‘The time lost in cases of indirect physical causation is consumed in the movements which take place among the parts of the mechanism in action, by which the active forces so transformed into momentum are transported over intervals of space to new points of action, the motion of matter in such cases being regarded as a mere carrier of force.’ (Quarterly Rev. No. cxxxv. p. 212.) ”

“ Thus we see that, if the three bodies were all moving in the same plane, they could, speaking generally, maintain more easily the requisite description of areas, that is, the requisite moment of momentum with smaller velocities than if they were moving in directions which were not so regulated; that is to say, the moment of momentum can be kept up with less energy when the particles move in the same plane. ”

“ In problems of impact we have to deal with cases of practically instantaneous impulse, where a very great and rapidly varying force produces an appreciable change of momentum in an exceedingly minute interval of time. ”

“ Reverting to the case of a body supported by the communication of momentum to a number of independent material particles, it is evident that the particles projected downwards eventually give up their momentum on striking the surface of the earth. ”

“ I represent the whole moment of momentum of our system by £1,000,000,000, the bulk of which is composed of the revolutional moments of momentum of the great planets, and the rotational moment of momentum of the sun. On this scale the rotational share which has fallen to our earth and moon does not even rise to the dignity of a single pound, it can only be represented by the very modest figure of 19s. ”

“ The settlement, therefore, of the definition of momentum, involved the determination of the question, What is that of [Pg 223] which a body, when it sets another body in motion, loses exactly as much as it communicates? And when experiment had shown that this something was the product of the velocity of the body by its mass, or quantity of matter, this became the definition of momentum. ”

“ In the production of motion by force, for instance, though the effect be cumulative with continued exertion of the cause, yet each elementary or individual action is, to our apprehension, instanter accompanied with its corresponding increment of momentum in the body moved. In all dynamical 195 reasonings no one has ever thought of interposing an instant of time between the action and its resulting momentum ”

“ Thus, when the body is pushed forward by the light, some other system must have lost the momentum the body acquires, and the only other system available is the wave of light falling on the body; hence we conclude that there must have been momentum in the wave in the direction in which it is travelling. ”

“ Still another scheme, far more important and attractive, was to derive power from the rotational energy of terrestrial bodies. I had discovered that objects on the earth's surface, owing to the diurnal rotation of the globe, are carried by the same alternately in and against the direction of translatory movement. From this results a great change in momentum which could be utilized in the simplest imaginable manner to furnish motive effort in any habitable region of the world. ”

“ I have been engaged,” continued he, “in explaining still farther the nature of momentum, and I now propose to exhibit an experiment of a different kind, in order to illustrate the same subject. You, no doubt, remember,” continued Mr. Seymour, “that velocity makes up for weight: although, therefore, a fluid, as air, or water, may, in a state of quiescence, be unable to support a body, yet, by giving it a certain velocity, it may acquire a sustaining power. ”

“ Thus in the case of a limited wave train, if it carry momentum, that momentum can be represented by some definite value of mv, and the remainder of the system with which the wave is associated must, relatively to the common mass centre, have an equal and opposite momentum at every instant of time. ”

“ Thus the present application of the principle of no momentum is in no way antagonistic to modern views and discovery as to the transference of momentum by light and other manifestations of electric radiation. ”

“ But although these circumstances be inefficient, a variation in the mean temperature would certainly occasion a corresponding change in the velocity of rotation: for in the science of dynamics, it is a principle in a system of bodies, or of particles revolving about a fixed centre, that the momentum, or sum of the products of the mass of each into its angular velocity and distance from the centre is a constant quantity, if the system be not deranged by an external cause. ”

“ And in this manner a material point, by the successive flowing of its ubication, describes a line in space, or evolves the intensity of its momentum into extension.
Hence, of local movement we can predicate both intensity and extension. The intensity is the formal principle, which, by actuating the inertia or mobility of the subject, evolves itself into extension. The extension is the actual evolution of the momentum, and constitutes the essence of local movement, which is always in fieri. ”

“ I am one of those who accept the proposition, and believe that a falling body acquires force [vires] in its descent, its velocity increasing in proportion to the space, and that the momentum [momento] of the falling body is doubled when it falls from a doubled height ”

“ And when the blow is violent enough to fracture the mass, we see, in the radial dispersion of its fragments, that the original momentum, in being distributed throughout it, has been divided into numerous minor momenta, unlike in their directions. ”

“ If we consider a finite train of waves thus sent back from a moving reflector, the time integral of the pressure must represent force transmitted along the cord, or a gain of longitudinal momentum in the reflected waves, or both together. ”

“ Remarks.—As the ubication, so also the momentum produced by accidental action, can be considered both absolutely and respectively. The momentum, considered absolutely, is an act received in a subject—an absolute momentum, an extrinsic term of the virtuality of the motive principle; and, as such a momentum is only one out of the innumerable acts which can proceed from the agent, it has an entity infinitely less than that of the agent. ”

“ Hence we may conclude that every such system as is presented to us in the infinite variety of Nature, must stand in intimate relation to some particular plane, being that which is known as the principal plane of moment of momentum. ”

“ The only ray of hope was to be found in the deficiency in one of the two great requisites of such a force. Momentum being weight, multiplied into velocity, there were some glimpses visible, of a nature to produce a slight degree of expectation that the last might yet be resisted. ”

“ The momentum, which is in the simple reciprocal ratio of the bodies, might consequently be increased at pleasure by the means proposed, in the subduplicate ratio of the bodies employed; and if momentum were an efficient force capable of reproducing itself, and of overcoming friction in proportion to its estimated magnitude, the additional force acquired by such a means of increase, might be employed for counteracting the usual resistances, and perpetual motion would be easily effected. ”

“ As to the energy we know already a number of different forms: the kinetic energy, gravitation, the elastic deformation energy, heat, chemical energy, and it is therefore not a principally new feature when to these different forms, the electromagnetic energy is attached as another form. Conversely, until now only a single form was known as momentum: mechanical momentum. ”

“ Then I say that all the momentum of the falling body is along the vertical and is a maximum when it falls in that direction; the momentum is less along DA and still less along EA, and even less yet along the more inclined plane FA. Finally on the horizontal plane the momentum vanishes altogether; the body finds itself in a condition of indifference as to motion or rest; has no inherent tendency to move in any direction, and offers no resistance to being set in motion. ”

“ The speed of a body which has in this manner acquired any momentum whatever will, when converted into uniform motion, retain precisely such a speed as, during a time-interval equal to that of the fall, will carry the body through a distance equal to twice that of the fall. ”

“ By the mutual action of particles on one another, by contraction, by tidal pulls, and so on, some energy of motion is constantly being changed into heat and thus dissipated away. Energy of motion, therefore, is slowly being lost to the system, and the only stable state for the bodies composing it is when their energy of motion has decreased to the minimum consistent with the initial moment of momentum. This principle we shall find very fecund in its application. It means that our whole system is evolving in a way to lessen its energy of motion while keeping its quantity of motion unchanged. ”

“ The force embodied as momentum in a given direction, cannot be destroyed; and if it eventually disappears, it re-appears in the reaction on the retarding body; which begins afresh to draw the now arrested mass back from its aphelion. The only conditions under which there could be absence of rhythm—the only conditions, that is, under which there could be a continuous motion through space in the same straight line for ever, would be the existence of an infinity void of everything but the moving body. ”

“ This will prevent one person from imagining it larger, another smaller, than it really is; so that in the composition of a given uniform motion with one which is accelerated different men may not obtain different values for the resultant. In order to determine and represent such a momentum and particular speed [impeto e velocità particolare] our Author has found no better method than to use the momentum acquired by a body in naturally accelerated motion. ”

“ It will thus be seen that Momentum and Energy are entirely different, although co-related; that momentum relates to velocity, which includes the element of time, whereas energy relates211 to the amount of work done, and may be represented by a force operating against a certain resistance, through a certain distance, entirely irrespective of time. The energy is the same with the same force operating against the same resistance, through the same distance whether the time consumed be great or small. ”