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a” O. # O' m O. #. 127. Phop. The momentum, or quantity of motion, ge. nerated by a single impulse, or any momentary force, is as the generating force. That is, m is as f: where m denotes the momentum, and f the force. For every effect is proportional to its adequate cause. So that a double force will impress a double quantity of motion; a triple force, a triple motion ; and so on. That is, the motion impressed, is as the motive force which produces it.
128. PRop. The momenta, or quantities of motion, in moving bodies, are in the compound ratio of the masses and velocities.
That is, m is as br.
For, the motion of any body being made up of the mo. tions of all its parts, if the velocities be equal, the momenta will be as the masses; for a double mass will strike with a double force ; a triple mass with a triple force ; and so on. Again, when the mass is the same, it will require a double force to move it with a double velocity, a triple force with a triple velocity, and so on ; that is, the motive force is as the velocity; but the momentum impressed, is as the force which produces it, by art. 127; and therefore the momentum is as the velocity when the mass is the same. But the momentum was sound to be as the mass when the velocity is the same. Consequently, when neither are the same, the momentum is in the compound ratio of both the mass and velocity.
129. Prop. In uniform motions, the spaces described are in the compound ratio of the velocities and the times of their description.
That is, s is as tw. For, by the nature of uniform motion, s : s : : T : t, when v is constant :
Corol. 1. In uniform motions, the time is as the space directly, and velocity reciprocally; or as the space divided by the velocity. And when the velocity is the same, the time is as the space. But when the space is the same, the time is reciprocally as the velocity.
Corol. 2. The velocity is as the space directly and the time reciprocally; or as the space divided by the time. And when the time is the same, the velocity is as the space. But when the space is the same, the velocity is reciprocally as the time.
130. In uniform motions generated by momentary impulse,
let b = any body or quantity of matter to be moved, = force of impulse acting on the body b,
v = the uniform velocity generated in b,
m = the momentum generated in b,
s = the space described by the body b,
t = the time of describing the space s with the veloc. v.
Then from the last three propositions and corollaries, we have these three general proportions, namely, f or m, m of bv, and s of tw; from which is derived the following table of the general relations of those six quantities, in uniform motions, and impulsive or percussive forces:
By means of which, may be resolved all questions relating to uniform motions, and the effects of momentary or impulsive forces.
131. Prop. The momentum generated by a constant and uniform force, acting for any time, is in the compound ratio of the force and time of acting.
That is, m is as fl.
For, supposing the time divided into very small parts, by art. 127, the momentum in each particle of time is the same, and therefore the whole momentum will be as the whole time, or sum of all the small parts. But by the same prop. the momentum for each small time, is also as the motive force. Consequently the whole momentum generated, is in the compound ratio of the force and time of acting.
Corol. 1. The motion, or momentum, lost or destroyed in any time, is also in the compound ratio of the force and time. For whatever momentum any force generates in a given time; the same momentum will an equal force destroy in the same or equal time ; acting in a contrary direction.
And the same is true of the increase or decrease of motion, by forces that conspire with, or oppose the motion of bodies.
Corol. 2. The velocity generated, or destroyed, in any time, is directly as the force and time, and reciprocally as the body or mass of matter.—For, by this and art., 128, the compound ratio of the body and velocity, is as that of the force and time ; and therefore the velocity is as the force and time divided by the body. And if the body and force be given, or constant, the velocity will be as the time.
132. Prop. The spaces passed over by bodies, urged by any constant and uniform forces, acting during any times, are in the compound ratio of the forces and squares of the times directly, and the body or mass reciprocally.
Or, the spaces are as the squares of the times, when the
let v denote the velocity acquired at the end of any time t, by any given body b, when it has passed over the spaces. Then, because the velocity is as the time, by the last corol. therefore #0 is the velocity at ot, or at the middle point of the time; and as the increase of velocity is uniform, the same space s will be described in the same time t, by the velocity +1, uniformly continued from beginning to end. But, in uniform motions, the space is in the compound ratio of the time and velocity; therefore s is as tw, or indeed s =
3tv. But, by the last corol. the velocity v is a', or as the force and time directly, and as the body reciprocally. Therefore s, or tv, is *:: that is, the space is as the force and square of the time directly, and as the body reciprocally,
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Or s is as to, the square of the time only, when b and f are given. Corol. 1. The space s is also as to, or in the compound ratio of the time and velocity; b and f being given. For, s = 4tv is the space actually described. But tv is the space which might be described in the same time t, with the last velocity v, if it were uniformly continued for the same or an equal time. Therefore the spaces, or 4tv, which is actually described, is just half the space tw, which would be described with the last or greatest velocity, uniformly continued for an equal time t. Corol. 2. The space s is also as vo, the square of the ve. * locity; because the velocity v is as the time t.
133. The last four propositions give theorems for resolv. ing all questions relating to motions uniformly accelerated. Thus, put b = any body or quantity of matter, f = the force constantly acting on it,
w t the time of its acting,
m = the momentum at the end of the time. Then, from these fundamental relations, m or br, m or ft,
t - s of tv, and v of 4. we obtain the following table of the general relations of uniformly accelerated motions:
134. From the above relations those quantities are to be left out which are given, or which are proportional to each other. Thus, if the body or quantity of matter be always the same, then the space described is as the force and square of the time. And if the body be proportional to the sorce, as all bodies are in respect to their gravity ; then the space described is as the square of the time, or square of the velo
city ; and in this case, if F be put = % the accelerating force; then will
135. Prop. If a body strike or act obliquely on a plain surface, the force or energy of the stroke, or action, is as the sine of the angle of incidence. Or, the force on the surface is to the same if it had acted perpendicularly, as the sine of incidence is to radius. Let AB express the direction and the absolute quantity of the oblique force on the plane De; or let a given body A, moving with a certain ve1ocity, impinge on the plane at B ; then its force will be to the action on the plane, as radius to the sine of the angle Abd, or as Ab to Ad or fic, drawing AD and Bc perpendicular, and Ac parallel to de. For, by art. 29, the force Ab is equivalent to the two forces Ac, cb ; of which the former Ac does not act on the plane, because it is parallel to it. The plane is therefore only acted on by the direct force cB, which is to AB, as the sine of the angle BAC, or ABD, to radius.
Corol. 1. If a body act on another, in any direction, and be any kind of force, the action of that force on the second body, is made only in a direction perpendicular to the surface on which it acts. For the force in AB acts on DE only by the force cB, and in that direction.
Corol. 2. . If the plane de be not absolutely fixed, it will *move, after the stroke, in the direction perpendicular to its