Consequently, when neither are the same, the momentum is in the compound ratio of both the mass and velocity. PROPOSITION IV. 22. In Uniform Motions, the Spaces described are in the Com. pound Ratio of the Velocities and the Times of their Descrip. tion. That is, s is as tv. For, by the nature of uniform motion, the greater the velocity, the greater is the space described in any one and the same time; that is, the space is as the velocity, when the times are equal. And when the velocity is the same, the space will be as the time ; that is, in a double time a double space will be described ; in a triple time, a triple space ; and so on. a Therefore universally, the space is in the compound ratio of the velocity and the time of description. 23 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. 24 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 (ime. Scholium. 25. In uniform motions generated by momentary impulse, let 6 any body or quantity of matter to be moved, f force of impulse acting on the body b, o the uniform velocity generated in b, m = the momentum generated in b, 8 = 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 « m, ma bu, and 8 & tv ; from which is derived the following table of the general relations of those six quantities, in uniform motions and impulsive or percussive, forces : VOL. II. of of 172 By means of which, may be resolved all questions relating lo uniform motions, and the effects of momentary or impulsive forces. PROPOSITION V. 26. 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 ft. For, supposing the time divided into very small parts, by prop. 2, 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. 27. Corol. 1. The motion, or momentum, lost or destroyed in any time, is also in the compound ratio of the force and time. For wha ver momentum any force generates in a given time; the same momentum will an equal force destroy e 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. 28. 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 the 3d prop. 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. PROPOSITION PROPOSITION VI. 29. 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 direct ly, and the Body or Mass reciprocally. Or, the Spaces are as the Squares of the Times, when the Force and Body are given, 9 or as fii Teat, is, o is as 6, or as 12 when f and b are given. For, let v donote the velocity acquired at the end of any time t, by any given body b, when it has passed over the space s. Then, because the velocity is as the time, by the last corol. therefore { v is the velocity at į t, 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 įv, uniformly continued from beginning to end. But, in uniform motions, the space is in the compound ratio of the time and velocity ; therefore 8 is as į lv, or indeed 8 = jtv. But, by the last corol. the velocity v is as . . v 용, b the force and time directly, and as the body reciprocally. Therefore s, or s tv, is as 5 space as the force and square of the time directly, and as the body reciprocally. Or : is as 12, the square of the time only, when 6 and f are given 30. Corol. 1. The space s is also as tv, or in the compound ratio of the time and velocity ; b and f being given. For, 8 = fev is the space actually described. But iv 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 tiine. Therefore the space s, or fiv, which is actually described, is just half the space iv, which would be de. scribed, with the last or greatest velocity, uniformly continu. ed for an equal time t. 31. Corol. 2. The space s is also as v2, the square of the velocity ; because the velocity v is.as the time t. ܪ 8 Scholium. 32. Propositions 3, 4, 5, 6, give theorems for resolving all questions relating to motions uniformly accelerated. Thus, put put b = any body or quantity of matter, f = the force constantly acting on it, (=the time of its acting, v= the velocity generated in the time t, s = the space described in that time, m = the momentum at the end of the time. Then, from these fundamental relations, m o bv, m & ft, ft 3 C tv, and vac we obtain the following table of the 6' general relations of uniformly accelerated motions : bs fs f12 v me bv a ft oc o v bfs v bfrv. fs bo fiv m2 m2 bol f2 st vin m2 b02 8 & Iu f f2 t m2 o Ng OC V 8 mt ४ 8 ms 8 სე ms m mo fo 12 m ms m bft m2 o 33 And from these proportions 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 force, 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 f city; and in this case, if p be put = the accelerating force; then will 8 & 10 Ft THE COMPOSITION AND RESOLUTION OF FORCES. 34. COMPOSITION OF Forces, is the uniting of two or more forces into one, which shall have the same effect; or the finding of one force that shall be equal to several others taken together, in any different directions. And the resolu. tion of Forces, is the finding of two or more forces which, acting in any different directions, shall have the same effect as any given single force. PROPOSITION VII. 35. If a Body at a be urged in the Directions AB and ac, by any wo Similar Forces, such that they would separately cause the Body to pass over the Spaces AB, AC, in an equal then if both Forces act together, they will cause the Body to move in the same Time, through ad the Diagonal of the Parallelogram ABCD. Time ; AC, Draw cd parallal to AB, and bd pa. rallel to And while the body is ъ ъ В AF carried over ab, or cd by the force in that CH direction, let it be carried over bd by the Hz force ir, that direction ; by which means Ch. it will be found at d. Now, if the forces be impulsive or momentary, the motions D will be uniform, and the spaces described will be as the times of description : theref. Ab or cd: AB or CD :: time in ab: time in AB, and bd or Ac: BD or ac :: time in ac: time in ac; but the time in Ao : time in Ac, and the time in AB = time in ac ; therefore ab: bd :: AB : BD by equality : hence the point d is in the diagonal ad. And as this is always the case in every point d, d, &c. there. fore the path of the body is the straight line ado, or the di. agonal of the parallelogram. But if the similar forces, by means of which the body is moved in the directions AB, AC, be uviformly accelerating ones, then the spaces will be as the squares of the times; in which case, call the time in bd'or cd, t, and the time in al or AC, T ; then it will be Ab or cd : AB or CD: :12 : 12, bd or AC : BD or AC :: 12 : To, theref. by equality, ab; bd :: AB:BD ; and so the body is always found in the diagonal, as before. 36. Corol. |