oc oc oc t OC t b bs f By means of which, may be resolved all questions relating 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 tinie, 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. 2 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 directly, and the Body or Mass reciprocally. Or, the Spaces are as the Squares of the Times, when the Force and Body are given. ft, That is, s is as or as t when f and b are given. For, b let v denote 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 it, 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 {v 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 įtv, or indeed s= ft tv. But, by the last corol. the velocity v is as b the force and time directly, and as the body reciprocally. fi b and square of the time directly, and as the body reciprocally. Or s is as t, the square of the time only, when b 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, s = tv 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 space s, or įtv, which is actually described, is just half the space tv, which would be described with the last or greatest velocity, uniformly continued for an equal time t. 31. Corol. 2. The space s is also as v', the square of the velocity; because the velocity v is as the time t. or as Therefore s, or ļ tv, is as ; that is, the space is as the force 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, Then, from these fundamental relations, m o bv, m e ft, se to, and ya 7 we obtain the following table of the general relations of uniformly accelerated motions : bs ifs ftv 173 c bo e ft * V bfs » bftv. t 8 C EC 8 8 <!>|~*~*~ S ft ma m2 fs fs ftv m m buo bs tv bs btv ta' fs 12? t'st b bft b mt ftv inv bu? mi?o sa tu C b f bf f f2t m? &c. f fi bfo 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 x be put = , the accelerating force; then will 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 Resolution 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 two Similar Forces, such that they would separately cause the Body to pass over the Spaces AB, AC, in an equal Time; then if both Forces act together, they will cause the Body to move, in the same Time, through as the Diagonal of the Parallelogram ABCD. 1 DRAW cd parallel to AB, and bd pa 7 b B rallel to AC. And while the body is carried over ab or cd by the force in that direction, let it be carried over bd” by the force in that direction; by which means it will be found at d. Now, if the forces be impulsive or momentary, the motions 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 ab = time in Ac, and the time in AB = time in Ac; therefore ab: bd :: AB : PD by equality : hence the point d is in the diagonal AD. And as this is always the case in every point d, d, &c, therefore the path of the body is the straight line Add, or the diagonal of the parallelogram. But if the similar forces, by means of which the body is moved in the directions AB, AC, be uniformly 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 AB or AC, T; then it will be Ab or cd : AB or CD ::ť: T?, bd or AC : BD or AC :: t: T?, 36. Corol. E 36. Corol. 1. If the forces be not similar, by which the body is urged in the directions AB, AC, it will move in some curved line, depending on the nature of the forces. 37. Corol. 2. Hence it appears, that the body moves over the diagonal AD, by the compound motion, in the very same time that it would move over the side AB, by the single force impressed in that direction, or that it would move over the side ac by the force impressed in that direction. 38. Corol. 3. The forces in the directions AB, AC, AD, are respectively proportional to the lines AB, AC, AD, and in these directions. 39. Corol. 4. The two oblique forces AB, AC, are equivalent to the single direct force AD, which may be compounded of these two, by drawing the diagonal of the parallelogram. Or they are equivalent to the double of AE, drawn to the middle of the line BC. And thus any force may be compounded of two or more other forces ; which is the meaning of the expression composition of forces. 40. Exam. Suppose it were D. required to compound the three forces AB, AC, AD; or to find the direction and quantity of one single force, which shall be equivalent to, and have the same effect, as if a body A were acted on by three forces in the directions AB, AC, AD, and proportional to these three lines. First reduce the two ac, AD to one AE, by completing the parallelogram ADEC. Then reduce the two AE, AB to one af by the parallelogram AEFB. So shall the single force Af be the direction, and as the quantity, which shall of itself produce the same effect, as if all the three AB, AC, AD acted together. 41. Corol. 5. Hence also any single direct force AD, may be resolved into two oblique forces, whose quantities and directions are AB, AC, having the same effect, by describing any paral E!... lelogram whose diagonal may be and this is called the resolution of forces. So the force AD may be resolved into the two AB, AC, by the parallelogram ABDC AD |