PROP. VIII. 15. If bodies strike one another obliquely, it is proposed to determine their motions after the stroke. Let the two bodies B, b, move in the oblique directions BA, bA, and strike each other at A with velocities which are in proportion to the lines BA, bA; to find their motions after the impact. Let CAH represent the plane in which the bodies touch in the point of concourse; to which draw the perpendiculars BC, bD, and complete the rectangles CE, DF. Then the motion in BA is resolved into the two B E D G H BC, CA; and the motion in bA is resolved into the two bD, DA; of which the antecedents BC, bD, are the velocities with which they directly meet, and the consequents CA, DA are parallel; therefore, by these the bodies do not impinge on each other, and consequently the motions, according to these directions, will not be changed by the impulse; so that the velocities with which the bodies meet, are as BC and D, or their equals EA and FA. The motions, therefore, of the bodies B, b, directly striking each other with the velocities EA, FA, will be determined by Prop. vi. p. 837, according as the bodies are elastic or nonelastic; which being done, let AG be the velocity, so determined, of one of them, as A; and since there remains also in the body a force moving in the direction parallel to BE, with a velocity as BE; make AH equal to BE, and complete the rectangle GH; then the two motions in AH and AG, or HI; are compounded into the diagonal AI, which therefore will be the path and velocity of the body B after the stroke. And after the same manner is the motion of the other body b determined after the impact. FUNDAMENTAL EQUATIONS OF MOTION. PROP. IX. 16. In uniform motion, the space s described with a velocity v in time t is s = tv. For v is the space described in each unit of time, and t is the number of units of time; therefore the whole space described is 17. In uniformly accelerated or retarded motion, the velocity ▾ generated by the force f in time t is v = ft. For the velocity generated in each second is ƒ, and hence in t seconds, it is ft, and therefore when it Cor. If u = velocity when t = 0, the velocity at the end of the time t is v=uft, the + being used when the force is accelerating, and the is retarding. PROP. XI. 18. The space described from rest by a body, acted on by a uniformly accele rating force is s = vt. Let s = space described, in the time t, by the action of the uniformly accele rating force ƒ, and v = velocity acquired in time t. Divide t into n equal intervals, and therefore each = -; then velocity acquired in each interval = and t n the space described in time t with the velocity at the end of each interval continued uniform during that interval is and the space described in the time t with the velocity at the beginning of each interval continued uniform during that interval is Now s manifestly lies between these two spaces, being described with velocities intermediate to the velocities with which these are described. Increase n, the number of intervals without limit, and we have Scholium.-If the body, instead of beginning to move from rest, be projected with velocity u; then the space described in time t is s = ut ± ft2 the applying when the force accelerates, and the when it retards the motion. For ut is the space described with the velocity of projection alone, and therefore the space described from both causes is s = ut ± }ft3. . . . (7) PROP. XII. 19. If the accelerating or retarding force be variable, then ds = vdt. Let S1 and s, be the spaces described in the times t、 and t2, where t1 = t − 0, = t + 0; then whatever is the nature of the motion, the space will be a function of the time; and hence by Taylor's theorem and t2 ds d2s 02 d's 03 dt dt2 1.2 + 1.2.3 But the space described with the uniform velocity v in the time is ve, which is always intermediate to the spaces s―s, and s -s, however small may be; therefore we have Cor. 1. In the same manner it is proved, that if ƒ be the accelerating force at the end of the time t; then 20. In variable motion we have therefore the general equations (10) = d's or f= dt (11) dv = fdt.... (a) vdv=fds. (c) (d) which are applicable either to the case of the motion being accelerated or retarded; but in the latter case, du is a negative quantity. THE LAWS OF GRAVITY; THE DESCENT OF HEAVY BODIES; AND THE MOTION OF PROJECTILES IN FREE SPACE. PROP. XIII. 21. All the properties of motion delivered in Proposition x1., its corollaries and scholium, for constant forces, are true in the motions of bodies freely descending by their own gravity; namely, that the velocities are as the times, and the spaces as the squares of the times, or as the squares of the velocities. For, since the force of gravity is uniform, and constantly the same, at all places near the earth's surface, or at nearly the same distance from the centre of the earth, and that this is the force by which bodies descend to the surface; they therefore descend by a force which acts constantly and equally; consequently, all the motions freely produced by gravity, are as above specified by that proposition, &c. 22. SCHOLIUM.-Now it has been found, by numberless experiments, that gravity is a force of such a nature that all bodies, whether light or heavy, fall perpendicularly through equal spaces in the same time, abstracting from them the resistance of the air—as lead or gold and a feather, which in an exhausted receiver fall from the top to the bottom in the same time. It is also found, that the velocities acquired by descending are in the exact proportion of the times of descent; and farther, that the spaces descended are proportional to the squares of the times, and therefore to the squares of the velocities. And hence it follows that the weights, or gravities of bodies near the surface of the earth, are proportional to the quantities of matter contained in them; and that the spaces, times, and velocities generated by gravity, have the relations contained in the proposition to which we have above referred. Moreover, as it is found, by accurate experiments, that a body, in the latitude of London, falls nearly 16', feet in the first second of time, and consequently that at the end of that time it has acquired a velocity double, or of 32 feet; hence it is obvious, if g denote 16 feet, the space fallen through in one second of time, or g the velocity generated in that time; then, because the velocities are directly proportional to the times, and the spaces to the squares of the times, therefore, 1" : t" :: g: gt = v, the velocity, and 12 t2: 19: 19t2 = 8, the space. And hence, for the descents of gravity we have these general equations, namely, Hence, because the times are as the velocities, and the spaces as the squares of either, therefore namely, as the series of the odd numbers, which are the differences of the squares denoting the whole spaces. So that, if the first series of natural numbers be seconds of time, namely, The times in seconds 4", &c. 3", 2", 1", 321, 64, 96, 128, &c. h e The velocity in feet will be The spaces in the whole times And the space for each second These relations may be aptly represented by the abscisses and ordinates of a parabola. Thus, if PQ be a parabola, PR its axis, and RQ its ordinate; and Pa, Pb, Pc, &c., parallel to RQ, represent the times from the beginning, or the velocities, then ae, bf, cg, k &c. parallel to the axis PR, will represent the spaces described by a falling body in those times; for, in a parabola, the abscisses Ph, Pi, Pk, &c., or ae, bf, cg, &c., which are the spaces described, are as the squares of the ordinates, he, if, kg, &c., or Pa, Pb, Pc, &c., which represent the times or velocities. R 23. And because the laws for the destruction of motion are the same as those for the generation of it, by equal forces, but acting in a contrary direction; therefore, 1st. A body thrown directly upwards, with any velocity, will lose equal velocities in equal times. 2nd. If a body be projected upwards, with the velocity it acquired in any time by descending freely, it will lose all its velocity in an equal time, and will ascend just to the same height from whence it fell, and will describe equal spaces in equal times, both in rising and falling, but in an inverse order; and it will have equal velocities at any one and the same point of the line described, both in ascending and descending. 3rd. If bodies be projected upwards, with any velocities, the height ascended to, will be as the squares of those velocities, or as the squares of the times of ascending, till they lose all their velocities. 24. When the body, instead of being permitted to fall from rest, is projected upwards or downwards with a given velocity u; then by Art, 17 and the scholium to Prop. XI. we have where the must be employed when the projection is vertically upwards, and the+when the projection is vertically downwards. PROP. XIV. 25. Let two weights W and w hang over a fixed pulley, to determine their motion, neglecting the inertia of the pulley, and the weight of the string. Here the moving force of W is Wg, and that of w = wg; also the mass resisting motion is W+w; hence the accelerating force on W = (W―w)g W + w (1.) Find the space descended by a body in 7 seconds of time, and the velocity acquired. s=gt2 = 16 × 72 = 16 × 49 = 788ft = space descended. v=gt = 324 X 7 = (2.) A body is projected vertically from the bottom of a tower 200 feet in height, with a velocity of 120 feet per second; in what time will it reach the top, and what will be its velocity at that time? Also, to what height above the top of the tower will it rise? the former of these values of t is the time at which the body in its ascent passes the top of the tower; and the latter, the time at which the body in its descent passes the top. Again, v = u — gt = 120 · 324 × 2·513′′ = 39·165 feet per second, the velocity of the body at the top of the tower. Hence height above top = 223.83 · 200 23.83 feet. =223.83 feet. |