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Miscellaneous Eramples on Uniform Motions, &c. i If I see the flash of a cannon, fired by a ship in distress at sea. and hear the report 33 seconds afterwards, how far is she from me:
Ans. 7 '' nuiles. 2 The diameter of an iron shot is 6-o inches; what is its weight, it being known from experiment, that a cast-iron ball of 4 inches diameter weighs 9 lbs ?
Ans. 42,294 lbs. 3. What is the weight of a leaden ball of 65 inches in diameter, the weight of a leaden ball of 44 inches in diameter being 17 lbs. ?
Ans. 63,888 lbs. 4. It is proposed to determine the proportional quantities of matter in the earth and moon, the density of the fornier being to that of the latter as 10 to 7, and their diameters 7930 and 2160 miles respectively.
Ans. as 71 to I nearly. 5. A body weighing 20 lbs. is impelled by such a force as to send it through 100 feet in a second; with what velocity would a body of g lbs. move if it were impelled by the same force
Aus. 250 feet per second. 6. The body A weighs 100 lbs., another body B weighs 60 lbs., but the body B is impelled by a force 8 times greater than A ; required the proportion of the velocities with which they move ?
Aps. the velocity of A is to that of B as 3 to 40. 7. The body A las passed over 50 miles, the body B only 5, but A moves with 5 times the velocity of B; what is the ratio of the times that they have been in motion ?
Ans. 2 to 1. 8. The body A moves 30 times swifter than B, and A has moved 12 minutes but B only 1, what difference will there be between the spaces described by then, supposing B to have moved over a space of 5 feet?
Ans. 1795 feet. 9. The hour and minute hands of a clock are together at 12 o'clock; when are they next together ?
Ans. 53 minutes past 1.
ON THE MOTIONS OF BODIES ACCELERATED OR
RETARDED BY THE ACTION OF CONSTANT AND UNIFORM FORCES.
28. 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.
For, suppose the time divided into very small parts, then (by article 21) the momentum generated 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 sanie article the momentum for toch small time, is also as the motive force. Consequently the whole momentum generated, is in the compound ratio of the force and tiins of acting. Or, retaining the same notation as before,
.mft. 29 Cor. 1. The momentum, or motiou lost or destroyed in any time is also
in the compound ratio of the force and time. For, whatever monientum any force generates in a given time, the same momentun will an equal force destroy in the same or equal time, if acting in a contrary direction. And the same is true of the increase or decrease of motion, by forces that conspire will, or oppose the motion of bodies.
30. Cor. 2. The relocity generated, or destroyed in any time, is as the force and time directly, and the body or mass of matter reciprochlly. For by the present article mast, and, by 22, in any sort of motion mocbo; therefore fiicbr, or osen And if we suppose b and 5 constant, the velocity is simply as the time.
31. If a lody be moved from a state of rest by an uniform force, the space described, reckoning from the beginning of the motion, varies as the square of the time, or as the square of the last acquired velocity.
Let AB. fig. 1, represent the tiine of the body's motion ; draw BC at right angles to AB, and let BC represent the last acquired velocity ; join AC; divide the time AB into small equal portions AD, DE, EF, FG, &c. and from the points D, E, F, G &c. draw DK, EL, FM, GN, &c. parallel to BC, meeting AC in the points K, L, M, N, &c. complete the parallelograms DX, EW, FV, GT, &c.
Then, in the similar triangles ABC, ADK, we have AB : AD :: BC: DK ; and, since BC represents the velocity acquired in the tiine AB, DK will represent the velocity acquired in the time AD; because, (by Cor. 2, Ari. 30,) when the mass and force are constant, the velocity is as the time; in the same manner, it appears that EL, FM, GN, &c. represent the velocities generated in the times AE, AF, AG, &c. Now, if the body move with the uniform velocity DK, during the time AD, and with the uniform velocities EL, FM, GN, &c. during the times DE, EF, FG, &c, respectively, the spaces described, may be properly represented by the rectangles DX, EW, FV, GT, &c. (because in uniform inotion s is always as vt); therefore, the whole space described, on this supposition, will be represented by the sum of these rectangles, or by the triangle ABC, together with the sum of the triangles AXK, KWL, LVM, MTN, &c. or because the bases of these small triangles are respectively equal to IB, and the sum of their alti!udes is equal to BC, the whole space described
be represented by the triangle ABC, together with half the rectangle BQ. Let now the intervals AD, DE, EF, FG, &c. be diminished without limit with respect to AB, and the rectangle BQ is diminished without limit with respect to the triangle ABC; or, in other words, ABC+BQ approaches to ABC as its limit; therefore, when the motion of the body is constantly accelerated, the space described is represented by the area of the triangle ABC.
described in any other time AG, reckoning from the beginning of the motion, is represented on the same scale by the area of the triangle AGN, and because these triangles are similar, the space described in the time AB : the space described in the time AG :: AB? : AGʻ. And therefore, generally, sa 12.
32. Again BG, GN, represent the velocities generated in the times AB, AC; and therefore from the same similar triangles the space described in the time AB : the space described in the time AG :: BGR : GN”. And consequently we have, in general, socvs.
Er. If a body be accelerated from a state of rest by a uniform force, and describe m feet in the first second of time, it will describe 4m, 9m, 16m....mtfeet, in the 2, 3, 4, ...t first seconds.
33. Cor. 1. The space described, reckoning from the beginning of the motion, is half that which wouli be described in the same time with the last acquired velocity continued uniform.
For completing the parallelogram BD, fig. 2, then, from the present article it appears, that the space described in the time AB, reckoning from the beginning of the motion, may be represented by the triangle ABC. But the space that would be described by a body moving uniformly for the time AB, with the last acquired velocity, may be represented by the rectangle BD; and the area of the triangle ABC, is, by a well-known theorein, equal to one balf that of the rectangle BD; hence the proposition is manifest, and in the case of wiform forres we have, generally, s=flo.
34. Cor. 2. JIS, T, V, represent the space, time, and last acquired velocity, in any otber case of uniform forces, we shall also have, Serv. But
8:8:: tv: TV :: to: TV; therefore, generally, $Ito; or when bodies are put in motion hy uniform furces, the spucts described in any times, reckoning from the beginning of the motion is euch case, are as the time and last acquired relocity jointly.
35. Cor. 3. The space described in the time GB, fig. 2, is represented by the area GBCN; or, if NM be drawn parallel to GB, the space may be represented by the rectangle GM, together with the triangle NMC. Now GM represents the space which a body would describe in i he time GB, with the uniform velocity GN; and the triangle NMC, which is similar to the triangle ABC, represents the space through which the body would be moved from a state of rest by the action of the force, in the time GB ; thus, the space described in any time, when a body is projected in the direction of the force, is equal to the space which it would have described in that time, with the first velocity continued uniform, together with the space through which it would have beeu moved from a state of rest, in the same time by the action of the force.
36. Cor. 4. If a body be projected in a direction opposite to that in which the uniform force acts, with the velocity BC, and move lill that velocity is destroyed, the whole time of its motion is represented by BA, and the space descrihed by the area ABC. For the time required to destroy any velocity by the action of a uniform force, is equal to the time that would be required to generate the same velocity by the action of the same force (29); and, since the wbole times of motion in the two cases are equal, and also if equal times be taken, from the beginning of the motion in oue case, and from the end of the motion in the latter, the velocities at those instants are equal. Since, then, the whole times of motion are equal, and also the velocities at all corresponding points of times, the whole spaces described are equal.
Also the space described in the time BG is represented, on the same scale, by the area BGNC; that is, by the rectangle BL diminished by the triangle CLN, or CNM. Thus it appears, that the space described in the time BG, is eqnal to that which would have been described with the first velocity continued uniform during that time, diminished by the space through which the body would have been moved from a state of rest in the same time, by the action of an uniform force.
37. The spaces passed over by bodies, urged by any constant and uniform forces, acting during any times, are in the compound rutio of the forces and squares of the times directly, and the lody or mass reciprocally, or as the mass and square of the velocity directly, and the force reciprocally.
For, by art. 34, sactv ; and, by Cor. 2, art. 30, vo; therefore, sahip Also, since van, we have to
and consequently by sublivo stitution, sa
58. Cor. The same expressions which represent the relations of the forces, spaces, times, and velocities, in accelerated motions, represent them, also, when the motions are retarded, and the bodies move till their whole velocities are destroyed. For the time in wbich any velocity is destroyed is equal to the time in which it would be generated by the same force; also, the spaces described, on the supposition that the body in the latter case is moved from a state of rest, has been shown to be equal
39. Articles 28, 29, &c. give theorems for resolving all questions relating to notions uniformly accelerated or retarded. Thus, let
b = any body or quantity of matter.
ft Then, from the fundamental relations moc bv, muft, sI tv, voci, we obtain the following Table of the general relations of uniformly accelerated or retarded motions.
ms fs oftv bv
bve bs fuoc OC
เบ bs btv ft
mo tac OOC
ON FON Fo Tful
-OC OC &c.
m 40. 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 inatter be always the same, then, instead of the relation sochy, we shall have, more simply, safe®, or the space described as the force and square of the time. And if the body be proportional to the force, as all bodies are ia respect of their gravity ; then sa 1 Olv?, or the space described is as the square of the time, or as the square of the velocity.
, substituting this in the foregoing Table, we shall have,
If F be put = 4, then will F represent the accelerating force, and
ON THE LAWS OF GRAVITY, AND OF THE DESCENT
OF HEAVY BODIES.
41. The force of gravity, at any given place, is an uniform force, which always acts in a direction perpendicular to the horizon, and accelerates all bodies equally.
The same body will, by its gravity, always produce the same effect under the saine circumstances : thus it will, at the same place, bend the same spring in the same degree ; it will also fall through the same space in the same time, if the resistance of the air be reinoved ; therefore, the force of gravity is uniform. Also, all bodies which fall freely by this force, descend in lines perpendicular to the horizon; and in an exhausted receiver, they all fall through the same space in tlie same time; consequently, gravity acts in a direction perpendicular to the horizon, and accelerates all bodies equally.
The result of numberless experiments made on the descent of heavy bodies is, that every body which falls freely in vacuo by the force of gravity, descends from rest through 1610 feet in the first second.
This fact being established, every thing relating to the descent of bodies when they are accelerated by the force of gravity, and to their ascent when they are retarded by that force, supposing the motions to be in vacuo, may be deduced from the foregoing propositions relating to constant forces.
Since in the first second of time a body falls through 16t's feet, at the end of that time it will have acquired a velocity which, without any further action of gravity would carry it through 32 feet in the next second (33). Therefore, if g denote 16, the space fallen through in one second of time, 2g will denote the velocity generated in that time, then (30 and 31) because the velocities are directly proportional to the times, and the spaces to the squares of the time, we shall have for any other time t.
As 1" : t" :: 2y : 2gt=v, the velocity acquired in t",
and 18 : 1 :: g :g7e=s, the space passed over in t". So that, for the descents of heavy bodies by the force of gravity, we have these general equations :