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time E in which the velocity 1 is extinguished by the uniform action of the corresponding resistance, or by 2 a, which is the space uniformly described during this time, with the velocity 1. And E and 2 a must be expressed by the same number. length. Thus, having ascertained these leading circumstances for a unit of velocity, weight, and bulk, we proceed to deduce the similar circumstances for any other magnitude; and, to avoid unnecessary complications, we shall always suppose the bodies to be spheres, differing only in diameter and density. First, then, let the velocity be increased in the ratio of 1 to v.
2 The resistance will now be # = r 2 a.
t), and e v = 2 a.; so that the rule is general, that the space along which any velocity will be extinguished by the uniform action of the corresponding resistance is equal to the height necessary for communicating the terminal velocity to that body by gravity. Fore v is twice the space through which the body moves while the velocity v is extinguished by the uniform resistance 2dly, Let the diameter increase in the proportion of 1 to d. The aggregate of the resistance changes in the proportion of the surface similarly resisted, that is, in the proportion of 1 to do. But the quantity of matter, or number of particles among which this resistance is to be distributed, changes in the proportion of 1 to do. Therefore the retarding power of the resistance changes in
The extinguishing time will be 7, -e,
- 1 - the proportion of 1 to d” When the diameter
2 this body. We must still have g = and
tu 2 a d’ therefore w” = 2 gad, and w = V 2 g a d, = v 2 g a V d. But u = V2 ga. Therefore the terminal velocity w for this body is = u Vod; and the height necessary for communicating it is ad. Therefore the terminal velocity varies in the subduplicate ratio of the diameter of the ball, and the fall necessary for producing it varies in the simple ratio of the diameter. The extinguishing time for the velocity v must now be Ed. v 3dly, If the density of the ball be creased in the proportion of 1 to m, the number -ticles among which the resistance is to be distributed is increased in the same proportion, and therefore the retarding force of the resistance is equally diminished; and, if the density of the air is inVol. XVIII.
It is a number of units, of time, or of .
v 72 Thus we see that the chicf circum-conces are regulated by the terminal velocity, or are conveniently referred to it. To communicate distinct ideas, and render the deductions from these premises perspicuous, it will be proper to assume some convenient units, by which all these qualities may be measured; and, as this subject is chiefly interesting in the case of military projectiles, we shall adapt our units to this purpose. Therefore let a second be the unit of time, a foot the unit of space and velocity, an inch the unit of diameter of a ball or shell, and a pound avoirdupois the unit of pressure, whether of weight or of resistance: therefore g is thirty-two feet. The great difficulty is to procure an absolute measure of r, or u, or a ; any one of these will determine the others. Sir Isaac Newton attempted to determiner by theory, and employed a great part of the second book of the Principia in demonstrating, that the resistance to a sphere moving with any velocity is to the force which would generate or destroy its whole motion in the time that it would uniformly move over eight-thirds of its diameter with this velocity as the density of the air is to the density of the sphere. This is equivalent to demonstrating, that the resistance of the air to a sphere, moving through it with a velocity, is equal to half the weight of a column of air having a great circle of the sphere for its base, and for its altitude the height from which a body must fall in vacuo to acquire this velocity. This appears from Newton's demonstration; for, let the specific gravity of the air be to that of the ball as 1 to m; then, because the times in which the same velocity will be extinguished by the uniform action of different forces are inversely as the forces, the resistance to this velocity would extinguish it in the time of describing eight-thirds m d, d being the diameter of the ball. Now 1 is to m as the weight of the displaced air to the weight of the ball, or as two-thirds of the diameter of the ball to the length of a column of air of equal weight. Call this length a ; a is therefore equal to two-thirds m d. Suppose the ball to fall from the height a in the time t, and acquire the velocity u. If it moved uniformly with this velocity, during this time, it would describe a space = 2a, or four-thirds m d. Now its weight would extinguish this velocity, * this motion, in the same time, that is, in the time of describing four-thirds m d, but the resistance of the air would do this in the time of describing eight-thirds m d: that is, in twice the time. The resistance therefore is equal to half the weight of the ball, or to half the weight of the column of air whose height is the height producing the velocity. But the resistance to different velocities are as the squares of the velocities; and therefore as their producing heights, and, in general, the resistance of the air to a sphere moving with any velocity, is equal to the half weight of a column of air of equal section, and whose altitude is the height producing the velocity. The result of this investigation has been acquiesced in by all Sir Isaac Newton's commentators. Many faults have indeed been found with his reasoning, and even with his principles; and it must be acknowledged that although this investigation is by far the most ingenious of any in the Principia, and sets his acuteness and address in the most conspicuous light, his reasoning is liable to serious objections, which his most ingenious commentators have not completely removed. Yet the conclusion has been acquiesced in, but as if derived from other principles, or by more logical reasoning. The reasonings or as: sumptions, however, of these mathematicians are no better than Newton's ; and all the causes of deviation from the duplicate ratio of the velocities, and the causes of increased resistance, which the latter authors have valued themselves for discovering and introducing into their investigations, were actually pointed out by Sir Isaac Newton, but purposely omitted by him to facilitate the discussion in re difficillima (See Schol. prop. 37. b. 2). The weight of a cubic foot of water is 62 lbs. and the medium density of the air is son of water; therefore let a be the height producing the velocity (in feet), and d the diameter of the ball (in inches), and r the periphery of a circle whose diameter is 1 ; the resistance of the air will be =
Erample.—A ball of cast iron weighing twelve pounds is four inches and a half in diameter. Suppose this ball to move at the rate of 257, feet in asecond. The height which will produce this velocity in a falling body is 97 feet. The area of its great circle is 0-11044 feet, or o, of one foot. Suppose water to be 840 times heavier than air, the weight of the air incumbent on this great circle, and 97 feet high, is 0.081151 lbs. half of this is 0 0405755 or 1%g, or nearly 3, of a pound. This should be the resistance of the air to this motion of the ball.
It is proper, in all matters of physical discussion, to confront every theoretical conclusion with experiment. This is particularly necessary in the present instance, because the theory on which this proposition is founded is extremely uncertain. Newton speaks of it with the most cautious diffidence, and secures the justness of the conclusions by the conditions which he as
sumes in his investigation. He describes with the greatest precision the state of the fluid in which the body must move, so as that the demonstrations may be strict, and leaves it to others to pronounce whether this is the real constitution of our atmosphere It must be granted that it is not; and that many other suppositions have been introduced by his commentators and foilowers to suit his investigation (for little or nothing has been added to it) to the circumstances of the case. Sir Isaac Newton himself, therefore, attempted to compare his proportions with experiment. Some were made by dropping balls from the dome of St. Paul's cathedral; and all these showed as great a coincidence with his theory as they did with each other: but the irregularities were too great to allow him to say with precision what was the resistance. It appeared to follow the proportion of the squares of the velocities with sufficient exactness; and, though he could not say that the resistance was equal to the weight of the column of air having the height necessary for communicating the velocity, it was always equal to a determinate part of it; and might be stated= n a, n being a number to be fixed by numerous experiments. One great source of uncertainty in his experiments seems to have escaped his observation: the air in that dome is almost always in a state of motion. In summer there is a very sensible current of air downwards, and frequently in winter it is upwards: and this current bears a very great proportion to the velocity of the descents. Sir Isaac takes no notice of this. He made another set of experiments with pendulums; and pointed out some very curious and unexpected circumstances of their motions in a resisting medium. There is hardly any part of his noble work in which his address, his patience, and his astonishing penetration, appear in greater lustre. It requires the utmost intenseness of thought to follow him in these disquisitions. Their results were much more uniform, and confirmed his general theory; and it has been acquiesced in by the first mathematicians of Europe. But the deductions from this theory were so inconsistent with the observed motions of military projectiles, when the velocities are prodigious, that no application could be made which could be of any service for determining the path and motion of cannon shot and bombs; and although John Bernouilli gave, in 1718, a most elegant determination of the trajectory and motion of a body projected in a fluid which resists in the duplicate ratio of the velocities (a problem which even Newton did not attempt), it has remained a dead letter. Mr. Benjamin Robins was the first who suspected the true cause of the imperfection of the usually received theories; and in 1737 he published a small tract, in which he showed clearly that even the Newtonian theory of resistance must cause a cannon ball, discharged with a full allotment of powder, to deviate farther from the parabola, in which it would move in vacuo, than the parabola deviates from a straight line. But he farther asserted, from good reasoning, that in such great velocities the resistance must be much greater than this theory assigns; because, besides the resistance arising from the inertia of the air which is put in motion by the ball, there must be a resistance arising from a condensation of the air on the anterior surface of the ball, and a rarefaction behind it: and there must be a third resistance, arising from the statical pressure of the air on its anterior part, when the motion is so swift that there is a vacuum behind. Even these causes of disagreement with the theory had been foreseen and mentioned by Newton (see the Scholium to prop. 37, Book II. Princip.); but the subject seems to have been little attended to. Some authors, however, such as St. Remy, Antonini, and Le Blond, have given most valuable collections of experiments, ready for the use of the profound mathematician.
Sect. IV.-OBSERVATIONs by MR. Robins, on VELocITY AND RESISTANce.
Two or three years after the appearance of his first publication, Mr. Robins discovered that ingenious method of measuring the velocities of military projectiles which has handed down his name to posterity with great honor: and, having ascertained these velocities, he discovered the prodigious resistance of the air, by observing the diminution of velocity which it occasioned. This made him anxious to examine what was the real resistance to any velocity whatever, in order to ascertain what was the law of its variation; and he was equally fortunate in this attempt likewise. From his Mathematical Works, vol. i. p. 205, it appears that a sphere of four inches and a half in diameter, moving at the rate of twenty-five feet one-fifth in a second, sustained a resistance of 0.04914 lb. or 1% of a pound. This is a greater resistance than that of the Newtonian theory, which gave los in the proportion of 1000 to 1211, or very nearly in the proportion of five to six in small numbers. And we may adopt as a rule, in all moderate velocities, that the resistance to a sphere is equal to # of the weight of a column of air having the great circle of the sphere for its base, and for its altitude the height through which a heavy body must fall in vacuo to acquire the velocity of projection. The importance of this experiment is great, because the ball is precisely the size of a twelve pound shot of cast iron; and its accuracy may be deH.". on. There is but one source of error.
e whirling motion must have occasioned some whirl in the air, which would continue till the ball again passed through the same point of its revolution. The resistance observed is therefore probably somewhat less than the true resistance to the velocity of twenty-five feet one-fifth, because it was exerted in a relative velocity which was less than this, and is, in fact, the resistance competent to this relative and smaller velocity. Accordingly, Mr. Smeaton places great confidence in the observations of Mr. Rouse of Leicestershire, who measured the resistance by the effect of the wind on a plane properly exposed to it. He does not tell us how the velocity of the wind was ascertained ; but our opinion of his penetration and experience leads us to believe that this point was well determined. The resistance observed by Mr. Rouse exceeds that
resulting from Mr. Robins's experiments nearly in the proportion of seven to ten. Chev. de Borda made experiments similar to those of Mr. Robins, and his results exceeded those of Robins in the proportion of five to six.
We must content ourselves, however, at present with the experimental measure mentioned above. To apply to our formulae, therefore, we reduce this experiment, which was made on a ball of four inches and a half diameter, moving with the velocity of twenty-five feet and one-fifth per second, to what would be the resistance to a ball of one inch, having the velocity a foot.
0.04919 being dimi
15:35.5, being diminished in the duplicate ratio of the diameter and velocity. This gives R= 0.00000381973 pound,
1000000 4.58204. The resistance here determined is the same whatever substance the ball be of; but the retardation occasioned by it will depend on the proportion of the resistance to the vis insita of the ball; that is, to its quantity of motion. This in similar velocities and diameters is as the density of the ball. The balls used in military service are of cast iron, or of lead, whose specific gravities are 7.207 and 11:37 nearly, water being 1. There is considerable variety in cast iron, and this density is about the medium. These data will give us,
This will give R =
of a pound. The ogarithm is,
For Iron. For Lead. W, or weight of a ball one
These numbers are of frequent use in all questions on this subject. Mr. Robins gives an expeditious rule for readily, finding a, which he calls F, by which it is made 900 feet for a castiron ball of an inch diameter. But no theory of resistance which he professes to use will make this height necessary for producing the terminal velocity. His F, therefore, is an empirical quantity, analogous indeed to the producing height, but accommodated to his theory of the trajectory of cannon-shot, which he promised to publish, but did not live to execute. We need not be very anxious about this; for all our quantities change in the same proportion with R, and need only a correction by a multiplier or divisor, when R shall be accurately established.
The use of these formulae may be illustrated by an example or two.
Ex. 1. To find the resistance to a twenty-four pound ball moving with the velocity of 1670 feet in a second, which is nearly the velocity communicated by sixteen pounds of powder. The diameter is 5603 inches.
tions geometrically, in the manner of Sir Isaac Newton. As we advance, we shall quit this track, and prosecute it algebraically, having by this time acquired distinct ideas of the algebraic quantitles. We must remember the fundamental theorems of varied motions. 1. The momentary variation of the velocity is roportional to the force and the moment of time jointly, and may therefore be represented by + v = ft, where v is the momentary increment or decrement of the velocity v, f the accelerating or retarding force, and t the moment or increment of the time t. 2. The momentary variation of the square of the velocity is as the force, and as the increment or decrement of the space jointly; and may be represented by + v v = f's. The first proposition is familiarly known. The second is the 39th of Newton's Principia, B.I. It is demonstrated in the article Optics, and is the most extensively useful proposition in mechanics. Having premised these things, let the straight line AC (fig. 2) represent the initial velocity V, and let CO, perpendicular to AC, be the time
in which this velocity would be extinguished by the uniform action of the resistance. Draw through the point A an equilateral hyperbola A e B having O F, OCD, for its assymptotes; then let the time of the resisted motion be re
resented by the fine C B, C being the first instant of the motion. If there be drawn perpendicular ordinates re, fg, DB, &c., to the hyperbola, they will be proportional to the velocities of the body at the instant; r, g, D, &c., and the hyperbolic areas AC re, AC, fg, A CD B, &c., will be proportional to the spaces described during the times Cr, C g, CB, &c. For suppose the time divided into an indefinite number of small and equal moments, C c, D d, &c., draw the ordinates ac, b d, and the perpendiculars b 6, a a. Then, by the nature of the hyperbola, A C : a c E O c : O C. and A C – a c : ac- Oc — OC : O C, that is, A a a c-C c : O C, and A a . C c = a c : O C, - A C- a c : A C-O C ; in like manner, B 3 : D d = BD’b D : B D.O.D. Now D d = C c, because the moments of time were
taken equal, and the rectangles AC-CO, BD-DO, are equal by the nature of the hyperbola; therefore Áa : B 6 = A.C. a c : BD-bd: but as the points c, d, continually approach, and ultimately coincide with C, D, the ultimate ratio of A.C. a c to B. D. b d is that of A C* to B D*; therefore the momentary decrements of A C and BD are as AC’ and BD”. Now, because the resistance is measured by the momentary diminution of velocity, these diminutions are as the squares of the velocities; therefore the ordinates of the hyperbola and the velocities diminish by the same law; and the initial velocity was represented by AC; therefore the velocities at all the other instants x, g, D, are properly represented by the corresponding ordinates. Hence, 1. As the abscissa of the hyperbola are as the times, and the ordinates are as the velocities, the areas will be as the spaces described, and AC re is to A c g f as the space described in the time Cr to the space described in the time C& (first theorem on varied motions).
2. The rectangle ACOF is to the area ACDB as the space formerly expressed by 2 a, or E to the space described in the resisting medium during the time CD; for AC being the velocity V, and O C the extinguishing time e, this rectangle is - e V, or E, or 2 a, of our former disquisitions; and because all the rectangles such as AC OF, BDO G, &c., are equal, this corresponds with our former observation, that the space uniformly described with any velocity during the time in which it would be uniformly extinguished by the corresponding resistance is a constant quantity, viz. that in which we always had v= E, or 2 a. 3. Draw the tangent Ar; then, by the hyperbola Cr- CO: now C r is the time in which the resistance to the velocity A C would extinguish it; for the tangent coinciding with the elemental arc Aa of the curve, the first impulse of the uniform action of the resistance is the same with its first impulse of its varied action. By this the velocity. A C is reduced to a c. If this operated uniformly, like gravity, the velocities would diminish uniformly, and the space described would be represented by the triangle A C r. This triangle, therefore, represents the height through which a heavy body must fall in vacuo, in order to acquire the terminal velocity. 4. The motion of a body resisted in the duplicate ratio of the velocity will continue without end, and a space will be described which is greater than any assignable space, and the velocity will grow less than any that can be assigned; for the hyperbola approaches continually to the assymptote, but never coincides with it. There is no velocity BD so small, but a smaller ZP will be found beyond it; and the hyperbolic space may be continued till it exceeds any surface that can be assigned. 5. The initial velocity AC is to the final velocity BD as the sum of the extinguishing time and the time of the retarded motion is to the extinguishing time alone; for A C : B D = OD (or O C x C D): O C : or V : v = e : ex t. 6. The extinguishing time is to the time of the retarded motion as the final velocity is to the velocity lost during the retarded motion: for the rectangles A F O C, B D OG, are equal; and therefore AVG F and B V CD are equal and
times and velocities, and the areas exhibiting the relations of both to the spaces described., But we may render the conception of these circumstances much more easy and simple, by expressing them all by lines, instead of this combination of lines and surfaces. We shall accomplish this purpose by constructing another curve LKP, having the line M L 3, parallel to O D for its abscissa, and of such a nature that if the ordinates to the hyperbola A C ex, fg, BD, &c. be produced till they cut this curve in L, p, n, K, &c., and the abscissa in L, s, h, 6, &c., the ordinates s, p, h, n, Ö, K, &c., may be proportional to the hyperbolic areas e A c <, f A cig, 3 A c K. Let us examine what kind of curve this will be. Make O C : Or = Or : Og; then (Hamilton's Comics, IV. 14. Cor.) the areas A Cre, e k g fare equal: therefore drawing ps, n t, perpendicular to OM, we shall have (by the assumed nature of the curve Lp K), M s = st; and if the abscissa O D be divided into any number of small parts in geometrical progression, (reckoning the commencement of them all from O), the axis Vi of this curve will be divided by its ordinates into the same number of equal arts; and this curve will have its ordinates M, ps, n t, &c., in geometrical progression, and its abscisso in geometrical progression. Also, let KN, MV, touch the curve in K and L, and let O C be supposed to be to Oc, as O D to Od, and therefore C c to D d as O C to OD; and let these lines Cc, D d, be indefinitely small; then (by the nature of the curve) Lo is equal to Kr; for the areas a A C c, b B D d are in this case equal. Also lo is to k r, as LM to KI, because c C : d D = CO : D O : Therefore I N : I K = r K : rok IK : M L = r k : o l M L : M V = ol : o L and IN : M N = r K : o L. That is the subtangent IN, or M. V, is of the same magnitude, or is a constant quantity in every part of the curve. Lastly, the subtangent IN, corresponding to the point K of the curve, is to the ordinate K 3 as the rectangle B D O G or AC OF to the parabolic area B DC A. For let fghn be an ordinate very near to B D & K ; and let h n cut the curve in n, and the ordinate KI in g; then. we have - Kq : q n = KI: IN, or Dg: q n = DO : IN; but B D : AC = CO : DO; therefore B D . D g : A.C. q n = CO : I.N.: Therefore the sum of all the rectangles B D.D.g is to the sum of all the rectangles A. C. q n, as CO to IN; but the sum of the rectangles B D : D g is the space AC DB; and, because A C is given, the sum of the rectangles A.C. q n. is the rectangle of AC, and the sum of all the lines q n : that is, the rectangle of AC and R. L.; therefore the space A C D B : A C. R. Lit CO : IN, and Åo. IN = AC . CO - R. L.; and therefore IN : R L = AC . CO : A CD B. Hence it follows that QL expresses the area BVA, and, in general, that the part of the line arallel to O'M, which lies between the tangent & N and the curve Lp K, expresses the corresponding area of the hyperbola which lies with