19 Bodies pro ! 20 of gravity as the unit of comparison. This renders the VG of the parabola are as the portions BC, BH of expressions much more simple. In this way, v expresses the tangent, or as the portions AD, Ad of the direcnot the velocity, but the lieight necessary for acquiring trix, intercepted by the same vertical lines AB, CT, it, and the velocity itself is expressed by Vē. To re HIG; for the times of describing BV, BVG are the duce such an expression of a velocity to numbers, we' same with those of describing the corresponding parts must multiply it by Vzg, or by 2 g, according as we BC, Bll of the tangent, and are proportional to these make g to be the generated velocity, or the space fallen parts, because the motion along BH is uniform; and BC, Bll are proportional to AD, Ad. Therefore the motion estimated horizontally is uni form. 5. The velocity in any point G of the curve is the which renders this an interesting subject, is, that the same with that which a heavy body would acquire by Night of cannon shot and shells are instances of such falling from the directrix along dg. Draw the tangen: motion, and the art of gunnery must in a great measure GT, cutting the vertical AB in T; take the points a, depend on tliis doctrine. f, equidistant from A and d, and extremely near them, Fig. 2. Let a body B (lig. 2.), be projected in any direc. and draw the verticals ab, jg; let the points e, f, cortion BC, not perpendicular to the horizon, and with tinually approach A and, and ultimately coincide with them. It is evident that B b will ultimately be to any velocity. Let AB be the height producing this velocity ; ihat is, let the velocity be that which a gG, in the ratio of the velocity at B to the velocity at heavy body would acquire by falling freely through AB. G; for the portion of the tangent ultimately coincide It is required to determine the path of the body, and with the portions of the curve, and are described in all the circumstances of its motion in this path? equal times ; but B b is to gG as BH to TG: there 1. It is evident, that by the continual action of fore the velocity at B is to that at G as BII to TG. gravity, the body will be continually deflected from the But, by the properties of the parabola, Bil' is to line BC, and will describe a curve live BVG, concave TG: as AB to dG; and AB is to dG as the square towards the earth. of the velocity acquired by falling through AB to the Describes 2. This curve line is a parabola, of which the verti square of the velocity acquired by falling through d G; a parabo- cal line ABE is a diameter, B the vertex of this dia and the velocity in BH, or in the point B of the parala. meter, and BC a tangent in B. bola, is the velocity acquired by falling along AB; Through any two points V, G of the curve draw therefore the velocity in TG, or in the point G of VC, GH parallel to AB, meeting BC in C and H, the parabola, is the velocity acquired by falling along d G. These few simple propo-itions contain all the theory to the body would arrive at the points V, G of the curve of the motion of projectiles in vacuo, or independent in the same time that it would have uniformly described on the resistance of the air ; and being a very easy and BC, BH, with the velocity of projection ; or that it neat piece of mathematical philosophy, and connected; e's: would have fallen through BE, BK, with a motion uni with very interesting practice, and a very respectable patet formly accelerated by gravity; therefore the times of profession, they have been much commented on, and describing BC, BH, uniformly, are the same with the have furnished matter for many splendid volumes. But times of falling through BE, BK. But, because the the air's resistance occasions such a prodigious diminumotion along BH is uniform, BC is to Bll as the time tion of motion in the great velocities of military proof describing BC to the time of describing BH, which jectiles, that this parabolic theory, as it is called, is we may express thus, BC : BH=T, BC: 'T, BH, = bardly of any use. A musket ball, discharged with the T, BE : T, BK. But, because the motion along BK ordinary allotment of powder, issues from the piece is uniformly accelerated, we have BE : BK=T? with the velocity of 1670 feet per second : this veloPE : To, BK, =BC: BHI', = EV* : KG"; there. city would be acquired by falling from the height of fore the curve BVG is such, that the abscissæ BE, eight miles. If the piece be elevated to an angle of BK are as the squares of the corresponding ordinates 45°, the parabola shonld be of such extent that it would EV, KG; that is, the curve BVG is a parabola, and reach 16 miles on the horizontal plain ; whereas it does BC, parallel to the ordinates, is a tangent in the not reach above half a mile. Similar deficiencies are observed in the ranges of cannon shot. 3. If through the point A there be drawn the ho We do not propose, therefore, to dwell much on this sebes rizontal line AD d, it is the directrix of the theory, and shall only give such a synoptical view of it viena! parabola. as shall make our readers understand the more general Lit BE be taken equal to AB. The time of falling circumstances of the theory, and be masters of the lanthrough BE is equal to the time of falling through guage of the art. AB; but BC is described with the velocity acquired Let OB (fig. 3.) be a vertical line. About the Figs by falling through AB: and therefore by N° 4. of per Centres A and B, with the distancc AB, describe the pendicular descents, BC is double of AB, and EV is semicircles ODB, AHK, and with the axis AB, and double of BE; therefore EV2=4 BE”, =4 BE XAB, semiaxis GE, equal to AB, describe the semi-ellipse =BEX 4AB, and 4 AB is the parameter or latus rec AEB : with the focus B, vertex A, diameter AB, and tum of the parabola BVG, and AB being one-fourth of tangent AD, parallel to the horizon, describe the para bola APS. the parameter, AD is the directrix. 4. The times of describing the different arcbes BV, Let a boily be projected from B, in any direction BC point B. BC, with the velocity acquired by falling through AB. ranges on any planes BP, BS, &c. and no point lying 8. The greatest range on any plane BP is produced 1. ADL parallel to the horizon is the directrix of when the line of direction BC bisects the angle OBP every parabola which can be described by a body pro formed by that plane with the vertical : for the parajected from B with this velocity. This is evident. bola described by the body in this case touches APS in 2. The semicircle AHK is the locus of all the foci P, and its focus is in the line BP, and therefore the of these parabolas: For the distance BH of a point B tangent BC bisects the angle OBP. of any parabola from the directrix AD is equal to its Cor. The greatest range on a horizontal plane is made distance BF from the focus F of that parabola ; there with an elevation of 45°. fore the foci of all the parabolas which pass through 9. A point M in any plane BS, lying between B and B, and have AD for their directrix, must be in the cir S, may be struck with two directions, BC and Bc; cumference of the circle which has AB for its radius, and these directions are equidistant from the direction and B for its centre. B t, which gives the greatest range on that plane: for 3. If the line of direction BC cut the upper semi- if about the centre M, with the distance ML from the circle in C, and the vertical line CF be drawn, cutting directrix AL, we describe a circle LF/, it will cut the the lower semicircle in F, F is the focus of the para circle All in two points F and f, which are evident-' bola BVPM, described by the body which is projected ly the foci of two parabolas BVM, B v M, having the in the direction BC, with the velocity acqired by fall. directrix AL and diameter ABK. The intersection ing through BA: for drawing AC, BT, it is evident of the circle ODB, with the verticals FC, fc, deterthat ACFB is a rhombus, and that the angle ABF is mine the directions BC, Bc of the tangents. Draw bisected by BC, and therefore the focus lies in the line At parallel to BS, and join t B, Cc Ff: then OB t BF; but it also lies in the circumference AFK, and = GBS, and Bt is ihe direction which gives the therefore in F. greatest range on the plane BS: but because Ff is the I C is in the upper quadrant of ODB, F is in the chord of the circles described round the centres B and upper quadrant of AFK; and if C be in the lower M, Ff is perpendicular to BM, and Co to A t, and quadrant of ODB (as when BC is the line of direction) the arches Ct, ct are equal; and therefore the angles then the focus of the corresponding parabola B v M is CB t, c B i are equal. in the lower quadrant of ATIK, as at f. Thus we have given a general view of the subject, 4. The ellipsis AEB is the focus of the vertex of which shows the connection and dependence of every all the parabolas, and the vertex V of any one of them circumstance which can influence the result; for it is eviBVPM is in the intersection of this ellipsis with the dent that to every vclocity of projection there belongs vertical CF: for let this vertical cut the horizontal a set of parabolas, with their directions and ranges ; lines AD, GE, BN, in o, a, N. Then it is plain that and every change of' velocity has a line AB correspondNa is half of No, and a V is balf of Co; therefore ing to it, to which all the others are proportional. As NV is balf of NC, and V is the vertex of the axis. the height necessary for acquiring any velocity increa If the focus is in the upper or lower quadrant of the ses or diminishes in the duplicate proportion of that circle AHK, the vertex is in the upper or the lower velocity, it is evident that all the ranges with given elequadrant of the cilipse AEG. vations will vary in the same proportion, a double ve5. If BFP be drawn through the focus of any one locity giving a quadruple range, a triple velocity giving of the parabolas, such as BVM, cutting the parabola a noncuple range, &c. And, on the other hand, when APS in P, the parabola BV\I touches the parabola the ranges are determined beforchand (which is the APS in P: for drawing P dx parallel to AB, cutting usual case), the velocities are in the subduplicate prothe directrix 0% of the parabola APS in x, and the di- portion of the ranges. A quadruple range will require rectrix AL of the parabola BVM in 8, then PB=Px; a double velocity, &c. but BF=BA, =AO, =xd: therefore Id=PF, and the point P is in the parabola BVM. Also the tan On the principles now established is founded the or- Experience gents to both parabolas in P coincide, for they bisect dinary theory of gunnery, furnishing rules which are to principally the angle x PB; therefore the two parabolas having a direct the art of throwing shot and shells, so as to bit directs the common tangent, touching each other in P. the mark with a determined velocity. practical Cor. All the parabolas which can be described by a But we must observe, that this theory is of little ser gunner. body projected from B, with the velocity acquired by vice for directing us in the practice of cannonading. falling through AB, will touch the concavity of the pa Here it is necessary to come as near as we can to the rabola APS, and lie wholly within it. olject aimed at, and the hurry of service allows no time 6. P is the most distant point of the line BP which for geometrical methods of pointing the piece after each can be hit by a body projected from B with the veloci- discharge. The gunner either points the cannon dity acquired by falling through AB. For if the direction rectly to the object, when within 200 or 300 yards of is more elevated than BC, the focus of the parabola de- it, in which case be is said to shoot point blank (pointer scribed by the body will lie between F and A, and the au blanc, i. e. at the white mark in the middle of the parabola will touch APS in some point between P and gunners target); or, if at a greater distance, he estiA; and being wholly within the parabola APS, it mates to the best of his judgment the deflection corresmust cut the line BP in some point within P. The ponding to his distance, and points the cannon accord. same thing may be shown when the direction is less ele- ingly. In this he is aided by the greater thickness at vated than BC. the breech of a piece of ordnance. Or lastly, when 7. The parabola APS is the focus of the greatest the intention is not to batter, but to rake along a line occupied 23 occupied by the enemy, the cannon is elevated at a con The angle DBA, made by the vertical line and the the angle of ELEVATION of the piece above the plane The principal use of it is to direct the bombardier in AB, e. throwing shells. With these it is proposed to break The angle ZAD, made by the vertical line, and the down or set fire to buildings, to break through the vault direction of the piece, may be called the ZENITH dised roofs of magazines, or to intimidate and kill troops tance, z. by bursting among them. These objects are always un The relations between all the circumstances of relo. der cover of the enemy's works, and cannot be touched city, distance, position, elevation, and time, may be inby a direct shot. The bombs and carcasses are there cluded in the following propositions. fore thrown upwards, so as to get over the defences and I. Let a shell be projected from A, with the velocity Relais produce their effect. acquired by falling through CA, with the intention of beters These shells are of very great weight, frequently ex hitting the mark B situated in the given line AB. ceeding 200 lbs. The mortars from which they are Make ZA=4AC, and draw BD perpendicular to the discharged must therefore be very strong, that they may borizon. Describe on ZA an arch of a circle ZDA, resist the explosion of gunpowder which is necessary for containing an angle equal to DBA, and draw AD to throwing such a mass of matter to a distance ; they are the intersection of this circle with DB; then will a consequently unwieldy, and it is found most convenient body projected from A, in the direction AD, with the to make them almost a solid and immoveable lump. velocity acquired by falling through CA, hit the mark Very little change can be made in their elevation, and B. therefore their ranges are regulated by the velocities For, produce CA downwards, and draw BF parallel given to the shell. These again are produced by the to AD, and draw ZD. It is evident from the conquantities of powder in the charge; and experience (con struction that AB touches the circle in B, and that the firming the best theoretical notions that we can form of angles ADZ, DBA, are equal, as also the angles AZD, the subject) has taught us, that the ranges are nearly DAB; therefore the triangles ZAD, ADB are simi. proportional to the quantities of powder employed, only lar. not increasing quite so fast. This method is much ea Therefore BD: DA-DA: AZ, sier than by differences of elevation ; for we can select DA’=BDXAZ; the elevation which gives the greatest range on the given Therefore BF'=AFXAZ,=AFX 4AC. . plane, and then we are certain tliat we are employing the smallest quantity of powder with which the service Therefore a parabola, of which AF is a diameter, and can be performed and we have another advantage, that AZ its parameter, will pass through B, and this parathe deviations which unavoidable causes produce in the bola will be the path of the shell projected as already "real directions of the bomb will then produce the small mentioned. est possible deviation from the intended range. This is Remark. When BD cuts this circle, it cuts it in tero the case in most matbematical maxima. points D, d; and there are two directions which wil The mov In military projectiles the velocity is produced by the solve the problem. If B'D' only touches the circle in ing force explosion of a quantity of gunpowder; but in our theory D', there is but one direction, and AB' is the greatest in thcory different it is conceived as produced by a fall from a certain height, possible range with this velocity. If the vertical line from that by the proportions of which we can accurately determine through B does not meet the circle, the problem is imin practice its quantity. Thus a velocity of 1600 feet per second possible, the velocity being too small. Wben B'Dy is produced by a fall from the height of 4000 feet, or touches the circle, the two directions AD' avd Ad 1333 yards. coalesce into one direction, producing the greatest range The height CA (Gg. 4.) for producing the velocity and bisecting the angle ZAB ; and the other two diof projection is called, in the language of gunnery, the rections AD, Ad, producing the same range AB, are IMPETUS. We shall express it by the symbol h. equidistant from AD', agreeably to the general propoThe distance AB to which the shell goes on any sition. It is evident that AZ: AD=S,ADZ : S,AZD, S,DBA : S,DAB,=S,P: Se Sp: Se Se : S,% Or 4h:1=S,p: S, ex $, %, and 4h XS, XS,=rXS,P 24 Fig. 4. H Hence we obtain the relations wanted. rox S ,p 4h x 4S,e X Sz' rX S, P, 4h XS,e' 46 x 8,2 and S, ed is the time of the flight. A knowledge of this is to each rx Se 27 with expe the same with the time of falling through DB, and 16 more from a parabola than the parabola itself deviatos rxS,e from the straight line. feet : DB=1" : t''2. Hence t'" and we have It is for such cogent reasons that we presume to say, the following easy rule. that the voluminous treatises which have been published From the sum of the logarithms of the range, and of on this subject are othing but ingenious amusements the sine of elevation, subtract the sum of the logarithms for young mathematicians. Few persons who have been of 16, and of the sine of the zenith distance, half the much engaged in the study of mechanical philosophy remainder is the logarithm of the time in seconds. have missed this opportunity in the beginning of their This becomes still easier in practice; for the mortar studies. The subject is easy. Some property of the should be so elevated that the range is a maximum : in parabola occurs, by which they can give a neat and which case AB-DB, and then half the difference of systematic solution of all the questions; and at this time the logarithms of AB and of 16 is the logarithm of the of study it seems a considerable essay of skill. They are time in seconds. tempted to write a book on the subject; and it finds. The theory Such are the deductions from the general propositions readers among other young mechanicians, and employs cf gunnery which constitute the ordinary theory of gunnery. It all the mathematical knowledge that most of the young compared remains to compare them with experiment. gentlemen of the military profession are possessed of. riment In such experiments as can be performed with great But these performances deserve little attention from the accuracy in a chamber, the coincidence is as great as practical artillerist. All that seems possible to do for can be wished. A jet of water, or mercury, gives us his education is, to multiply judicious experiments on the finest example, because we have the whole parabola real pieces of ordnance, with the charges that are used. in actual service, and to furnish him with tables calcu- These observations will serve to justify us for having given so concise an account of this doctrine of the pa- But it is the business of a philosopher to inquire into Causes of mark is less than the distance between it and the spout, the causes of such a prodigious deviation from a well- this deficiand that the descending branch of the curve is more founded theory, and having discovered them, to ascer. ency. perpendicular than the ascending branch. And this tain precisely the deviations they occasion. Thus we difference is more remarkable as the jet is made with shall obtain another theory, either in the form of the greater velocity, and reaches to a greater distance. This parabolic theory corrected, or as a subjuct of indepenis evidently produced by the resistance of the air, which dent discussion. This we shall now attempt. diminishes the velocity, without affecting the gravity of The motion of projectiles is performed in the atmo. Effect of the projectile. It is still more sensible in the motion of sphere. The air is displaced, or put in motion. What the atmo. bombs. These can be traced through the air by the ever motion it requires must be taken from the bullet. sphere, light of their fuzes; and we see that their highest point The motion communicated to the air must be in the prois always much nearer to the mark than to the mortar portion of the quantity of air put in motion, and of the on a borizontal plane. velocity communicated to it. If, therefore, the displaced The greatest horizontal range on this plane should air be always similarly displacedl, wbatever be the velobe when the elevation is 45°. It is always found to be city of the bullet, the motion communicated to it, and much lower. lost by the bullet, must be proportional to the square of The ranges on this plane should be as the sines of the velocity of the bullet and to the density of the air twice the elevation. jointly. Therefore the diminution of its motion must be A ball discharged at the elev. 19o. 5' raaged 448 yards greater when the motion itself is greater, and in the 9. 45 330 very great velocity of shot and shells it must be prodiIt should have ranged by theory digious. It appears from Mr Robins's experiments that The range at an elevation of 45° should be twice the a globe of 4 inches in diameter, moving with the veloimpetus. Mr Robins found that a musket-ball, disclar city of 25 feet in a second, sustained a resistance of 315 ged with the usual allotment of powder, had the velo. grains, nearly £ of an ounce. Suppose this ball to move city of 1700 feet in a second. This requires a fall of 800 feet in a second, that is 32 times faster, its resist45156 feet, and the range should be 90312, or 17% ance would be 32X32 times į of an ounce, or 768 miles; whereas it does not much exceed half a mile. A ounces or 48 pounds. This is four times the weight of 24 pound ball discharged with 16 pounds of powder a ball of cast iron of this diameter; and if the initial should range about 16 miles; whereas it is generally velocity had been 1600 feet per second, the resistance sliort of 3 miles. would be at least 16 times the weight of the ball. It is Such facts show incontrovertibly bow deficient the indeed much greater than this. 31 is COND ison parabolic theory is, and how unfit for directing the prac This resistance, operating constantly and uniformly compared ws the tice of the artillerist. A very simple consideration is on the ball, must take away four times as much from with that ciency sufficient for rendering this obvious to the most unin- its velocity as its gravity would do in the same time. of gravity, structed. The resistance of the air to a very light body We know that in one second gravity would reduce the y. may greatly exceed its weight. Any one will feel this velocity 800 to 768 if the ball were projected straight but as the velocity diminishes continually by the resist- city, 30 at 241 28 he 33 2a' 201 2 city, the real diminution will be somewhat less than 128 force of resistance. It follows from this remark, that feet. We shall, however, see afterwards that in one this velocity is the greatest that a body can acquire by second its velocity will be reduced fram 800 to 687. the force of gravity only. Nay, we shall afterwards see From this simple instance, we see that the resistance of that it never can completely attain it; because as it apo the air must occasion great deviation from parabolic mo proaches to this velocity, the remaining accelerating 32 tion. force decreases faster than the velocity increases. It and consi. In order to judge accurately of its effect, we must may therefore be called the limiting or TERMINAL velodered as a consider it as a retarding force, in the same way as we city by gravity. retarding consider gravity. The weight W of a body is the ag Let a be ihe height through which a heavy body force. gregate of the action of the force of gravity g on each must fall, in vacun, to acquire its terminal velocity in particle of the body. Suppose the number of equal par- air. If projected directly upwards with this velocity, ticles, or the quantity of matter, of a body to be M, it will rise again to this height, and the height is half then W is equivalent to g M. In like manner, the re the space which it would describe uniformly, with this sistance R, which we observe in any experiment, is the velocity, in the time of its ascent. Therefore the reaggregate of the action of a retarding force R on each sistance to this velocity being equal to the weight of particle, and is equivalent to R'M : and as g is equal to the body, it would extinguish this velocity, by its uniR , after the same dis- We shall keep this distinc. forn action, in the same time, and tance, that gravity would. Now let g be the velocity which gravity generates letter R or r, which expresses the aggregate resistance. or extinguishes during an unit of time, and let u le The resist If we', in this manner, consider resistance as a retard. the terminal velocity of any particular body. The theoance of the ing force, we can compare it with any other such force rems for perpendicular ascents give us g= u and a air not uni- by means of the retardation which it produces in similar circumstances. We would compare it with gravity by being both numbers representing units of space; tberecomparing the diminution of velocity which its uniform For the whole action produces in a given time with the diminution fore, in the present case, we have r'= produced in the same time by gravity. But we have resistance r', or w'M, is supposed equal to the weight, or no opportunity of doing this directly; for when the resistance of the air diminishes the velocity of a body, it to g M; and therefore r' is equal to g, = and 2 a= diminishes it gradually, which occasions a gradual diminution of its own intensity. This is not the case with There is a consideration which ought to have place gravity, which has the same action on a body in motion or at rest. We cannot, therefore, observe the uniform here. A body descends in air, not by the whole of its action of the air’s resistance as a retarding force. We weight, but by the excess of its weight above that of must fall on some other way of making the comparison. the air which it displaces. It descends by its specific We can state them both as dead pressures. A ball may gravity only as a stone dues in water. Suppose a bobe fitted to the rod of a spring stillyard, and exposed to dy 32 times heavier than air, it will be buoyed up by impulse of the wind. This will compress the stillyard to the mark 3, for instance. Perhaps the weight of the a force equal to of its weight; and instead of ac 32 quiring the velocity of 32 feet in a second, it will only We account this equal to the pressure of the air, because they ba acquire a velocity of 31, even though it sustained no lance the same elasticity of the spring. And in this way weight of the body and a that of an equal bulk of air: resistance from the inertia of the air. Let p be the we can estimate the resistance by weights, whose pres the accelerative force of relative gravity on each particle sures are equal to its pressure, and we can thus compare it with other resistances, weights, or any other pressures. will be gXI-; and this relative accelerating force In fact, we are measuring them all by the elasticity of the spring. This elasticity in its diferent positions is might be distinguished by another symbol y. But in supposed to have the proportions of the weights which all cases in which we liave any interest, and particularkeep it in these positions. Thus we reason from the na ly in military projectiles, is so small a quantity that ture of gravity, no longer considered as a dead pressure, but as a retarding force ; and we apply our conclusions it would be pedantic affectation to attend to it. It is to resistances which exhibit the same pressures, but much more than compensated when we make g=32 which we cannot make to act aniformly. This sense of feet instead of 321's which it should be. the words must be carefully remembered whenever we Let e be the time of this ascent in opposition to graspeak of resistances in pounds and ounces. vity. The same theoremis give us eu=2 a; and since "Gravity The most direct and convenient way of stating the the resistance competent to this terminal velocity is and resist- comparison between the resistance of the air and the equal to gravity, e will also be tbe time in which it ance com- accelerating force of gravity, is to take a case in which would be extinguished by the uniform action of the repared when they are we know that they are equal. Since the resistance is sistance; for which reason we may call it the extinguishequal. here assumed as proportional to the square of the velo- ing time for this velocity. Let R and E mark the re city, it is evident that the velocity may be so increased sistance and extinguishing time for the same body mox- R will be P 34 1 20 |