ral. a a hlap: a 8; If there were no resistance, the smallest velocity would process we have only learned how to compute the moIts applica- be at the vertex of the curve, and it would immediately tion from the vertex in the descending branch till the tion made increase by the action of gravity conspiring (in however ball bas acquired a particular direction, and the motion more gene small degree) with the motion of the body. But in a to the vertex from a point of the ascending branch The mode of application is this: Suppose a trajecto- by the fall a, and for a particular velocity at the vertex, The velocity in the descending branch augments con- which will be characterized by n, and that the velocity tinually: but it cannot exceed a certain limit, if the ve- at that point of the ascending branch where the inclilocity at the vertex has been lees than the terminal velo- nation of the tangent is 30° is 900 feet per second. city; for when the curve is infinite, p is also infinite, and Then, we are certain, that if a ball, whose terminal ve locity is that produced by the fall a, be projected with P city and time corresponding to every point will be such as is bere determined. an infinity of cases : for its characteristic is the propor- jectories for a sufficient variety of these proportions, we In either case we see that the curve on this side will shall find a trajectory that will nearly correspond to any have a perpendicular assymptote. It would require a case that can be proposed : and an approximation suffilong and pretty intricate analysis to determine the place ciently exact will be had by taking a proportional meof this assymptote, and it is not material for our present dium between the two trajectories which come nearest purpose. The place and position of the other assymp- to the case proposed. 87 tote LO is of the greatest moment. It evidently di- Accordingly, a set of tables or trajectories have been computed stinguishes the kind of trajectory from any other. Its computed by the English translator of Euler's Com- tables or position depends on this circumstance, that if p marks mentary on Robins's Gunnery. They are in number 18, trajectothe position of the tangent, n—P, which is the deno. distinguished by the position of the assymptote of the minator of the fraction expressing the square of the ve- ascending branch. This is given for 5°, 10, 15°, &c. °& locity, must be equal to nothing, because the velocity to 85°, and the whole trajectory is computed as far as it is infinite: therefore, in this place, P = n, or n= can ever be supposed to extend in practice. The followipVi+p++ { log.p + vitp. In order, therefore, ing table gives the value of the number n corresponding to each position of the assymptote. Through to find the point L, where the assymptote LO cuts the lhe whole horizontal line AL, put P=n, then will AL= y.r pp OLB OLB P -P 0,00000 45 1,14779 sions are the natural or hyperbolic. But the operations 5 0,08760 50 1,43236 0,17724 55 may be performed by the common tables, by making 1,82207 15 0,27712 60 2,39033 the value of the arch M m of the curve = 0,37185 65 3,29040 4,88425 30 0,60799 75 8,22357 35 0,75382 80 17,54793 mon logarithms, or 0,43429; also the time of descri 40 0,92914 85 67,12291 fe and Since the path of a projectile is much less incurvated, n+P ntQ' and more rapid in the ascending than in the descending Vā branch, and the difference is so much the more remarkand making the time = x log. able in great velocities; it must follow, that the range 86 on a horizontal or inclined plane depends most on the Such then is the process by which the form and mag- ascending branch : therefore the greatest range will not applying nitude of the trajectory, and the motion in it, may be be made with that elevation which bisects the angle of this process determined. But it does not yet appear how this is to position, but with a lower elevation; and the deviation be applied to any question in practical artillery. In this from the bisecting elevation will be greater as the initial 3 G 2 velocities ries. of this ar. I n n ticlef = (film) . n I 10 a 20 M X log. T ) Са с cos. e. I COS. to be very velocities are greater. It is very difficult to frame an But let us see what advantage we are likely to derive dence, and almost every case requires an appropriated 1. Balls of equal density, projected with the same Let e be the angle of elevation, a the height elevation, and with velocities wbich are as the square producing the terminal velocity, h the height produ- roots of their diameters, will describe similar curves.cing the initial velocity, and c the number whose hy. This is evident, because, in this case, the resistance will perbolic logarithm is i (i, e. the number 2,718). be in the ratio of their quantities of motion. Therefore Then, all the homologous lines of the motion will be in the aa proportion of the diameters. tan, et 2. If the initial velocities of balls projected with the &c. Make y = v, and take the maximum by vary same elevation are in the inverse subduplicate ratio of the whole resistances, the ranges, and all the homoinge, we obtain Sine+a sin. e = hyperbol. log. logous lines of their track, will be inversely as those re 2h sistances. 2 h These theorems are of considerable use: for by means which gives us the angle e. a sine e of a proper series of experiments on one ball projected The numbers in the first column, multiplied by the with different elevations and velocities, tables may be terminal velocity of the projectile, give us the initial constructed which will ascertain the motions of an infivelocity; and the numbers in the last column, being nity of others. 89 multiplied by the height producing the terminal veloci But when we take a retrospective view of what we shown from ty, and by 2,30 26, give us the greatest ranges. The have done, and consider the conditions which were as. various conmiddle column contains the elevation. The table is not sumed in the solution of the problem, we shall find that siderations computed with scrupulous exactness, the question not re much yet remains before it can be rendered of great lide quiring it. It may, however, be depended on within practical use, or even satisfy the curiosity of the man of science The resistance is all along supposed to be in one part of 2000. To make use of this table, divide the initial velocity the duplicate ratio of the velocity; but even theory by the terminal velocity w, and look for the quotient in points out many causes of deviation from this law, such the first column. Opposite to this will be found the ele as the pressure and condensation of the air, in the case vation giving the greatest range ; and the number in the of very swift motions ; and Mr Robins's experiments are last column being multiplied by 2,30 26 xa (the height ceedingly great in such cases. Mr Euler and all sub sufficient to show us that the deviations must be ex. producing the terminal velocity) will give the range. sequent writers have allowed that it Table of Elevations giving the greatest Range. may be three times ler gives a rule for ascertaining with tolerable accuracy what this increase and the whole resistance may amount to. Let H be the height of a column of air whose weight is equivalent to the resistance taken in the du0,6909 439.40 plicate ratio of the velocity. The wbole resistance will 0,1751 HP 0,7820 0,2169 be expressed by H+ This number 28845 is the 28845 beight in feet of a column of air wbose weight balances 1,5641 0,5789 its elasticity. We shall not at present call in question , 1,7291 0,6551 his reasons for assigning this precise addition. They 2,0726 39.50 0,7877 are rather reasons of arithmetical conveniency than of physical import. It is enough to observe, that if this measure of the resistance is introduced into the process of investigation, it is totally changed; and it is not too much to say, that with this complication it requires the 1,2298 knowledge and address of a Euler to make even a par- tial and very limited approximation to a solution.- Any law of the resistance, therefore, which is more complicated than what Bernoulli has assumed, namely, that of a simple power of the velocity, is abandoned by all the mathematicians, as exceeding their abilities; and they have attempted to avoid the error arising from the Advantage assumption of the duplicate ratio of the velocity, either to be de rived from Such is the solution which the present state of our by supposing the resistance throughout the whole trathe solution mathematical knowledge enables us to give of this cele- jectory to be greater than what it is in general, or of the prob- brated problem. It is exact in its principle, and the ap- they have divided the trajectory into different porlem. plication of it is by no means difficult, or even operose. tions, and assigned different resistances to each, which 43 .20 a 40.20 34 .20 a 3,4581 ous. vary, through the whole of that portion, in the dupli- Academy of Paris for 1769) corresponds better with After all, the practical artillerist must rely chiefly on Necessity ing to precision, by any theory of the action of gun- the records of experiments contained in the books of of attend. powder ihat we are yet possessed of. In the next place, practice at the academies, or those made in a more pub-ing to exour theories of the resisting power of the air are en lic manner. Even a perfect theory of the air's resis-periments. } } to a motion exceeding 1100 feet in a second, is almost Cause of its What are we to conclude from all this? Simply this, three times greater than in the duplicate ratio to the reinutility. that we have bardly any knowledge of the air's resist- sistance to moderate velocities. Euler's translator, in ance, and that even the solution given of this problem his comparison of the author's trajectories with experi- There is another essential defect in the conditions as- corresponds to our a, almost double of what these au- in many cases which frequently occur, than it is physical theory is of no use to the practical artillerist. is still of at the beginning and end of the flight. This is an- It plainly shows him the impropriety of giving the pro- some use in other latitude given to authors in their assumptions of jectile an enormous velocity. This velocity is of no ef practice. the air's resistance. The Chevalier de Borda has, with fect after 200 or 300 yards at farthest, because it is so considerable ingenuity, accommodated his investigation rapidly reduced by the prodigious resistance of the air. to this circumstance, by dividing the trajectory into Mr Robins has deduced several practical maxims of the portions, and, without much trouble, has made one greatest importance from what we already know of this equation answer them all. We are disposed to think subject, and which could hardly have been even conjecthat bis solution of the problem (in the Memoirs of the tured without this knowledge. See GUNNERY. a a a а 90 a And 2 I 21 428 456 159 93 And it must still be acknowledged, that this branch ral days, not only do the experiments of one day differ and may be of physical science is highly interesting to the philoso- among themselves, but the mean of all the experiments brought to pher; nor should we despair of carrying it to a greater of one day differs from the mean of all the experiments eliter per perfection. The defects arise almost entirely from our of another no less than one-fourth of the whole. The fection. ignorance of the law of variation of the air's resistance. experiments in wbich the greatest regularity. may be I. II. III. 92 420 4 we should suppose the resistance as the square of the ve 9 locity, this comparison will give in each experiment an 18 470 exact determination of the initial and final velocities, 32 272 479 II III. IV. V. 416 106 360 400 16641121 enabled him to give it. 600 606 1859 94 Tables cal We shall conclude this article, by giving two or three 800 6649 2373 866 2435 culated on tables, computed from the principles established above, 1000 10300 2845 1138 2919 the prece- and which serve to bring into one point of view the 149613259 3259 1378 3343 ding prin- chief circumstances of the motion in a resisting medium. ciples. 1400 20364 3640 1606 3734 Although the result of much calculation, as any person 1600 26597 3950 1814 4050 who considers the subject will readily see, they must not 1800 336634235 1992 4345 be considered as offering any very accurate results ; or 41559 4494 2168 4610 that, in comparison with one or two experiments, the 2200 50286 4720 2348 4842 differences shall not be considerable. Let any person 2400598464917 2460 1 5044 peruse the published registers of experiments which have 2600 51062630 5238 been made with every attention, and he will see such 2800 5293 2760 enormous irregularities, that all expectations of perfect 3000 5455 28625596 In the experiments 5732 The 200 200 338 1150 3740 1812 I 200 2000 5430 |