Jurij Vega and Ballistics Jurij Vega in balistika Dr. Mihael Perman and Dr. Marko Razpet Institute for Mathematics, Physics and Mechanics University of Ljubljana Abstract The ballistic curve is the solution of a system of nonlinear differential equations if the resistance of the air is assumed to be proportional to the square of the velocity. In Chapter 4. vol 4. of his lectures on mathematics Jurij Vega (1754-1802) presents a careful analysis of the problem. The underlying assumptions fmm physics are examined, and then differential equations are derived. There is no closedform solution but the equations can be transformed into simpler forms that are more amenable to finding approximate solutions. Vega suggests various numerical procedures. Perhaps the most original part is the use of a result that can be traced back to Eider's work on ballistics. The arvlength covered by the cannon ball can be expressed explicitly as a function of the angle of the trajectory at a given moment. Vega develops an ingenious way to discretize the differential equations and find a numerical solution. The comparison to numerical solutions obtained by computer using Runge-Kutta methods shows that Vega's method gives superb results. 395 396 Jl'RIJ VEGA ANI) BALLISTICS Povzetek Balistična krivulja je rešitev sistemu nelinearnih diferencialnih enačb, če privzatnemo. ila je zračni upor sorazmeren kvadratu hitrosti. V Četrtem poglavju eetrte knjige svojih matematičnih predavanj Jurij Vega (1754 -1802) predstavlja podmhno analizo problema. Razdelilne so osnovne fizikalne predpostavke, nato sta izpeljani diferencialni enačbi. Rešitev v sklenjeni obliki ni. vendar pa je enačbe mogoče pretvoriti v preprostejšo obliko, ki ji je lažje najti približne rešitve. Vega predlaga različne numerične postopke. Morda najbolj izvirna je uporaba rezultata, ki ga lahko zasledimo v Hitlerjevem delu na področju balistike. Dolžino loka. ki ga opiše topovska krogla, lahko izrazimo eksplicitno kot funkcijo kota, pod katerim se giblje krogla v danem trenutku. Vega razvije inovativen način, s katerim je mogoče diskretizirati diferencialni enačbi in najti numerično rešitev. Primerjava z numeričnimi rešitvami. kijih pridobimo z računalnikom z uporabo metode Runge-Kutta, pokaže, da daje Vegova metoda izvrstne rezultate. 1. Introduction The mathematician Jurij Vega 1754-1802 was a prolific mathematical writer. He is best known for his monumental work on logarithmic tables and his military exploits. From the perspective of a mathematician, however, his lectures on mathematics that were published between 1782 and 1802 in four volumes are perhaps even more important. Shortly after starting his teaching career at the imperial Artillery School in Vienna in 1782 Vega set out to write systematic and clear textbooks. They became an instant success and were reprinted many times over the following decades. See Vega's bibliography in [3]. It is interesting to note that the last edition of volume 2 was published in 1848. The informal style and carefully chosen examples are enjoyable to read even today. In this paper we examine Vega's work on the ballistic curve, which appears as Chapter 4 in the last volume of his lectures published first in 1800 and then again in 1819. The volume is dedicated to hydrostatics, the motion of fluids and to the motion of objects in fluids. In his treatment of ballistics Vega sets out with a careful examination of the physical assumptions about the resistance of the medium to the motion of an immersed object. He acknowledges that there is some disagreement about the exact magnitude of the force opposing the motion, but finally settles for the quadratic law. He limits his investigation to spherical bodies, and as experimental evidence presents an experiment that was conducted at Newton's urging in London in 1710 and 1719. The most intriguing part is the treatment of the ballistic curve. The basic equations are readily derived from Newton's laws. They are then skillfully transformed 2. PHYSICAL ASSUMPTIONS 397 in such a way as to obtain a single differential equation. Vega correctly observes that there is no closed form solution and sets out to suggest various approximations. Perhaps the most surprising result is the derivation of an explicit formula for the arclength along the trajectory as a function of its slope. This result can be traced back to Euler's work on ballistics |2|. This analytic representation is then exploited to give a surprisingly accurate numerical procedure for finding the ballistic curve. One of the main aims of this paper is to translate the text into modern notation and to check the results from a modern viewpoint. Along the way it is necessary to unravel older notation, bul once that is done the presentation is wonderfully clear, and. we dare say. better than many calculus textbooks used today. Vega also provides fully worked-out examples illustrating the solutions from a practical viewpoint. At the end we reproduce Vega's calculations and examine the accuracy of his approximations using modern computers. 2. Physical assumptions Vega's computations are based on the assumption that the force exerted by the resistance of the medium on a moving object is proportional to the square of its velocity. If the density of the medium is denoted by pa, the velocity by v and the cross-section by S, the force R is given as R=~kpaSv2l~ where v = |t7|. The proportionality factor k depends on the geometry of the moving body and is in general difficult to compute. In §117. however. Vega derives the explicit form of the resistance law for a moving ball. If the diameter of the ball is denoted by D. the force is given by R = -^-paD2 vv (2.1) lo In order to further simplify the formulae, the mass of the cannon ball is denoted by M. Clearly where /> is the mass density of the ball, and introducing the ratio N = />//>„ one gets M = JD*Npa. It follows that R 3 — =--cc. M SDN Introducing the parameter a = ^ DN (2.2) JURIJ VEGA AND BAI.l.lSTICS which has the dimension of length, the force can finally be expressed as R 1 ^ — = -—vv. (2.3) M 2d It is interesting to note that Vega gives experimental evidence for the laws of resistance. First he solves the elementary differential equations describing the velocity of a ball moving vertically under the inlluence of gravity. In equation (4.34) in [4] the distance x covered by the falling ball by the time t is given as / l + /,-'"/« \ x = bt + 2ii log I---I (2.4) where h is the notation for Euler's e and b is an abbreviation for m l4ga(N-l) N The calculations are then compared to data recorded in two experiments conducted at Newton's urging in London in 1710 and 1719. Balls were dropped from the dome of St. Paul's cathedral and the times it took them to reach the ground were recorded. In the first case glass balls were chosen and dropped from 220 feet. Vega computes the known height from the times using (2.4). Table 1 reproduces the results. Weight Diamet. Recorded Duration Computed of Ball of Ball of Fall Height Lon. Grain Lon.Inches Second. Tertia. Feet Inches 510 5.1 8 12 226 11 642 5.2 7 42 230 9 599 5.1 7 42 227 10 515 5.0 7 57 224 5 483 5.0 8 12 225 5 641 5.2 7 42 230 7 Tabela / Table 1. Rezultati eksperimenta v Londonu iz leta 1710 / The data from the 1710 experiment in London From the table it follows that the computed heights slightly overstate the height. In the second experiment, swine bladders were blown up in a wooden mold to give them the shape of a near perfect ball. The bladders were then dropped from 272 feet. Table 2 gives the data in this case. In this second case the agreement between the computed and known height is more convincing except in the last case, which can most probably be attributed to measurement error. Vega offers no further comment on the experimental evidence 3. VEGA'S DIFFERENTIAL EQUATIONS 399 Weight Diarnct. Observed Duration Computed of Ball of Ball of Fall Height Grain Inch. Seconds Feet Inches 128 5.28 19 271 11 156 5.19 17 272 1.05 137.5 5.3 18.5 272 7 97.5 5.26 22 277 4 99.125 5 21.125 282 0 Tabela / Table 2. Rezultati eksperimenta v Londonu iz leta 1719/ The data from the 1719 experiment in London in his §164. It is quite likely that this was deliberate because such discrepancies would defeat the purpose of finding accurate numerical methods to approximate the solutions of subsequent equations. 3. Vega's differential equations Having dealt with physical assumptions. Vega turns to ballistics. The simple cases of motion in a straight line are dealt with first. In particular all the differential equations for vertical fall taking into account the resistance of the medium and the buoyancy are carefully solved. Then section 4.3 turns to the main problem: the ballistic curve. It is assumed that a cannon ball with mass density p and diameter D is fired at an angle // and initial velocity v. The density of the air is denoted by pa- The problem is to find the trajectory of the cannon ball under the quadratic law of air resistance. A few simplifying assumptions are also made. The air is assumed to be homogenous and at rest, the buoyancy is assumed to have a negligible effect, and there is no correction for the rotation of the earth. The question is translated into mathematics by placing the cannon at the origin of a coordinate system. The trajectory will lie in a plane determined by the vector of the initial velocity. The x-axis will represent the ground and the force G of gravity will point in the direction opposite to the (/-axis as G — —Mgj. where j = (0.1). The trajectory will be described by a vector function ?=if(t) = (x(t),y(t)) of time. As before. M is the mass of the cannon ball and g is the gravity constant. By Newton's second law we obtain the equation: Mf = R + G. Dividing by the mass M, the equation becomes r = -—vv-gj. (3.1) hi 6 Jl'RIJ VEGA ANI) BALLISTICS Taking into account that v = (;i\ y) and v = \/x2 + />-, and writing (3.1) componentwise, we obtain a system of differential equations 1 . x _ -—vx (3.2) 2d 1 . y = ~7rvy ~ a (3.3) 2d with the initial conditions x(0) = 0. //(()) = 0. ,r(0) = ccos/i, y(0) = csiu//. The system of differential equations (3.2) and (3.3) has a unique solution satisfying the initial conditions. The solution, however, cannot be expressed in closed form using elementary functions. Approximate solutions can be found, bul first the basic equations have to be transformed. From (3.2) and (3.3) one can easily derive the identity Let p be the slope of the trajectory at time t. We have dx x From (3.4) one obtains P-i- = ~(J- (3.5) Denote the arc length from the starting point () to the position at time I by .s = s(t) = I i>(t) (It. From = \/l + p2 and š = ~ = v ax df and taking into account (3.2) we obtain r—i—t d.s d.s (It vx 2ax yjl+p2 = — = —— = — = ——■ d.r at dx x2 .r2 Multiplying the equations /:-t 2ax Vi+P2 = rs- (3.6) xz (J P = ~ (3.7) x we obtain x/lT^/i = 2ag^ = -„,ryjL ^ . 3. VEGA'S DIFFERENTIAL EQUATIONS 401 In other words, the equation is >/1 + p~ dj) = -agd j . (3.8) Because dp _ dp d t _ P _ p& _ 9 dx d/ dx x x2 x2' the differential equation can be rewritten as: y/l+p2dp = ad . (3.9) The initial conditions imply that p(0) = tan a and ^(0) = —„ 9 , . (3.10) dx cz cos- /i One can. in principle, integrate the equation (3.9), but the resulting differential equation linking p and dp/dx is too cumbersome for further calculations. Vega suggests various approximations to the solution of the equation above. For small initial angles (Vega considers angles up to 15° to be small) one can argue that the term \r is small enough so that the term \J 1 + p2 in (3.9) can be ignored. Integrating the simplified equation dp = ad^) (3.11) one obtains a nonhomogeneous first order linear differential equation /dp, 9 \ p - tan/i = a — + -. \d.r c- cos- // J Rewriting as dyj d;r p -tan n -jr^i a and using the initial conditions in (3.10). the solution is obtained as los ) = f • V r- cos- // / ° Rewriting again we find dy ay ag * — =p = tan/i + —-^---5-5—en. dx c- cos- // c- cos- // Because y(0) = 0, yet another integration gives Vega's approximation to the ballistic trajectory as y = (tan // + , x - ./' 9 (e« - l) . (3.13) \ cr cos- /i J c cos- // V / 402 Jl'RIJ VEGA ANI) BALLISTICS An interesting question arising at this point is the quality of this approximation. Figure 1 shows the numerical solution of the system of equations (3.2) and (3.3) using the Runge-Kutta algorithm with step size 0.1 s (solid line) and the data // = 15°, c = 400 m/s, and a = 800 m. The dashed line shows Vega's approximate trajectory. As is to be expected, Vega's approximation slightly overstates the range of the cannon. Numerical calculations show that the range of the cannon ball given the physical assumptions is 2.323.5 in. The highest elevation is 24(i m attained at x — 1.437 m. Vega's range is 2,353.5 m with the highest elevation 249.3 m attained when x = 1,447 in. As an improvement to his first approximations - in particular, when the angles are not small - Vega suggests taking p = tan(///2) in equation (3.9). He argues that this quantity is approximately the average angle of the trajectory. The basic-equation is which means that one only has to change the parameter a to u cos(/t/2) in all the calculations. In § 152 in [4] Vega sets out to calculate the range of a cannon from the characteristics of the cannon ball, the angle, and the initial velocity. To this end one sets y = 0 in (3.13), which leads to the transcendental equation —t— = 1 -\--sin // cos p = 1 + -— sin2u. f "9 2ay ' For practical purposes Vega compiled a ballistic table giving the values of the function n i-*, (e" - 1 )/n for n ranging from 0.01 to 10.00 with step size 0.01. For purposes of interpolation, the table also gives the differences of subsequent entries. Taking n = x/a or n = x/(acos(p/2)), and using the table, one finds with a, p and c given the equation ew ~ 1 - , c2 en — 1 i r-2 sin 2// -= 1 + -— sin 2/1. or -= 1 H--- " 2ag n 2ag cos ^ 4. AN ALTERNATIVE NUMERICAL PROCEDURE Once ii is found, the range is computed as x = na or x = na cos///2. 403 4. An alternative numerical procedure The equations (3.2) and (3.3) cannot be solved in closed form. Surprisingly, however, in § 173 in [4| Vega finds a simple expression for the arc-length as a function of the slope of the trajectory. Let p be the angle between the a.--axis and the tangent to the trajectory at a given point. Recall that p = tan d-,; d.s d tan p dp ds 1 d^ d.r d^ d.s d.r dip ds d.r cos3 p ds Taking derivatives with respect to I on both sides and using (3.9) gives (4.1) cos * \ cos'1 p d.s J For convenience, introduce the function F as / m ir\ ir 2 \ 2 4 / 2 Obviously. F(0) = 0, dF/dp = l/cos3p and F(-p>) = -F(^). At the beginning when s = 0 one has p = // and consequently the expression dp/dx = cl<^/(cos3 tpds) takes the value —fj/(r2 cos2 //)). Integrating, one obtains dt? a d^J ... > i > ■■■ Denote the increments of the arelength between the subsequent angles by As/.: = s(