Halley's Comment-Projectiles with Linear Resistance Author(s): C. W. Groetsch and Barry Cipra Source: Mathematics Magazine, Vol. 70, No. 4 (Oct., 1997), pp. 273-280 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2690864 Accessed: 29/11/2010 10:44 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=maa. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact
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VOL. 70, NO. 4, OCTOBER 1997
Halley'sComment WithLinearResistance Projectiles C.W. GROETSCH
ofCincinnati UniversiFty Cincinnati,OH 45221-0025
BARRY CIPRA
305 OxfordSt. MN 55057 Northfield,
If only gravitywere working,the path would be symmetrical
it is the wind resistance that produces the tragic curve
Norman Mailer The Naked and the Dead
1. Introduction of the motionof projectilesin a ModerndynamicsbeganwithGalileo'sinvestigations compoundedtheacceleratedverticalmotion nonresisting medium.Galileoingeniously of a projectile(obtained fromhis law of fallingbodies) with its unaccelerated horizontalmotion(an expressionof his principleof inertia)to concludethatthe path of the projectileis parabolic.Projectilemotionis now a primeapplicationin virtually everycalculustext.Indeed, the powerof analyticgeometryand calculus,alongwith the elementsof Newton'sdynamics,has reduced Galileo's greatachievementto a mere exercise.If a projectileof unitmass is projectedfromthe originat an angle 0 gravitational withthe horizontal withinitialspeed v, underthe influenceof a uniform accelerationg, thenthe equationsof motion y(O) = 0 j(t) = -g, 9 (O) = vsin 0, x(0) =v cos 0, x(O) = 0 x(t) =0, integrated to yield: maybe immediately y(t)
=-g
t2 + (v sin 6)t;
x(t) = (vcos 6)t.
is then revealedwhen the timeparameteris The parabolicnatureof the trajectory eliminated: Y(__)
2V2cS2
+ (tan 0)x.
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MATHEMATICS MAGAZINE
The rangeof the projectile,R(6), is the positivex-intercept of the trajectory: R(0) = g-sin20, (2) g and thereforethe maximumrange is attainedwhen the angle of elevationis IT/4 radians,a factthatwas provedby Galileo [5] and observedmuchearlierby Tartaglia [3]. Air resistancehas so farbeen neglected.The precisenatureof air resistanceis a verycomplicatedmatter,but it is the commonexperienceof thosewho have ridden motorcyclesthat resistanceincreases with velocity.The linear model, in which resistanceis takento be proportional to velocity, is the acceptedfirstapproximation to resistivebehavior[9], [10] and under some circumstancesgives predictionsthat square quite well with observation[1], [4]. The equationsof motionin the linear resistancemodelare y(t) = -g - kzj(t), x(t) = -kx(t),
y(O) = v sin 0, y(0) = 0 x(O) = vcos 0, x(0) = 0()
3
where k is the resistanceconstant.These equations,while more complicatedthan thosein the nonresistive case (1), are linearand therefore solvedto maybe routinely give
x(t) = (vcos 0)(1 -e-kt)/k = inJ y(t)Ykt k2 )1-e)-k 0+kk -ekteJ (k k
g
The rangeR(6) is againthatpositivevalue of x givingy = 0. As before,thismaybe foundby eliminatingt, settingy = 0, and solvingforthe positiveroot x. A little rearranging showsthat Cos6 AORO a (1-eA(O)R(O)) (4) R( 0) =
where A(6) = a sec 0 + b tan0, a = k/v, b = k2/g [7]. Unlike in the nonresistive is definedexplicitly as a case, wherethe rangefunction by (2), now it is characterized fixedpoint of the transformation definedin (4). In [7] it is shown that for each iterationconvergesmonotonically to R( 6). Modernnumeri0 E (0, 17r72)fixed-point cal and graphicalsoftware can therefore be used to quicklyand reliablycalculateand fora projectilein a linearlyresistingmedium. displaythe rangefunction In this note we show how computationalstudies of Equation (4) and some associatedrelationships can be used to discoverinteresting featuresoftheanglegiving maximumrange in a linearlyresistingmedium.Rigorousanalyticalproofsfor the are provided.We also provea theoremon conjecturessuggestedbythe computations the rangefunction thatwas suggestedbya rathercrypticremarkof EdmondHalleyin 1686 and is supportedbythe graphicalevidencein the nextsection.Our pointis that stronggraphicalevidencecan do morethanjust suggestresults;it can also spurthe searchforanalytical proofs.Galileo said as muchin thewordsof his characterSalviati [6, p. 60]: Pythagoras,a long time before he found the demonstrationfor the Hecatomb,had been certainthatthe square of the side subtendingthe rightangle in a rectangular trianglewas equal to the square of the other of the conclusionhelped not a littlein the search two sides; the certainty fora demonstration.
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VOL. 70, NO. 4, OCTOBER 1997
2. GraphicalObservations FIGURE 1 shows the range functionfor k = 1 and for various muzzle velocities (v = 100, 200, 500, bottomto top). The plot was producedby a simple MATLAB iterationfor100 equally-spacedanglesin routinethatcomputedR(6) by fixed-point can [0, ir/2]. Similarplots,forfixedmuzzlevelocityand variousresistanceconstants, be foundin [7].
500 450 400 350 300 200 150 100 50 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
FIGURE 1
Rangeversusangleofelevation In a nonresistive mediumthe maximumrangeoccursat 0 = -I/4, by (2). The plotsin 1 (as well as those in [7] and the trajectory curvestracedin [4] and [9]) all indicatethatin a linearlyresistingmediumthe maximumrange always occursat an angle of elevation0 < Ir/4. The angle at whichthe maximumrangeoccurshas no knownexplicitrepresentation, but it can be characterized in termsoftheunique fixed To see this,we notethatthe maximumrangeoccursat an pointof a certainfunction. R'(6) = 0 (see [7] fora proofthatR is differentiable). Differentiatangle 0 satisfying ing(4) and settingR'(6) = 0 we obtain 0-sin 0 i A0R0)+COS60-(OR a ( eA(o)R(o)) + a (asec 0 tan0 + b sec26)R(O)eA(O)R(O) ?0=However,by (4) FIGURE
1 -eA(o)R(O) = a sec OR( 06) thisabove and rearranging, we findthat Substituting sin 0 = (sin 0 + c)e-A()R(o)
(5)
wllerewe have set c = b/a = vk/g.By (4) we also have e-A(0)R(0)
1-a
sec 0 R( 0)
and therefore by (5) R(6)-
(cla)cos sin 6 + c
and hence A(6) R(6) =(a sec60+ btan60)R(6) =
C + c2 sin 0
sin6+
(6)
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MATHEMATICS MAGAZINE
Settings = sin 0, and substituting (6) into(5) we findthats satisfiesthe fixedpoint equation C +C
.2S
= (s + c)e- s?c . (7) Equation (7) allowsthe possibility of computings, and hence the optimalangle of elevation,0 = sin-s, directlyby a numericalmethod,ratherthantracingtrajectories and selectingvisuallythatwhichseems to have maximumrange.More importantly, the equationsuggeststhe possibility of an analyticalstudyof the behaviorof s = sin 0 withregardto k and v, the mainphysicalparametersof the problem. A plot of the functionvalues s(c), definedby Equation (7), versusc in whichthe values s(c) were computedfor1,000equally-spacedvaluesof c in (0, 50) by a simple MATLAB routine(based on fixed-point iteration)is givenbelow. s
0.7 0.6 0.5 0.4 0.3 0.2 0
5
10
15
20 25 30 FIGURE 2
35
40
45
50
Sizeofangleofelevation versusc = vk/g What do the plots suggest?Clearly,FIGURE 1 indicatesthatthe angle of elevation givingmaximum rangeis alwaysless thanIrr/4,thatis, s(c) < 1/ 4 forall c > 0. This evidenceis bolsteredin FIGURE 2. Furthermore, theplotin FIGURE 2 suggeststhats(c), and hence the optimalangle, decreases with an increase in c. In particular,the optimalangle of elevationdecreases as the resistanceconstantincreases,forfixed muzzlevelocity;it also decreasesas the muzzlevelocityincreases,forfixedresistance constant.All ofthesepropertieshaveinterpretations as statements aboutthevariables the relationship(7). In the next sectionwe use a parametric s and c satisfying representation derivedfrom(7) to giveanalyticproofsof thesestatements. Recall thatTartagliaand Galileo pointedout thatin a nonresisting mediumthe optimal angle of elevationis Irr/4. Furthermore,Galileo [5] proved that equal deviationsabove and below the optimalangleproducethe same range,thatis, R(il/4--b)) = R(1T/4+ 4) for 0 < 4? < rr/4,
a factwhichis evidentfrom(2). In 1686 Edmond Halley [8] noted thatin certain ofsmallshotbya crossbowa consistent due evidently experimental firings asymmetiy, to air resistance,occursin thisrelationship. To quote Halley: ". . . in smallerbrassschott... constantlyand evidently,the under ranges otttventthe upper." What his statementto saythat Halleymeantis notentirelyclear,but we interpret R(T/4- 4)) > R(T/4 + 4) for 0 < 4 0 and hencethe anglegivingmaximalrangeis always less than 17-74. We also note that equations (11) and (13) give an alternative, parametric wayof producingthe graphin FIGURE 2.
4. The Halley Problem In thissectionwe proveanalytically thatthe relation(8), suggestedby FIGuRE 3, must hold in a linearlyresistingmedium.We showed in the previoussectionthat the maximumof the ranigefunctionR occursat some anglein (0, -r/4);hence (8) holds forsomneangle 4 ( (0, irr/4). To showthat(8) holdsforall 4 ( (0, 1-r/4), it is therefore sufficient to showthatthe equation R( -rT4- 4) = R(-TI4 + 41)) (4) in termsofthe has no solution4 E (0, mrr/4). We accomplishthisbyreformulating convert(4) to functionQ(6) = aR(6)/cos 0. In fact,routinemanipulations (6()
= 1 -e-B(O)Q(O)
(14)
VOL.
70, NO. 4, OCTOBER
279
1997
where B(6) =1 +csin 6 and c = b/a. Note that 0 < Q(6) < 1. Given 4) z (0, Ir/4), let 0= Ir/4 - 4) and 6' = 17r-4 + 4). Then sin 0 = cos 6', cos 0 = sin6' and we wishto showthat R(6) = R(6') is impossible.Now,if R(6) = R(6'), then c sin 0 Q(6) However,c sin0 = B(6)
c cos 6' sin6' aR( 06)
c sin 0 cos 0 aR(6) -
c sin6' Q(60)
1, and by (13) ln(
B(O)
-
Q())
(16) (6
Q(6)
Therefore, c sin 0 =
-
Q( 0) + ln(I - Q( 0)) Q(6)
and a similarformulaholdsfor6'. Substituting theseresultsinto(15), we obtain Q(6) + ln(l-Q(6)) Q( 0)2
Q(0') + ln(1-Q(6')) Q(6,)2
However,for0 < x < 1, the function = f(X)=x+ln(I-x) Ax)
x2
1
1X
2 3
IX92
4~
is clearlystrictlydecreasingand hence one-to-one.Therefore,Q(6)= Q(6') and hence,by (16), B(6) = B(O') and, sin 0= sin 6'. Since 0, 6' E (0, -i/2), 0 = 6', so iT/4
- 4)= 0=
' = iT/4+
4),
whichis impossiblefor4)E (0, 17r74). We concludethatin a linearlyresisting medium Halley'sobservation (8) holdsforall 4)E (0, -i/4). forsupport from CWGis grateful received theNational ScienceFoundation. Acknowledgment.
REFERENCES 1. A. Battye,Modellingair resistancein the classroom,TeachingMathematicsanfdIts Applications10 (1991), 32-34. 2. R. Courant,Differential and IntegralCalculuis,Vol. II, Interscielnce, New York,NY, 1961. 3. S. Drake and I. Drabkin, Mechanicsin. Sixteenth-Century Italy, Universityof WisconsinPress, Madison,WI, 1969. 4. H. Erlichson,Maximumii dragand lift,wvith projectilerangewvith particularapplicationto golf,Allier. 51 (1983), 357-361. Jour.of Phiysics and notes, 5. G. Galilei,TwvoNew Sciences(Elzevirs,Leyden,1638), trainslated wvith a new introductioin by Stillmnan Drake, Second Edition,Wall and Thoimipson, Toronto,Canada, 1989. WorldSystemiis 6. G. Galilei,Dialogateon theGCreat (Florence,1630), translated by Giorgiode Santillana, of ChicagoPress,Chicago,IL, 1953. University
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7. C.W. Groetsclh, Tartaglia'sinverseproblemin a resistivemediunm, Am-er.M11ath.Mllonthl,/ 103 (1996), 546-551. 8. E. Halley,A discourseconcerninggravity, and itspropertieswhereinthe descentof heavybodies,aind but fullyhandled:togetherwZitlh the motionofprojectsis briefly, of a problemof greatuise the sollution in gunnery,PhilosophicalTransactions oftheRoyal Society, of Londlon16 (1686), 3-21. 9. K. Symon,Mechanics,Addison-Wesley, Reading,MA, 1953. and D. Young,Advalnced Dyniiamiics, 10. S. Timoslheinko McGraw-Hill,New York,NY, 1948.
Carl B. AllendoerferAwards - 1997 ofexpository in 1976,aremadetoauthors Awards,established TheCarlB. Allendoerfer mathea distinguished Magazine.CarlB. Allendoerfer, articlespublishedinMathematics AssociaoftheMathematical ofWashington, servedas President attheUniversity matician attheAugust1997Mathfest, tionofAmerica,1959-60.Thisyear'sawardswerepresented follow. inAtlanta.Thecitations Colm Mulcahy,"Plottingand SchemingwithWavelets,"MathematicsMagazine 69 thatmaysoon be a visibility andpopularity (December1996). "Waveletsareacquiring a fewyearsback."So beginsColmMulcahy's"Plotting enjoyedbyfractals onthescalefirst developed, andSchemingwithWavelets."Inotherwords,waveletsarea hottopic,recently research aboutbythemathematics discussedandwritten widelyapplied,andincreasingly thistopicnowat pieceof work,presenting Mulcahy'sarticleis an admirable community. of mathematicians students andtotheuninitiated a levelfullyaccessibleto undergraduate sophistiand usingclearand increasingly fromfirst principles Starting all backgrounds. intoproblems theauthorleadsthereaderfromsimpledatamanipulation, catedgraphics, and finally andpowerful to theefficient graphics, andquestionson 1- and2-dimensional effected ofwavelets.Mulcahywriteswithclarbythetechniques datastorageandretrieval problems drawnintofundamental ity,charm,and humorso thatthereaderis irresistibly inaddition hehasresearched andprovideda forbothhumansandcomputers; andsolutions contribution forfurther waveletreading.Thispaperis a significant thorough bibliography ofmathematics ofthetwenty-first literature century. totheexpository Lin Tan, "The Group ofRationalPointson theUnitCircle,"MathematicsMagazine rejoicesin thesolutionof Fermat'sLast community 69 (June1996). The mathematics theredevelopedandapplied.Whathappensif mathematics Theoremandin thepowerful insteadof focusingon ellipticcurvesandcubicequationsas was necessaryforthelatter Intheinnovaaboutconicsandtheunitcircleinparticular? work,oneaskssimilarquestions Tan Lin on the Unit of Rational Points "The simplyandnaturally Circle," tivepaper Group and familiar on more rational about related manageable some points addresses questions develops curves.Takinga group-theoretic pointof view,theauthorclearlyandenticingly forthecircleand therational-point neededto determine groupstructure themathematics led fromelementary x2 - 2 = 1. The readeris skillfully theory therelatedhyperbola, towardandstimulatandimportant intoprofound mathematics, pointing andtrigonometry andcubiccurves,among betweengroups,quadraticforms, in therelationship inginterest the from EulertoWiles,supporting stretches Thescholarly bibliography otherapplications. inmovingthereader toneofthepaperthatsucceedsadmirably andmotivational historical at ofmodern mathematics tothefrontiers andsimpleproblem froma nicelyunderstandable itsbest.
