DTIC ADA574898: A Classic Chase Problem Solved from a Physics Perspective

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INSTITUTEOFPHYSICSPUBLISHING EUROPEANJOURNALOFPHYSICS
Eur.J.Phys.26(2005)985–990 doi:10.1088/0143-0807/26/6/005
A classic chase problem solved from a
physics perspective
Carl E Mungan
PhysicsDepartment,USNavalAcademyAnnapolis,MD21402-5040,USA
E-mail:[e-mail]
Received6April2005,infinalform22May2005
Published8August2005
Onlineatstacks.iop.org/EJP/26/985
Abstract
Thetrajectory,traveltimeandrelativeapproachvelocityofapursuertracking
a prey along a simple curve of pursuit are deduced using basic principles
of two-dimensional kinematics. While such curves are well known in the
mathematics literature, little attention has been paid to this problem by the
physicseducationalcommunity, despitethefactthatithasabundantphysical
applications. Italsomakesaninterestingalternativetothetraditionalproblems
ofintroductorykinematics.
(Somefiguresinthisarticleareincolouronlyintheelectronicversion)
1. Introduction
Imagine that you are the commander of a submarine and spot an enemy warship travelling
perpendiculartoyourlineofsightatdistanceHwithconstantvelocityofmagnitudeV.You
fire a torpedo having constant speed υ at the ship. The torpedo tracks the target, so that it
always travels directly toward the ship. What is the trajectory of the torpedo through the
water? If the torpedo impacts the warship, how long does it take to reach the target? If it
doesnotreachit,whatistheclosestdistancethatitgetstoit? Whatistherelativeapproach
velocityoftheshipalongthetorpedo’slineofsight?
Thisisanexampleofwhatisknownasachaseproblem[1]andtheresultingtrajectories
arecalledcurvesofpursuit[2,3]. Problemsofthisgeneralsortareofinteresttothemilitary
community[4]andtovideogamedesigners. Historically,theparticularprobleminthispaper
was first solved by the French mathematician Pierre Bouguer in 1732, and may have been
originallyproposedbyLeonardodaVinciintheformofacatandmousechase[5]. Itistreated
insomesecond-yearundergraduate-leveldifferentialequationstexts[6]. However,suchbooks
typicallyusetheformulaforarclengthtoobtainadifferentialequationinthefirstandsecond
spatialderivativesofthecoordinatesofthecurveandthenmakeachangeofvariabletosolve
it. The resulting nomenclature and analysis do not tie in well with undergraduate physics
courses. Thepresentpaperinsteadusestimederivativesandvectorcomponentstorecastthe
0143-0807/05/060985+06$30.00 (cid:1)c 2005IOPPublishingLtd PrintedintheUK 985
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A classic chase problem solved from a physics perspective
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13. SUPPLEMENTARY NOTES
14. ABSTRACT
The trajectory, travel time and relative approach velocity of a pursuer tracking a prey along a simple
curve of pursuit are deduced using basic principles of two-dimensional kinematics. While such curves are
well known in the mathematics literature, little attention has been paid to this problem by the physics
educational community, despite the fact that it has abundant physical applications. It also makes an
interesting alternative to the traditional problems of introductory kinematics.
15. SUBJECT TERMS
16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF
ABSTRACT OF PAGES RESPONSIBLE PERSON
a. REPORT b. ABSTRACT c. THIS PAGE Same as 6
unclassified unclassified unclassified Report (SAR)
Standard Form 298 (Rev. 8-98)
Prescribed by ANSI Std Z39-18
986 CEMungan
y
B V D ship
L
torpedo
θ
H
C
υ
x
A
Figure1. Trajectoriesofthetorpedoandwarship. PointAisthelocationofthetorpedowhen
fired;pointBindicatestheinitialpositionofthetarget;pointChascoordinates(x,y)anddenotes
thecurrentlocationofthetorpedo;andpointDwithcoordinates(Vt,H)isthecurrentlocationof
theship.
solutionintermsofthemorefamiliarconceptsofvelocityandacceleration. Itthenbecomes
anapplicationof2Dkinematics.
2. Firstintegralofthedifferentialequation
Setupacoordinatesystemwiththetorpedostartingattheoriginandtravellinginitiallyinthe
y-direction and the warship steaming in the x-direction. At some arbitrary instant in time t
(after firing the torpedo at t = 0), the positions of the ship and torpedo are as indicated in
figure1. ThedashedrighttrianglehasabaseoflengthVt−x,heightH −yandhypotenuse
(representingthedistancetothetarget)oflengthL.Theratiooftheriseandrunofthistriangle
isequaltotheratiooftheyandxcomponentsofthetorpedo’svelocity,becausethetorpedo
isheadeddirectlytowardtheship,sothat
υ
Vt −x =(H −y) x. (1)
υ
y
Differentiatingbothsideswithrespecttotimeandrearrangingleadsto
(cid:3)
(cid:1) (cid:2) (cid:1) (cid:2)
V d υ d υ 2 υ2
= x = −1=− (cid:4) a , (2)
H −y dt υy dt υy υ2 υ2−υ2 y
y y
usingthefactthatυ2 +υ2 =υ2isaconstantandintroducingthey-componentofthetorpedo’s
x y
centripetalacceleration. Inordertointegratethisequation,onecansubstitutethechain-rule
result,a =υ dυ /dy,familiarfromthederivationofthework-kinetic-energytheorem. The
y y y
variablescanthenbeseparatedandbothsidesintegratedtoobtain
(cid:5) (cid:5)
V y dy =− υy (cid:4)υdυy , (3)
υ 0 H −y υ υ υ2−υ2
y y
wherethelowerlimitsaretheinitialvalues. Computingthesestandardintegrals,onegets
 (cid:4)  (cid:4)
(cid:1) (cid:2) (cid:1) (cid:2)
−V ln 1− y =lnυ+ υ2−υy2 ⇒ 1− y −V/υ = υ+ υ2−υy2. (4)
υ H υ H υ
y y
Aclassicchaseproblemsolvedfromaphysicsperspective 987
Thequantityinthelastsetofparenthesesappearssofrequentlyintheremainderofthispaper
thatitisusefultogiveititsownsymbol,z≡1−y/H,intermsofwhichequation(4)canbe
rewrittenas
2υ
υ = . (5)
y z−V/υ +zV/υ
Thisisthesolutionforthey-componentofthetorpedo’svelocityalongitscurveofpursuit.
3. Solvingforothervariablesofinterest
Equation (5) can be used to readily solve for other physically interesting quantities. They
willbeexpressedintermsofthedimensionlessparameterz,whichrepresentsthenormalized
transversedistancefromtheshiptothetorpedo. First,υ2+υ2 =υ2leadsto
x y
υ 1−z2V/υ
cosθ = x = . (6)
υ 1+z2V/υ
Equations(5)and(6)canalternativelybecompactlyexpressedintermsofhyperbolicfunctions.
Next,togetthecoordinatesofthetorpedo,notethatυ =dy/dt sothatequation(5)can
y
againbeseparatedandintegratedtoget
(cid:5) (cid:10)(cid:1) (cid:2) (cid:1) (cid:2) (cid:11)
1 y y −V/υ y V/υ
t = 1− + 1− dy. (7)
2υ H H
0
Providedthatυ (cid:3)=V,theintegrandsareoftheformz±nwheren(cid:3)=1,sothattheresultis
t V +υ V −υ
=− z1−V/υ + z1+V/υ +1, (8)
τ 2υ 2υ
whereτ isacharacteristictimedefinedas
Hυ
τ ≡ (9)
υ2−V2
whosephysicalsignificancewillbecomeclearmomentarily. Equation(8)givesthetimethat
thetorpedoneedstoreachagivenposition. Onecansubstitutethatalongwithequations(5)
and(6)into(1)tofind
x 0.5 0.5 V/υ
= z1−V/υ + z1+V/υ − , (10)
H V/υ−1 V/υ+1 V2/υ2−1
which is the trajectory x(y) of the torpedo expressed in terms of the normalized quantities
x/H,y/H =1−zandV/υ. Thisisplottedinfigure2forseveralvaluesofthespeedratio.
Thetorpedostrikesthetargetwheny =H ⇒z=0,whichimplies
HV/υ
x = . (11)
impact 1−V2/υ2
This is positive if and only if υ > V. As expected, the torpedo only hits if it is travelling
fasterthantheship! Inthatcase,thetimebetweenlaunchandimpactisx /V =τ defined
impact
by equation (9). This impact time τ can be reduced if the torpedo leads [7] or, even better,
mirrors[8]theship’smotionratherthanheadingstraighttowardit.
Thedistancebetweenthetorpedoandtargetatanytimeis
(cid:12)
υ
L= (Vt −x)2+(H −y)2 =(H −y) , (12)
υ
y
usingequation(1)inthesecondstep. Substitutingequation(5)thengives
H
L= (z1−V/υ +z1+V/υ). (13)
2
988 CEMungan
1
1/5 1/2
1
0.8
2
0.6
H
y/ 5
0.4
0.2
0
0 0.5 1 1.5 2
x/H
Figure2. Graphsofequation(10)or(16)for0(cid:1)y(cid:1)H. Thenumberoneachcurvelabelsthe
valueofV/υ.ImpactoccursifV/υ<1,asisevidentforthetoptwotraces.
Thevaluez thatgivesthedistanceofclosestapproachL forV >υ isfoundbysetting
min min
dL/dz=0,
(cid:1) (cid:2) (cid:1) (cid:2)
V −υ υ/2V H V −υ υ/2V
z = ⇒ L = (cid:12) . (14)
min V +υ min 1−υ2/V2 V +υ
Asonemighthaveguessed,theminimumseparationisequaltoHinthelimitasV/υ →∞
andoccursatt =0.
On the other hand, as V → υ, equation (14) has the limiting value L → H/2 and
min
this occurs as t → ∞. Assuming the torpedo does not actually hit the target, it can never
getcloserthanthis,regardlessofitsspeed! Thetrajectoryinthiscaseisgivenbyintegrating
equation(7)withV/υ =1toget
H
t = (1−z2−2lnz). (15)
4υ
Then,followingthesamereasoningusedtoderiveequation(10),onededucesthat
x z2−1 √
= −ln z, (16)
H 4
plottedasthemiddlecurveinfigure2.
4. Relativeapproachvelocityofthetarget
Fromfigure1,onecanseethattheshipisrecedingfromthetorpedoatspeedV cosθ along
thelineofsight,whilethetorpedoisapproachingalongthesamelineatspeedυ. Hence,the
normalizedrelativeapproachvelocityofthewarshipis
υ V
rel =1− cosθ, (17)
υ υ
where cosθ is given by equation (6). This is plotted in figure 3 for the case of V/υ = 2.
A positive value of υ /υ indicates that the ship and torpedo are approaching one another;
rel
anegativevaluemeanstheyarereceding. Therefore, theinterceptalongthehorizontalaxis
representsthepositionofclosestapproachbetweenthetorpedoandtarget. Thiscanbeverified
mathematicallybysubstitutingz fromequation(14)into(6)todeducetheheadingofthe
min
Aclassicchaseproblemsolvedfromaphysicsperspective 989
1.0
0.6
υ 0.2
/
υrel
-0.2
-0.6
-1.0
0 0.2 0.4 0.6 0.8 1
y/H
Figure3. Plot of the normalized relative approach velocity from equations (17) and (6) as a
functionofthenormalizedtransversepositionofthetorpedo,forthecaseofV =2υ. Aty=0,
thetorpedo’sheadingisθ =90◦andthusυrel/υ=1. Ontheotherhand,asy→H,thetorpedo
finaltlesrcineplitniesbloechaitneddtahteys/hHip(=sin1c−eV3−>1/4υ=)∼so0.t2h4a.tθ =90◦andhenceυrel/υ=−1.Thehorizontal
torpedoatminimumseparation,
(cid:1) (cid:2)
υ
θ =cos−1 (18)
min
V
or60◦inthecaseoffigure3. Thissimplerelationcanalternativelybededucedbysettingthe
timederivativeofthefirstequalityinequation(12)tozero,andthensubstitutingequation(1)
intoit.
In practice, the curve in figure 3 could be experimentally mapped out if the torpedo
measurestheDopplershiftinthetrackingreturnsignal. TheclassicalDopplershiftedreflection
frequencyoffthetargetis
(u−V cosθ)(u+υ)
f(cid:9) =f , (19)
(u−υ)(u+V cosθ)
forasourceoffrequencyfaboardthetorpedowithasignalspeeduthroughthewater. (For
example,inthecaseofDopplersonar,fistypicallyoftheorderofhundredsofkilohertzand
thesoundspeeduisabout1500ms−1inseawater.) Thefourtermsinparenthesesarisefrom
theDopplershiftsinthesignals:emittedbythemovingsource—firstterminthedenominator;
received at the ship—first term in the numerator; echoed by the ship—second term in the
denominator; andreceivedbackatthetorpedo—secondterminthenumerator. Forrealistic
torpedo and warship speeds, υ and V are much smaller than u, so that equation (19) can be
approximatedas
(cid:1) (cid:2) (cid:1) (cid:2)
f(cid:9)=∼f 1−2V cosθ +2υ ⇒ (cid:4)f u =∼2 1− V cosθ , (20)
u u f υ υ
where(cid:4)f ≡ f(cid:9)−f. Theleft-handsideofthesecondequalityisthenormalizedfrequency
shift divided by the Mach number of the torpedo. Meanwhile, the right-hand side is twice
the normalized relative approach velocity of the target, according to equation (17). Hence,
figure3directlygivestheDopplershifttowithinaconstantscalingfactor. Inparticular,zero
shiftrepresentsthebestopportunityforthetorpedotoexplodeandattempttoinflictdamage
ontheship(assumingithasnotalreadyimpacteditbythen),whichisthebasisofproximity
fuses. Similar spectral effects are exploited by ‘Doppler bats’, such as the horseshoe and
pipistrelle,whilechasinginsects[9].
990 CEMungan
5. Extensionstorelatedapplications
As an introduction to chase problems, this paper restricted consideration to the simple case
ofaquarrytravellingatconstantvelocityfollowedbyapursuertravellingatconstantspeed.
In this concluding section, a number of variations on this basic theme are listed. Interested
readersarereferredtotheliteratureforfurtherdiscussionofthem. Manyofthemcouldform
thebasisforchallengingstudentprojects,wellsuitedtocomputersimulation.
Militarypursuit. Anticipating theprey’s futurelocation; targeting whenfiringbulletsat the
quarry; limiting the maximum acceleration (minimum radius of curvature); pursuit from a
constantangleofattack;optimumsearchtrajectorieswhenaquarry’slocationisonlypartially
known;controllingtheimpactdirectionsothatitisfromthefrontorside;three-dimensional
pursuitthroughtheairorwater.
Gamingstrategies. Chaseinaconfinedspace(suchasaboxingring);pursuitwhenthemotion
isconstrained(suchasonachessboard); preventingaquarryfromreachingarefuge(asin
tag);arunnerlaunchingatamovingtarget(asindodgeball).
Animalmotions. Bugsontheverticesofapolygoneachpursuingtheirnearestneighbourin
aclockwiseinwardspiral[10]; adoginflowingwaterwhoisfollowingarunneronland; a
spiderscurryingalonghisweb;afarmertryingtoroundupmultipleescapedpigs.
Mathematicalanalyses. Polarcoordinatesofthepreyrelativetothepursuer;quarryfollowing
a nonlinear plane curve and/or whose speed varies in a predictable manner; relationship to
specialcurves suchasthetractrix(shapeofaninitiallystraightchainwhoseendisdragged
in a direction other than its initial axis) or caustic (envelope of reflected rays from a curved
mirror);geodesicpursuitonacurvedsurface(e.g.acylinderorhillside);chaseonarotating
turntable.
Acknowledgments
IthankBrianJenkinsandMurrayKormanforinspiringthispaper.
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[8] O’ConnellJ1995PursuitandevasionstrategiesinfootballPhys.Teach.33516–8
[9] DennyM2004Thephysicsofbatecholocation:signalprocessingtechniquesAm.J.Phys.721465–77
[10] WattonAandKydonDW1969Analyticalaspectsofthen-bugproblemAm.J.Phys.37220–1