The dog-and-rabbit chase problem as an exercise in introductory kinematics

Transcription

1 The dog-and-rabbi chase problem as an exercise in inrodcor kinemaics O I Chashchina 1 Z KSilagadze 1 1 Deparmen of Phsics Novosibirsk Sae Universi 63 9 Novosibirsk Rssia Bdker Insie of Nclear Phsics 63 9 Novosibirsk Rssia silagadze@inpnsks (Received 3 Ags 9 acceped 19 Sepember 9) Absrac The prpose of his aricle is o presen a simple solion of he classic dog-and-rabbi chase problem which emphasizes he se of conceps of elemenar kinemaics and herefore can be sed in inrodcor mechanics corse The aricle is based on he eaching experience of inrodcor mechanics corse a Novosibirsk Sae Universi for firs ear phsics sdens which are js beginning o se advanced mahemaical mehods in phsics problems We hope i will be also sefl for sdens and eachers a oher niversiies oo Kewords: Phsics Edcaion Classical Mechanics eaching Resmen El propósio de ese aríclo es presenar na solción simple al problema clásico del perro el conejo el cal remarca el so de los concepos básicos de cinemáica por lo ano pede ser empleado en n crso inrodcorio de Mecánica Ese aríclo esá basado en la experiencia generada del crso inrodcorio de Mecánica en la Universidad Esaal de Novosibirsk para el primer año de Física dónde los esdianes comienzan con méodos avanzados de maemáicas en los problemas de Física Se espera qe el presene rabajo sea úil ano para los esdianes como para los almnos oras niversidades ambién Palabras clave: Edcación en Física Enseñanza de la Mecánica Clásica PACS: 14Fk 45D- ISSN I INTRODUCTION A rabbi rns in a sraigh line wih a speed A dog wih a speed > sars o prsi i and dring he prsi alwas rns in he direcion owards he rabbi Iniiall he rabbi is a he origin while he dog s coordinaes are x() = () = (see Fig 1) Afer wha ime does he dog cach he rabbi? This classic chase problem and is variaions are ofen sed in inrodcor mechanics corse [1 3] When one asks o find he dog s rajecor (crve of prsi) he problem becomes an exercise in advanced calcls and/or in he elemenar heor of differenial eqaions [4 5] However is reamen simplifies if he radiional machiner of phsical kinemaics is sed [6] The mahemaics of he solion becomes even simpler if we frher nderline he se of phsical conceps like reference frames vecor eqaions decomposiion of veloci ino radial and angenial componens II DURATION OF THE CHASE e r 1 be a radis-vecor of he dog and r a radis-vecor of he rabbi So ha FIGURE 1 Dog and rabbi chase The dog is heading alwas owards he rabbi r 1 = r = (1) As he dog alwas is heading owards he rabbi we can wrie r r k () 1 = () a Am J Phs Edc ol 3 No 3 Sep hp://wwwjornallapenorgmx

2 O I Chashchina and Z KSilagadze The proporionali coefficien k depends on ime Namel a he sar and a he end of he chase we obviosl have k() = kt ( ) = (3) Differeniaing () and sing (1) we ge = k () + k() (4) As he dog s veloci does no change in magnide i ms be perpendiclar o he dog s acceleraion all he ime: = (5) Formall his follows from d d = d = d = Eqaions (4) and (5) impl ( ) = k ( ) or = k () (6) We can inegrae he las eqaion and ge x x k() = (7) A he ime =T when he dog caches he rabbi and he chase erminaes we ms have x=t and remembering ha k(t)= we easil find T from (7): T = (8) III DOG S TRAJECTORY IN THE RABBIT S FRAME e s decompose he dog s veloci ino he radial and angenial componens in he rabbi s frame (see Fig ): r = + cos( π ) = cos = sin( π ) = sin FIGURE Componens of he dog s veloci in he rabbi s frame we ge Hence Ping r dr = d 1 dr + cos = rd sin r + cos ln d = sin π / z = cos and sing he decomposiion + z A B = + z z + z A = (1 + ) we can easil inegrae and ge π / 1 B = ( 1) B cos (1 cos ) 1 + d = ln + = ln[( cg ) ] A sin (1 cos ) sin he eqaion of he dog s rajecor in he rabbi s frame is B r = r and = r r = cos r = sin (9) r = ( cg ) (1) sin If we divide he firs eqaion on he second and ake ino accon ha I DOG S CURE OF PURSUIT e s now find he dog s rajecor in he laboraor frame From (9) and (1) we ge a Am J Phs Edc ol 3 No 3 Sep 9 54 hp://wwwjornallapenorgmx

3 which b sing ( cg ) = sin sin d 1 1 = 1 + d( cg ) cg sin can be recas in he form ( cg ) d ( cg ) = d cg 1 = 1 ( 1)( cg ) + ( + 1)( cg ) T (11) The dog-and-rabbi chase problem as an exercise in inrodcor kinemaics and herefore k () = (14) I remains o differeniae (1) o ge he derivaive of and hence he desired expression for k() : k 1 1 () 1 ( 1) + = + (15) T Sbsiing (1) and (15) ino (13) we finall ge he dog s crve of prsi in he form x = (16) Of corse his resl is he same as fond earlier in he lierare [4 5 6] p o applied convenions = and T was defined earlier hrogh (8) B b (1) = rsin = cg and herefore (11) reprodces he resl of Ref [6] (1 ) + (1 ) = + + (1) T THE IMIT DISTANCE FOR EQUA EO- CITIES e speeds of he dog and he rabbi are eqal in magnide and heir iniial posiions are as shown in he Fig 3 To wha limi converges he disance beween hem? I is convenien o answer his qesion in he rabbi s frame For eqal velociies = eqaions (9) ake he form r = (1 + cos ) r = sin d ( r r cos ) = r (1 cos ) + r sin = d 1 = = From (7) we have 3 x k() = + (13) 1 T T and o ge he eqaion of he dog s rajecor we have o express k() hrogh Taking he componen of he vecor eqaion () we ge = k() FIGURE 3 The chase wih eqal velociies a Am J Phs Edc ol 3 No 3 Sep hp://wwwjornallapenorgmx

4 O I Chashchina and Z KSilagadze We see ha r(1 cos ) = C C is a consan A = we have (see Fig 3) r = 1 + cos = + 1 (noe ha we are assming > π / as in Fig 3 so ha cos = 1 sin ) Now le s find he dog s rajecor We have (see Fig 3) B x= ( τ ) dτ cos( π ) = ( τ) dτ + cos = sin( π ) = sin C = In he rabbi s frame π when Hence he limi disance beween he dog and he rabbi is [5] r C + + = = 1 cosπ 1 I CHASE AONG THE TRACTRIX If he rabbi is allowed o change he magnide of his speed he can manage o keep he disance beween him and he dog consan e s find he reqired fncional dependence () and he dog s rajecor in his case If r = = cons so ha r = eqaions (9) ake he form = cos = sin (17) becase d d cos ( τ) dτ = = (19) sin 1 cos o d () = d sin according o he second eqaion of (17) B sing he decomposiion = cos 1 cos 1 cos he inegral in (19) is easil evalaed wih he resl ( τ) dτ = ln( cg ) ln( cg ) g = he parameric form of he dog s rajecor is x = = cos + ln( cg ) ln( cg ) sin which can be easil inegraed o ge To ge he explici form of he rajecor we se or sin ln = sin sin = sin e cos = 1 1+ cos cg = = sin which gives Then he firs eqaion of (17) deermines he reqired form of he rabbi s veloci () = 1 sin e (18) x cg = ln( ) ln This rajecor is a par of racrix he famos crve [7 8] wih he defining proper ha he lengh of is angen beween is direcrix (he x -axis in or case) and he poin of angenc has he same vale for all poins of he racrix a Am J Phs Edc ol 3 No 3 Sep 9 54 hp://wwwjornallapenorgmx

5 II CONCUDING REMARKS We hink he problem considered is of pedagogical vale for ndergradae sdens which ake heir firs ear corse in phsics I demonsraes he se of some imporan conceps of phsical kinemaics as alread sressed b Mngan in Ref [6] The approach presened in his aricle reqires onl minimal mahemaical backgrond and herefore is siable for sdens which js begin heir phsics edcaion However if desired his classic chase problem allows a demonsraion of more elaborae mahemaical conceps like Frene-Serre formlas [9] Mercaor projecion in carograph [1] and even hperbolic geomer (which is realized on he srface of revolion of a racrix abo is direcrix) [11] Ineresed reader can find some oher variaions of his chase problem in [ ] ACKNOWEDGEMENTS The work is sppored in par b grans SciSchool- 956 and RFBR a REFERENCES [1] Irodov I E Problems in General Phsics (NTTS ladis Moscow 1997) p 9 (in Rssian) [] Belchenko Y I Gilev E A and Silagadze Z K Problems in mechanics of paricles and bodies Par 1: relaivisic mechanics (Novosibirsk Universi Press Novosibirsk 6) p 9 (in Rssian) The dog-and-rabbi chase problem as an exercise in inrodcor kinemaics [3] Silagadze Z K Tes problems in mechanics and special relaivi Preprin phsics/6557 [4] Olchovsk I I Pavlenko Y G and Kzmenkov S Problems in heoreical mechanics for phsiciss (Moscow Universi Press Moscow 1977) p 9 (in Rssian) [5] Pák P and Tkadlec J The dog-and-rabbi chase revisied Aca Polechnica (1996) [6] Mngan C A A classic chase problem solved from a phsics perspecive Er J Phs (5) [7] Yaes R C The Caenar and he Tracrix Am Mah Mon (1959) [8] Cad W G The Circlar Tracrix Am Mah Mon (1965) [9] Pckee C C The Crve of Prsi Mah Gazee (1953) [1] Pijls W Some Properies Relaed o Mercaor Projecion Am Mah Mon (1) [11] Beroi B Caenacci R and Dappiaggi C Psedospheres in geomer and phsics: From Belrami o de Sier and beond Preprin mah/56395 [1] oka A J Conribion o he Mahemaical Theor of Capre I Condiions for Capre Proc Na Acad Sci (193) [13] alan Conribion á l ede de la corbe de porsie Compes Rends (1931) [14] oka A J Families of crves of prsi and heir isochrones Am Mah Mon (198) [15] Qing-Xin Y and Yin-Xiao D Noe on he dog-andrabbi chase problem in inrodcor kinemaics Er J Phs 9 N43 N45 (8) [16] Wnderlich W Über die Hndekrven mi konsanen Schielwinkel Monashefe für Mahemaik (1957) a Am J Phs Edc ol 3 No 3 Sep hp://wwwjornallapenorgmx