A Matemática Na NFL

8/17/2019 A Matemática Na NFL

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C KCRJK ²CRMLC HC HEA7Zkc Ch²camsj bn Pnhtn Jxtrc.

Cutnr7 Prne. Vnbrmin V. Inhĺ[email protected]

VJ\ZKN

N prjsjhtj crtmin nfgjtmvc jstmkuacr c cprjhbmzcijk kctjk²ctmlc ctrcv²js bc mhtjrbmslmpamhcrmbcbj, cam-chbn n lnhdjlmkjhtn c̀ E²ĳsmlc bj uk jspnrtj quj vjk sj tnrhchbn c lcbc bmc kcms mhtjrjsschtj jkhnssn pc²ĳs7 n Eutjfna Ckjrmlchn. \cˑn bjslrmtcs caiukcs ajms e²ĳsmlcs bj mkpnrtchtj jstubn pcrc c lnk-prjjhscˑn bn ejhn˅kjhn bn ldutj pcrc c kcrlcĺcˑn bn jxtrc pnmht , jxpchbmhbn n jstubn pcrc ldutjs bj

bmejrjhtjs bmstc˅hlmcs. \jrcˑn cfnrbcbns t²npmlns bj Ajms bj Hjwtnh, Knvmkjhtn Zhmenrkj, KnvmkjhtnZhmenrkjkjhtj Ucrmcbn, Achĺckjhtn Pcrcf²namln, Ajm bc Lnhsjrvcĺcˑn bj Jhjrimc j Ajm bc Lnhsjrvcĺcˑnbc Suchtmbcbj bj Knvmkjhtn, Vcmn bj Lurvcturc bj ukc Pcr²cfnac, sjiumbns bj caiuhs jxjkpans jcpamlcĺnˑjs pr²ctmlcs. Ns ir²cffilns enrck nftmbns ctrcv²js bj smkuacĺnˑjs hn snetwcrj IjnIjfrc. \upnˑj-sjhjljsscrmckjhtj quj n ajmtnr pnssuc c lnkprjjhscˑn bj Kctjk²ctmlc j E²ĳsmlc f²csmlcs.

Pcacvrcs-ldcvj7 Hctmnhca Enntfcaa Ajciuj? Jxtrc Pnmht? Vcmn bj Lurvcturc? Jqucĺnˑjs Bmejrjhlmcms?Achĺckjhtn Pcrcf²namln.

MHRVNBZĹ CˑN

Emiurc 37 Omlojrs kchbchbn vjr!

Cpjscr bn Eutjfna Ckjrmlchn cmhbc hcˑn sjr tcˑn pnpuacr hn Frcsma, pjrljfj-sj, rjljhtjkjhtj, quj jstjjspnrtj tjk ctrc²ĳbn lcbc vjz kcms cbkmrcbnrjs pnr tnbn n hnssn tjrrmt²nrmn. ²J uk jspnrtj jxtrjkckjhtjjstrct²jimln j vnatcbn pcrc ca²jk bc utmamzcĺcˑn bc enrĺc e²ĳsmlc. Ujhlj n tmkj quj utmamzcr c kjadnrjstrct²jimc bj ctcquj lnhsjiumhbn trcspnr c bjejsc cbvjrs²crmc j pjhjtrcr kcms vjzjs hc jhbznhj , j c

kjadnr jstrct²jimc bj bjejsc, mkpjbmhbn n ctcquj cbvjrs²crmn bj lnhvjrtjr pnhtns. Jhffik, ²j uk jspnrtjrjldjcbn bj Kctjk²ctmlc!

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²J uk jspnrtj gnicbn pnr 6 tmkjs lnk 33 gnicbnrjs jk lckpn lcbc uk. N tmkj quj tjk c pnssj bjfnac ²j ldckcbn bj tmkj bj ctcquj (nejhsmvn) j c prmhlmpca mhtjhĺcˑn bjssj tmkj ²j cvchĺcr pjan lckpnctrcv²js bj lnrrmbcs nu pcssjs ct²j lruzcr c amhdc bj ina j ldjicr c ukc ²crjc ldckcbc jhbznhj . Pcrcmssn, bjvj trchspnr c fcrrjmrc bn tmkj cbvjrs²crmn (bjejhsmvn).

Jk sukc, n nfgjtmvn bn gnin ²j snkcr kcms pnhtns. C prmhlmpca gnicbc ²j jhtrcr hc ²crjc cn euhbn bn

lckpn cbvjrs²crmn (jhbznhj ) lnk c pnssj bc fnac (tnuldbnwh ), kcrlchbn 5 pnhtns, j n bmrjmtn c ukldutj amvrj c ina pnr kcms uk pnhtn jxtrc, nu kjskn bnms pnhtns jxtrcs, sj ns gnicbnrjs tjhtcrjk, cnmhv²js bn ldutj amvrj cn ina, uk pcssj nu ukc lnrrmbc. Pnrtchtn, n nfgjtmvn prmknrbmca ²j chntcr 9 nu 4pnhtns jk ukc ‛bjslmbc‛bj ctcquj.

Suchbn n tmkj bj ctcquj kcrlc ns pnhtns cakjgcbns nu smkpajskjhtj pjrbj c pnssj bj fnac pnr jrrnspr²nprmns nu k²jrmtns bc bjejsc cbvjrs²crmc, ns pcp²jms scˑn mhvjrtmbns7 tca tmkj jhtrc pcrc c pr²nxmkc gnicbclnk n tmkj bjejhsmvn, j n cbvjrs²crmn lnk n nejhsmvn, j cssmk pnr bmchtj ct²j n ffihca bns 2 tjkpns bj 3:kmhutns (hcˑn lnrrmbns). 3

Emiurc 67 Bmkjhsnˑjs bj uk lckpn bj eutjfna ckjrmlchn.

(Enhtj71dttps7//fanibmcrmnhst.ffiajs.wnrbprjss.lnk/600=/33/bmcrmnhstlckpneutjfnackjrmlchn.phi8)

N prmhlmpca nfjgtmvn bjstj crtmin ²j eczjr ukc ch²camsj, jk pcrtmluacr, bn pnhtn jxtrc!

Cp²ns uk jknlmnhchtj j tcˑn cakjgcbn tnuldbnwh , n tmkj bj ctcquj tjk c npnrtuhmbcbj bj lnkpajkjhtcrsjus pnhtns ichdns lnk uk jxtrc pnmht , tntcamzchbn 9 pnhtns hc gnicbc! N jxtrc pnmht ²j uk ldutj bj


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Emiurc


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C vjanlmbcbj mhmlmca bn ldutj ²j lcaluacbc pjan calchlj bn knvmkjhtn. Bj ectn, pnbjkns lnhsmbjr²c-acsjkprj pnsmtmvc (hc kjskc bmrjĺcˑn j sjhtmbn bn knvmkjhtn). Uckns, bjssc enrkc, bjlnkpnr n vjtnrvjanlmbcbj mhmlmca, tca quj7

Emiurc :7 Bjlnkpnsmĺcˑn bn vjtnr vjanlmbcbj mhmlmca (v0).

lnsδ ; v0x

|v0| ⇖ v0x ; |v0|lnsδ ⇖ v0x ; v0 lns δ

sjhδ ; v0y|v0| ⇖ v0y ; |v0|sjhδ ⇖ v0y ; v0 sjh δ

Jk rjacĺcˑn c x, tjkns, pnrtchtn, uk Knvmkjhtn Zhmenrkj . Anin, c vjanlmbcbj bj prngjĺcˑn dnrmznhtcabc fnac nvca ²j lnhstchtj j miuca c̀ vjanlmbcbj mhmlmca bj suc lnkpnhjhtj dnrmznhtca. Jk tjrkns e²ĳsmlns, cbjffihmĺcˑn bj tca vjanlmbcbj ²j c vcrmcĺˑ cn bn jspcĺn dnrmznhtca pjrlnrrmbn pnr uk bjtjrkmhcbn mhtjrvcanbj tjkpn. Jk amhiuciujk kctjk²ctmlc, jslrjvjkns7

v0x ; vx ; bx

btKchmpuachbn c jqucĺcˑn clmkc j mhtjirchbn, nftjkns7

vx ; bx

bt ⇖ bx ; vx bt ⇖

∯ bx ;

∯ vx bt ⇖ x(t) ; vx t + L

Rjkns quj c lnhbmĺcˑn mhmlmca x(0) ; x0 mkpamlc L ; x0. Lnkn lnhsmbjrckns x0 ; 0, jhtcˑn lnhlau²ĳknsquj,

x(t) ; vx t

Bc²ĳ,

x(t) ; (v0 lns δ)t (3)

Pcrc n knvmkjhtn vjrtmlca, c fnac nvca snerj ukc cljajrcĺcˑn lnhstchtj j lnhtr²crmc cn knvmkjhtn,vjrtmlca, pcrc fcmxn, bjhnkmchbc cljajrcĺcˑn ircvmtclmnhca i. Pnrtchtn, c ; ∖i ²j c cljajrcĺcˑn bnknvmkjhtn, lugn lnhljmtn ²j n qunlmjhtj jhtrj c vcrmcĺˑ cn bc vjanlmbcbj pjan mhtjrvcan bj tjkpn. Jkamhiucijk kctjk²ctmlc, pnbjkns kchmpuacr j jslrjvjr7

c ; ∖i ; bvybt ⇖ bvy ; ∖ibt ⇖

∯ bvy ; ∖

∯ ibt ⇖ vy(t) ; ∖it + L

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Pcrc t ; 0, tjkns L ; v0y . Bc²ĳ, vy(t) ; v0y ∖ it ²j c jqucĺcˑn dnr²crmc bc vjanlmbcbj. Lnknbjsjgckns nftjr c jqucĺcˑn bc trcgjt²nrmc, j n lnhljmtn bj vjanlmbcbj ²j n qunlmjhtj bc vcrmcĺˑ cn bn jspcĺnpjan mhtjrvcan bj tjkpn, jslrjvjkns7

vy ; by

bt ⇖ by ; vy bt ⇖ by ; (v0y ∖ it)bt ⇖ ∯ by ; ∯ v0ybt∖ ∯ itbt ⇖

y(t) ; v0yt ∖ it6

6 + L

Pcrc t ; 0 tjkns L ; y0. Hjstj lcsn, lnhsmbjrckns quj y0 ; 0. Pnrtchtn c jqucĺcˑn bc trcgjt²nrmc ²jbcbc pnr7

y(t) ; v0y t ∖ it6

6

y(t) ; (v0 sjh δ)t ∖ it66

(6)

Pnrtchtn, (3) j (6) scˑn cs jqucĺnˑjs dnr²crmcs bn knvmkjhtn, bjslnhsmbjrchbn c rjsmstj˅hlmc bn cr.

UJANLMBCBJ MHMLMCA K²MHMKC PCVC N LDZRJ

Hjstj knkjhtn, mhmlmcrjkns lnhlausnˑjs mkpnrtchtjs snfrj n knvmkjhtn bc fnac nvca ct²j quj jac calchljc trcvj cbvjrs²crmc. Lnkn n tjkpn t ²j n kjskn pcrc ckfcs cs lnkpnsmĺnˑjs bn knvmkjhtn, msnackns tjk (3) j n sufstmt²ĳkns jk (6), tca quj7

y(t) ; (v0 sjh δ)

xv0 lns δ

∖ i6

x

6

v60lns6 δ

y(t) ; x ti δ ∖ i6

x

v0 lns δ

6(


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Kcs, bcbn quj i ≄ =, 43 k/s6, x ;


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cakjgcbn, eczjhbn lnk quj c fnac nvca nftjhdc uk mkpuasn kcmnr, lnhsjqujhtjkjhtj, ctmhimhbn n sjunfgjtmvn.

C UJANLMBCBJ MHMLMCA JK EZHĹ CˑN BN CALCHLJ, BC CARZVC J BN RJKPN

Hc bjlnkpnsmĺcˑn bj knvmkjhtns, lnkn n tjkpn ²j mbj˅htmln pcrc ckfns (vjrtmlca j dnrmznhtca), vcknsmsnacr t hc jqucĺcˑn bn knvmkjhtn dnrmznhtca, rjimbn pjac jqucĺcˑn x(t) ; (v0 lns δ)t. Kchmpuachbn c

jqucĺcˑn, ldjickns eclmakjhtj c lnsδ ; xv0 t

.

C jqucĺcˑn bn knvmkjhtn vjrtmlca hns b²c y(t) ; (v0 sjh δ) t∖ i t6

6 .

Bc mbjhtmbcbj trminhnk²jtrmlc, tjkns quj sjh6δ + lns6δ ; 3. Mstn mkpamlc quj7

sjh6δ + x6

v60 t6

; 3 ⇖ sjh6δ ; 3 ∖ x6

v60 t6

sjh6δ ; v60 t

6 ∖ x6v60 t

6

sjhδ ;

∗ v60 t

6 ∖ x6v0 t

\ufstmtumhbn n rjsuatcbn clmkc hc jqucĺcˑn dnr²crmc bn knvmkjhtn vjrtmlca, nftjkns7

y ; v0∗ v60 t6 ∖ x6v0 t

t ∖ i t6

6 ⇖ y ; v60 t6 ∖ x6 ∖

i t6

6

v60 t

6 ∖ x6 ; y + i t6

6

Jajvchbn ckfns ns kjkfrns cn qucbrcbn7

v60 t6 ∖ x6 ; y6 + y i t6 + i

6 t2

2

v60 t6 ; x6 + y6 + y i t6 + i6

t2

2 (9)

Kchmpuachbn c jqucĺcˑn clmkc, ldjickns c7

v0 ;

x6 + y6 +

i t6

2 (2y + i t6)

t (4)

C Jqucĺcˑn (4) hns b²c n quj prjtjhb²ĳckns, nu sjgc, c vjanlmbcbj mhmlmca jk euhĺcˑn cpjhcs bc bmstc˅hlmcdnrmznhtca, bc caturc j bn tjkpn (ajkfrchbn quj i

≄=, 43 k/s6).

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C Jqucĺcˑn bn Rjkpn

Rjkns quj x6 + y6 + y (i t6) + i6 t2

2 ; (v0 t)

6 ²j c jqucĺcˑn bn hnssn knvmkjhtn. Pcrc bjslnfrmrkns t,

trcfcadchbn-c caijfrmlckjhtj, nftjkns7

i

6

2

t2 + (yi ∖ v60)t6 + x6 + y6 ; 0

Rjkns ukc jqucĺcˑn fmqucbrcbc. Ecĺckns R ; t6. \ufstmtumhbn, pnbjkns rjsnavjr ukc jqucĺcˑn bnsjiuhbn ircu pjan k²jtnbn bj Fdc̀socrc 7

i6

2

R 6 + (yi ∖ v60)R + x6 + y6 ; 0

∅ ; (yi

∖v0)

6

∖2

i6

2 (x6 + y6)

∅ ; 6yi(yi ∖ v60) + v20 ∖ i6x6

R ; v60 ∖ yi ³

∗ 6yi(yi ∖ v60) + v20 ∖ i6x6

i6

6

R ; 6v0i

6

∖

6y

i ³

6

i6 6yi(yi ∖v60) + v

20

∖i6x6

Eczjhbn t ; ³∗ R , tjkns7

t ;

∗ 6

i

v60 ∖ yi ³

6yi(yi ∖ v60) + v20 ∖ i6x6 (=)

(vmstn quj n tjkpn hcˑn pnbj sjr hjictmvn)

C jqucĺcˑn clmkc hns enrhjlj n tjkpn bn ldutj jk euhĺcˑn bc vjanlmbcbj mhmlmca, bc caturc y bn ldutjjk rjacĺcˑn c trcvj, bn calchlj x j bc cljajrcĺcˑn bc ircvmbcbj.

Jqucĺcˑn bn Calchlj K²cxmknLnk n nfgjtmvn bj jhlnhtrcrkns tca jqucĺcˑn, utmamzjkns c Jqucĺcˑn (


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Rca jqucĺcˑn ²j eclmakjhtj rjsnav²ĳvja, vmstn quj7

(3) x ; 0 (nrmijk) nu,

(6)

i

6v6n lns6δx ; ti δ

x ; v60 sjh(6δ)

i (33)

C Jqucĺcˑn (33) ²j lnhdjlmbc lnkn bn calchlj k²cxmkn pcrc n ldutj.

C UJANLMBCBJ BC FNAC HN MH\RCHRJ t7 JSZCĹ CˑN BJ RNVVMLJAAM

Pcrc n l²caluan bc vjanlmbcbj bc fnac nvca hn mhstchtj t, bjsprjzchbn-sj c rjsmstj˅hlmc bn cr, vjrmffilcknsquj c ²uhmlc enrĺc quj ctuc snfrj jac ²j n sju pr²nprmn pjsn P , j c cljajrcĺcˑn c quj jst²c sufkjtmbc ²j cpr²nprmc cljajrcĺcˑn bc ircvmbcbj ∖i, lnhtr²crmc cn knvmkjhtn - lnhenrkj jxpamlcbn chtjrmnrkjhtj.

Emiurc 97 Vjprjsjhtcĺcˑn bc fnac nvca vmcgchbn hn mhstchtj t.

Bj clnrbn lnk c 6 c Ajm bj Hjwtnh , c rjsuatchtj bcs enrĺcs quj ctuck snfrj uk lnrpn ²j miuca cn prnbutnbc kcssc bjssj lnrpn pjac cljajrcĺcˑn c quca jst²c sufkjtmbn, pnssumhbn c kjskc bmrjĺcˑn j sjhtmbn bjssccljajrcĺcˑn. Pnrtchtn, pnbjkns jslrjvjr7

E r ; P ; ∖k i ⇖ k c ; ∖k i

Ajkfrchbn quj c cljajrcĺcˑn ²j c vcrmcĺcˑn bc vjanlmbcbj huk bjtjrkmhcbn mhtjrvcan bj tjkpn, bmvmbmhbnc jqucĺcˑn pnr k, j cpamlchbn jssj lnhljmtn, nftjkns7

bv

bt ; ∖i

Uckns rjsnavjr jssc jqucĺcˑn bmejrjhlmca7

bv

by

by

bt ; ∖i ⇖ bv

by v ; ∖i

=


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v bv ; ∖i by ⇖vy∯

v0y

v bv ; ∖iy∯

y0

by

36v

6vyv0y

; ∖i_y[y

y0

3

6(v6y ∖ v60y) ; ∖i(y ∖ y0)

v6y ; v60y ∖ 6i(y ∖ y0) (36)

C jqucĺcˑn clmkc ²j tckf²jk lnhdjlmbc lnkn Jqucĺˑ cn bj Rnrrmljaam . Umstn quj y0 ; 0, lnhlau²ĳkns bj(36) quj,

v6y ; v60y ∖ 6i y

Jk rjacĺcˑn c δ, rjjslrjvjkns c jqucĺcˑn clmkc, bj enrkc quj7

v6 ; v60 ∖ 6i y(lnssjl6δ) (3


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Emiurc 47 Pnsmĺcˑn bc fnac nvca jk uk bjtjrkmhcbn tjkpn t j c bjlnkpnsmĺcˑn vjtnrmca bc vjanlmbcbj jbc cljajrcĺcˑn.

Nfsjrvj quj c cljajrcĺcˑn ljhtr²ĳpjtc ²j bmrjlmnhcbc pcrc n ljhtrn bc trcgjt²nrmc lurvma²ĳhjc, j c cljajrcĺcˑnircvmtclmnhca vjrtmlca pcrc fcmxn - sjkprj. Jhtrj jacs d²c uk c˅hiuan γ, kjskn c˅hiuan jhtrj c vjanlmbcbjj suc lnkpnhjhtj dnrmznhtca. Pcrc jstcfjajljrkns n rcmn bj lurvcturc, bjvjkns prmkjmrn lnkprjjhbjrcaiuhs lnhljmtns7

3. N k²nbuan bc cljajrcĺcˑn ljhtr²ĳpjtc ²j bcbn pnr clp ; v6

V, nhbj v ²j c vjanlmbcbj hn mhstchtj t j V n

rcmn bj lurvcturc. Pnrtchtn, pnbjkns jslrjvjr,

V ; v6

clp

6. Lnkn n knvmkjhtn hc bmrjĺcˑn dnrmznhtca ²j uhmenrkj, suc vjanlmbcbj ²j lnhstchtj, n quj hns ajvc clnhlaumr quj v0x ; vx pcrc qucaqujr tjkpn t. Jssc nfsjrvcĺcˑn ²j kumtn mkpnrtchtj, pnms lnhlau²ĳknsquj7

vx ; v0x ; v0lnsδ


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Inhĺcajz.

v6 ; v60lns6δ + v60sjh

6δ ∖ 6v0itsjhδ + i6t6

v6 ; v60(sjh6δ + lns6δ)∖ 6v0itsjhδ + i6t6

v6 ; v60 ∖ 6v0itsjhδ + i6t6

v ;

v60 ∖ 6v0itsjhδ + i6t6

5. Prjlmsckns, cinrc, jstcfjajljr c rjacĺcˑn bc cljajrcĺcˑn ljhtr²ĳpjtc clp ; i lnsγ. Nfsjrvj, pjac ffiiurc,quj

lnsγ ; vx

v

; v0lnsδ∗

v60 ∖ 6v0itsjhδ + i6t6.

Anin,

clp ; i lnsγ ; i

v0lnsδ∗


(3:)

9. Cinrc, rjtnrhckns c̀ jqucĺcˑn bn rcmn bj lurvcturc quj bjsjgckns, nu sjgc,

V ;

v6

clp ;

v60

∖6v0itsjhδ + i

6t6

i

v0lnsδ∗


V ;

∗ (v60 ∖ (6i v0 sjh δ)t + i6t6)<

i v0 lns δ (35)

quj hns b²c n rcmn bj lurvcturc bc trcgjt²nrmc jk qucaqujr tjkpn t.

N Vcmn bj Lurvcturc Pcrc c Caturc K²cxmkcJk tjrkns e²ĳsmlns, quchbn c fnac nvca ctmhij c caturc k²cxmkc (y ; dkcx), suc vjanlmbcbj ffihca vjrtmlca(vy) ²j huac. Prnhtckjhtj, bc Jqucĺcˑn bj Rnrrmljaam (36), lnhlau²ĳkns quj7

0 ; v60y ∖ 6i(dkcx)

dkcx ; v60 sjh

6δ

6i (39)

Uckns sufstmtumr cinrc c Jqucĺcˑn (39) clmkc jk (6), bj enrkc quj7

36


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Inhĺcajz.

v60 sjh6δ

6i ; (v0sjhδ)t ∖ it

6

6

Kuatmpamlchbn ckfns ns acbns bc jqucĺcˑn pnr 6i, nftjkns7

v60 sjh6δ ∖ (6iv0sjhδ)t + i6t6 ; 0Lnkn sjh6δ ; 3 ∖ lns6δ,

v60(3 ∖ lns6δ) ∖ (6iv0sjhδ)t + i6t6 ; 0

v60 lns6δ ; v60 ∖ (6i v0 sjh δ)t + i6t6 (34)

Nfsjrvj ctjhtckjhtj quj n acbn bmrjmtn bc jqucĺcˑn ²j jxctckjhtj n hukjrcbnr bc Jqucĺcˑn (35).

Pnrtchtn, mrjkns sufstmtu²ĳ-an pnr v60 lns6δ, bj kchjmrc quj7

Vdkcx ;

∗ (v60 lns

6δ)<

i v0 lns δ

Vdkcx ; (v60 lns

6δ)∗

v60 lns6δ

i v0 lns δ

Vdkcx ; v60

i lns6 δ (3=)

Jxmstj ukc nutrc enrkc bj ldjicrkns jxctckjhtj c̀ kjskc lnhlauscˑn chtjrmnr, lnhsmbjrchbn quj hccaturc k²cxmkc c ²uhmlc enrĺc quj ctuc snfrj c fnac nvca ²j n sju pr²nprmn pjsn, pnrtchtn, miuca c̀ enrĺcljhtr²ĳpjtc. Anin, jslrjvjhbn kctjkctmlckjhtj,

E Vjs ; P ; E lp ⇖ k i ; k clp ⇖ i ; clp

i ; v6

V ⇖ V ; v

6

i

Vdkcx ; v60

i lns6 δ

SZCHRMBCBJ BJ KNUMKJHRN BN LDZRJ

Hjstj t²npmln trctcrjkns fcsmlckjhtj bj lnamscˑn j mkpuasn snermbns pjac fnac nvca burchtj n ldutj.Rjkns quj tjr jk kjhtj quj c fnac nvca bjmxc sju knkjhtn mhjrlmca mhmlmca kjbmchtj c cpamlc ĺcˑn bjukc enrĺc bj lurt²ĳssmkc burcĺcˑn. Ujrjkns quj n k²nbuan bj tca enrĺc ²j jajvcb²ĳssmkn, pcrc quj c fnacpnssc bjmxcr n ldcˑn j cbqumrmr vjanlmbcbj suffilmjhtj pcrc ctrcvjsscr cs trcvjs cbvjrs²crmcs.

3


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Inhĺcajz.

Knkjhtn Amhjcr

N knkjhtn amhjcr bj ukc pcrt²ĳluac bj kcssc k ²j ukc irchbjzc vjtnrmca bmrjtckjhtj prnpnrlmnhca c̀vjanlmbcbj cbqumrmbc pnr jac. N lnhljmtn e²ĳsmln kcms prjlmsn bmz quj c tcxc bj vcrmcĺˑ cn lnk n tjkpn bnknkjhtn bj ukc pcrt²ĳluac ²j miuca c enrĺc rjsuatchtj quj ctuc snfrj jac j tjk c kjskc nrmjhtcĺˑ cn quj

jssc enrĺc .

Jk amhiucijk kctjk²ctmlc, jslrjvjkns7

σ ; k v

nhbj σ ²j n knkjhtn amhjcr, k ²j c kcssc bc pcrt²ĳluac j v c vjanlmbcbj vjtnrmca. Pnrtchtn, n knkjhtnamhjcr jk uhmbcbjs \.M. ²j bcbn pnr (oi.k/s).

Bn lnhljmtn bj knkjhtn amhjcr, jslrjvjkns7

E V ; bp

bt ⇖ E V ;

b

bt

(k v)

⇖ E V ; k c

quj scˑn jxprjssnˑjs jqumvcajhtj rjejrjhtjs c̀ 6c Ajm bj Hjwtnh.

Lnamsnˑjs \mkpajs

Suca c mkpnrtc˅hlmc bj jhtjhbjrkns snfrj knkjhtn amhjcr> C rjspnstc sj b²c quchbn jstubckns lnamsnˑjs.Lnkn bmssj chtjrmnrkjtj, c fnac nvca sj jhlnhtrc pcrcbc hn knkjhtn chtjrmnr cn ldutj. ²J prjlmsnquj ukc enrĺc E sjgc cpamlcbc pcrc quj jssc fnac bjmxj suc mh²jrlmc c ffik bj quj pnssc vmcgcr ct²jcs trcvjs cbvjrs²crmcs. Nfsjrvj quj hn knkjhtn mhmlmca, chtjs bn ldutj, n knkjhtn bc fnac nvca ²j p ; k v ; k 0 ; 0. Kcs, hn knkjhtn quj n omlojr ldutc c fnac, tjkns c cĺcˑn bj ukc enrĺc jxtjrhcquj catjrc brcstmlckjhtj n knkjhtn mhmlmca.

C vcrmcĺcˑn bjssj knkjhtn ²j lcaluacbn pjan quj bjhnkmhchns mkpuasn, nu sjgc, n quchtn c enrĺc bnldutj burchtj uk mhtjrvcan bj tjkpn lurt²ĳssmkn bt catjrnu n knkjhtn mhmlmca bc fnac nvca.

Jk amhiucijk kctjk²ctmlc, jslrjvjkns quj c vcrmcĺcˑn bn knkjhtn (nu mkpuasn) ²j bcbn pnr7

E V ; bp

bt ⇖ bp ; E (t) bt ⇖

e ∯ m

bp ;

e ∯ m

E (t) bt ; M

Ns ²ĳhbmljs m j e mhbmlck ns knkjhtns mhmlmca j ffihca rjspjltmvckjhtj bj cpamlcĺcˑn bjssc enrĺc E . Lnknn mkpuasn ²j c vcrmcĺcˑn bn knkjhtn bj ukc pcrt²ĳluac, pnbjkns jslrjvj˅-an tckf²jk bj enrkc quj7

M ; ∅σ ; σe ∖ σm ; kve ∖kvm ⇖ M ; k(ve ∖ vm) ; k ∅v

Hc pr²ctmlc, lnkn sjkprj c fnac nvca jhlnhtrc-sj jk rjpnusn mhmlmcakjhtj, tjkns quj n mkpuasn sjr²clcaluacbn sjkprj fcsjchbn-sj jk v0, vmstn quj ∅v ; ve ∖ vm ; v0∖ 0 ; v0. Pnrtchtn, pcrc h²ns mkpnrtcquj σ ; k v0 j M ; k v0.

Bc Jqucĺcˑn (33), pnbjkns lnhlaumr quj7

v0

; ∗ i x sjh(6δ)sjh(6δ)

32


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Inhĺcajz.

Anin,

M ; σ ; k

sjh(6δ)

∗ i x sjh(6δ) (60)

Nfsjrvcĺˑ njs7 \jkprj quj enrkns rjsnavjr jxjrl²ĳlmns nu nftjr jxjkpans bj clnrbn lnk ukc rjcambcbj

kcms cprnxmkcbc, bjvjkns hns ajkfrcr quj c trcg²jtnrmc pcrcf²namlc ²j uk pnuln kcmnr quj x ;


16/20

Inhĺcajz.

jptk vx ; L

vx(t) ; L j∖ptk

Pcrc c lnhbmĺcˑn mhmlmca vx(0) ; v0x ⇖ L ; v0x. Anin7vx(t) ; v0x j

∖ptk (66)

j,

v(t)lnsδ ; v0lnsδj∖ptk

v(t) ; v0 j∖ptk (6


17/20

Inhĺcajz.

Knvmkjhtn Ujrtmlca

Bj clnrbn lnk c 6 c Ajm bj Hjwtnh , hn knvmkjhtn vjrtmlca, tjkns quj c enrĺc rjsuatchtj snfrj c fnac²j c snkc bc enrĺc bj rjsmst˅jhlmc bn cr j n pjsn bc fnac. Kctjkctmlckjhtj, jquclmnhckns7

E V ; E Vcr + P

⇖ k c ;

∖ p v

∖k i

Lnkn c ; by

bt, tjkns quj c jqucĺcˑn bc caturc bc fnac nvca mhmlmc-sj jk7

k bvy

bt ; ∖ p vy ∖k i ⇖

bvybt

+ p

k vy ; ∖i

Rjkns ukc jqucĺcˑn bmejrjhlmca nrbmh²crmc amhjcr rjsnav²ĳvja pnr ectnr mhtjirchtj. N ectnr mhtjirchtj ²jukc euhĺcˑn tca quj º(t) ; j

∯ p(t)bt. Pcrc rjsnavjrkns c jqucĺcˑn tjkns quj7

º(t) ; j∯ pkbt ; j

ptk

Kuatmpamlchbn tnbc c jqucĺcˑn pnr º(t), tjkns7

jptk

bvybt

+ jptk

p

kvy ; ∖i j

ptk

b

bt(j

ptk vy) ; ∖i j

ptk

jptk vy(t) ; ∖i

∯ j

ptk bt

jptk vy(t) ; ∖ki

p j

ptk + L

vy(t) ; ∖ki p

+ Lj∖ptk (65)

Lnkn lnhbmĺcˑn mhmlmca, tjkns quj v(0) ; v0. Pnrtchtn,

v0y ; ∖ki

p + L ⇖ L ; v0y +

ki

p

\ufstmtumhbn L7

vy(t) ;

v0y +

ki

p

j∖

ptk ∖ ki

p (69)

v(t) ;

v0(sjhδ) + ki p

(lnssjl δ)j∖

ptk ∖ ki

p (64)

39


18/20

Inhĺcajz.

c jqucĺcˑn dnr²crmc bc vjanlmbcbj bc fnac nvca.

C ffik bj quj nftjhdckns c caturc bc fnac fcstc mhtjircrkns c jquc ĺcˑn bc vjanlmbcbj, vmstn quj v ; by

bt.

by

bt ; v0y +

ki

p j∖ptk ∖ ki

p

y(t) ;

∯ v0y +

ki

p

j∖

ptkbt ∖

∯ ki

p bt

y(t) ;

v0y +

ki

p

∯ j∖

ptk bt∖ ki

p

∯ bt

y(t) ;

v0y +

ki

p

∖k

p j∖

ptk

∖ kit

p + L

y(t) ;

v0y +

ki

p

k

p

(3 ∖ j∖ptk )∖ kit

p + L (6=)

Bcbc c lnhbmĺcˑn mhmlmca y(0) ; y0 ; 0, tjkns7

y(t) ;

v0y +

ki

p

k

p

(3 ∖ j∖ptk ) ∖ kit

p (


19/20

Inhĺcajz.

Jstj tjrkn ²j kumtn mkpnrtchtj!

\jiumhbn n rclmnl²ĳhmn, lnhlau²ĳkns quj7

∖ ptk

; ah

pvy + ki

pv0y + ki

t ; ∖

k

p

ah

pvy + ki

pv0y + ki

t ;

k

p

ah

pv0y + ki

pvy + ki

(


20/20

Inhĺcajz.

td ; ∖k p

ah

ki

p v0y + ki

td ; k

p ah

p v0(sjhδ) + ki

ki (

A Matemática Na NFL

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