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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!
3
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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
2
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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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Inhĺcajz.
Kcs, bcbn quj i ≄ =, 43 k/s6, x ;
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Inhĺcajz.
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).
9
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Inhĺcajz.
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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Inhĺcajz.
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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Inhĺcajz.
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∗
v60 ∖ 6v0itsjhδ + i6t6
(3:)
9. Cinrc, rjtnrhckns c̀ jqucĺcˑn bn rcmn bj lurvcturc quj bjsjgckns, nu sjgc,
V ;
v6
clp ;
v60
∖6v0itsjhδ + i
6t6
i
v0lnsδ∗
v60 ∖ 6v0itsjhδ + i6t6
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.
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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)
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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 ;
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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
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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)
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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 (
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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
(
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Inhĺcajz.
td ; ∖k p
ah
ki
p v0y + ki
td ; k
p ah
p v0(sjhδ) + ki
ki (