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YpasHel!ne (6) |
H Bhipa:>KaeT 3atWH oeu:J~Cenu;z tnOIIKU M Bl\OJib Tpa- |
eKTopmi. |
• |
TaKHNI o6pa30\I, |
'IT06hi3J :J.aTh ,nBmKeHne TO'IKHecTeCTBeHHbiM crro- |
co6oM, na.no 3a,l\aTb: 1) TpaeKTopn:D TO'IKH;2) HaqaJio OTC'leTaHa TpaeKTvpnn C yKa3aJII!e'vlIIOJIOJ!OITeJibHOrO H OTpHUaTeJibHOfO HanpaBJiem:;l OTC'leTa; 3) 3aKOH ,nBmKeiiMI TO'IKH B,n0•1b TpaeKTOplll! B Blf)l.e s=f (t).
3al\!eTHM, 'lfOBeJIH'IHHaS B ypaBHeHHH (6) OIIpe,neJIHeT IIOJIO)KeHHe ,nBnMyr.uei:icH T04IOI, a He rrpof•.neHHhiH e10 rryTh. HanpHMep, ecmi ToqKa, .nsuraHcb H3 Ha4aJia 0', ,1\0XOJ,HT .no noJioJKeHnH M 1 (puc. 115),
a 3aTel\l, nepeMei.UaHCb B o6paTHOM HarrpaBJiei-IIHJ, IlpHXO)IJIT B IIOJIO.fKeHPe AI, TO B 3TOT \!O~.teP.T ee I<Oop)l.HHaTa s=O'M, a rrpoi1,neii!IhiH 3a
Bpe;,tH JlBmKeHHH |
nyTh 6:.;r,eT paseH O'M1 +M1A1, T. e. ne |
pasen s. |
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§ 38. |
BEI(TOP |
CI(OPOCTH |
TOlJk.H |
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0.J;HOH H3 OCIIOBHbiX |
KHHeMaTlflieCK!IX |
XapaKTepHCTHK |
)l.BHJKe- |
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HHH TO'IKH HBJIHeTCH BC'KTO!)HaH BeJIH4IIHa, |
Ha3biBae~taH CK0fJOCn1b/O |
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mo111\U. |
Bse)l.eM cHaYa.la rroHnT>re o cpen.Hci1 CKopocTH TOYKII 3a KaKoti- |
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HH6y)l.h |
npOMe)KYTOK BpeMeHH. I1yCTb ,nBH.lKYI.UaHCH TO'IKa HaXO,!l;HTCH |
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Pnc. |
l 16 |
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B MOM:eHT Bpel\IeHH t |
B liOJIOI!<eHl!H ;\1, orrpe,neJIHe:VIO'.lpa,nnycoM-BeKTO- |
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p0\1 r, |
a B MO\leHT tl |
rrpHXOJ,l!T B IIO.'IOlliei-IHeM1> Oi1pe,neJIHeMoe BeKTO- |
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pO\! t 1 |
(pnc. 116). Tor.J,a nepeMelQeHHe TO'IKH3a npo'<IeMyToKBpeMeHH |
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At=t1-t orrpe,neJHieTCH B~Kropo\t |
JH.\11, Koropr:,p"f 6y,ne'II Ha3hiBaTh |
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8CKtnOpO"H nepeAte~WHW! nlO'IIW. 3TOT BeKTOp HarrpaBJTeH IIO XOp,!l;e, eCJIH T04Ka ,ll,Bl'1KeTC5I KpiiBOJIHHeilHO (Dl1C. 116, a), H B)l,OJib CaMOi'r TpaeKTOpH!I AB, KOr,L(a .J:B:Ii-KeHne HBJIRercH rrpHMomme{IHbm (p11c. 116, 6). l1s Tpeyro.1hiiHKa 01H \11 Bli.J:Ho, qro r+.wiH1 =G"; cJie,!J;oBaTeJibHo,
MM1 = rl-r=~r.
0THo::neHHe BeKTopa nepeMer.uemiH T04KH K cooTBeTCTBYIOI.UeMy IIpOl<ieJKyTKY spe\reHH :I,aeT BeKTOpHyl-0 Be,TIH4IIHy, Hi13b!BaeMy10
1tCU ITO \10,!l;yJIJ-O 11 HanpasJieHIUO CKOpOCtnb/0 tnO'IXU Sa rrpoMe.tKyTOK BpeMeHH Af:
(7)
4* |
!79 |
HarrpasJieH BeKTOp Vcp TaK )Ke, KaK n BeKrop MMl! T. e. rrpn KpnsoJIHHeiiHOM JI:BH)KeHHH BJI:O.Tib xopJJ.bi MM1. B eTopony JI:BH)KeHHH TO'IKH,
a IIpii rrpHMOJIHHeHH0\1 )l.BH)KeHIUI- B)I:OJib eaMOH TpaeKTOpHII ( OT )l,e- JieHHH Ha ~f HarrpaBJieHHe BeKTOpa He H3McHHeTeH).
QqeBH)l.HO, 'ITO '!eM MeHhllie oy)I:eT rrpoMe)KyTOK BpeMeHII ~f, )l.JI51
KOTOporo Bbi'IHeJieHaepe)l.HHH |
eKopoeTb, TeM BeJIH'IHH.aVcpoy.n.eT TO'I- |
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nee xapaKTepH30BaTb )l.BI!iKefiHe TO'IKH. 4TOObi |
IIOJIY'IHTb TO'IHYfO |
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XapaKrepHeTHI<y |
)l.BHiKeHIIH, |
BBO)I:HT |
IIOHHTHe 0 |
C!WpOCtnU |
tnOiiKU 8 |
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iJaHHbtu MOJteHm epe.tteHu. |
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CKopoerbfO _2'04KHB )I:aHHhiii MOMeHT speMer-m t |
HasbrB~TCH |
seKTop- |
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naH BeJIH'IHHaV, |
K KOTOpoii erpeMHTeH epe)l.HHH eKopoeTb Vcp IIpH eTpeM- |
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JieHHl-1 rrpoMe)KyrKa speMenH 11t K nymo: |
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- |
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. |
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v= hm (vq,) = |
ltm |
M. |
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tl.t--..0 |
M->-0 |
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I |
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Tipe)I:eJI ornoUJeHHH L\rl M rrp11 M -+ 0 npeJI:eTaBJIHeT eo6oii rrepsy10 |
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npOH3BOJ\IIyfO OT BeKTOpa rITO apryMeHTY f |
H OOO:JHa1-IaereH, J{aK H npo- |
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l13BO)l.H3H OT eKaJIHpHOH tPYHKUHH, CHMBOJIOM drtdt. 0KOHqareJibHO IIOJiyqaeM
(8)
I1TaK, eeKtnOp CKOpOCtnU tnO'IXU 8 aaHHbLU MOMeHm Bpe.MeHU paeet-t nepeou npou3eoiJHou om paiJuyca-eeKmopa moqKu no epe.MeHu.
TaK KaK npe,7JgJihiibiM narrpasJieHHeM ceKyrueii MM1 HBJIHeTeH Kaeare.rrbHaH, 10 eeKmop cKopocmu moliKU 6 iJaHHbtu MOAteHm epeMeHu Ha- npae.llet-t no Kacame.llbHOu K mpaeKmopuu mottKu 6 cmopoHy iJem/CeHUJl.
<J>op111yJia (8) I10Ka3biBaeT TaKtKe, 'ITOBeKrop eKopoeTH V paBeH OTHOUJeHHIO 3JieMeHTapnoro nepeMerueHHH TO'IKH dr, HarrpasJieHHoro no
KaeaTe/IbHOH K rpaeKTOpHH, K eooTBeTCTBYIOIUeMy rrpoMe)KyTKY BpeMeHII cit.
f1p11 npHMOJIHHe}moM )l.BHJKeHHH ser<rop eKopoern vsee speMH narrpaBJien B)I:OJib npHMOH, ITO KOTOpOH )I:BH)KeTeH TO'IKa, H MOJKeT H3MeH5!TbeH JIHillb 'IHCJieHHO; npn KpHEOJIHHeHHOM ,I\BH)KeHIH1 KpoMe 'IHeJIOBOfO 3Ha'IeHH5! Bee BpeMH !13MeHHeTeH 11 HarrpaBJieHHe BeKTOpa eKOpoeTII TO'lKH. Pa3Mepnoerb eKopoeni LIT, r. e. )l.JIIIHa/spe\1H; B Ka-
'IeCTBe e)I:HHHU H3MepeHHH rrpHMeHH!OT oobi'IHO Mle HJIH KM/'1. Bonpoc o6 orrpe.n.eJieHHH MOJI:YJIH eKOpoeTH oy)I:eT paeeMOTpeH B § 40 II 42.
§ 39. BEI(TOP YCI(OPEHIHI TOlli(M
YeKopeHHeM TO'IKH Ha3biBaeTeH BeKTOpHaH BeJIH'IIIHa, XapaKTepH3YfOJUaH 113MeHeHHe e Te'IeHHeMBpeMeHH MO)I:yJIH H narrpaBJieHHH CKOpoeTH TO'IKH.
DyeTb B neKoTopbrii MOMeHr speMeHH ~ JI:BH.tKYIUaHeH TO'IKa Haxo- )l.HTeH B II0•10.tKeHHH M H HMeeT eKOpOCTb V, a B MOMeHT f1 rrpHXO,I\HT 1\
100
llOJIO)KeHHe Ml H HMeeT CKOpOCTb vl (pnc. 117). Tor,n:a sa npoMemyTOK BpeMenn M=t1- t cKopocTb TOIIKII noJiyiiaeT rrpnpaur.enne L1v=v1-V.
,UJI5l IIOCTpOeHH5I BeKTOpa f1v OTJIO)KHM OT TO'IKHM BeKTOp, paBHbHi Vlt
H IIOCTpOHM rrapaJIJieJiorpaMM, B KOTOpOM |
}l:HaroHaJib!O |
oy,n:eT Vlt a |
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OJJ;HOH H3 CTOpOH i). Tor,n;a, |
O'lel3HJJ;HO, BTopa5! CTOpona H oyJJ;eT H306pa- |
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maTb BeKTOp f1v. 3aMeTHM, |
'ITO BeKTOp |
M |
V |
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L1v acer,n:a narrpaaJien a cTopoHy Bor- |
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HYTOCTH TpaeKTOPHH. |
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0THO!Ileime IIpHparn:eHIUI BeKTopa |
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cKopocTn Sv K cooTBeTcTayiOW:eMy npo- |
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MemyTKY BpeMenn L1t onpe,n:eJI5leT eeK- |
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mop cpeiJHeco ycKopeHuJl mottKU 3a .3TOT |
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npoMemyToK BpeMenn: |
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acp= i1V;M. |
(9) |
PHc. |
117 |
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BeKTop cpe,n:nero ycKopenn5l nMeeT To me HarrpaBJieHne, 'ITOH BeKTop Sv, T. e. nanpaaJien a cTopony aornyToCTH TpaeKTopmr.
YcKopenueM TO'IKHB )J.aHHbiH MOMeHT Bpe'\1eHHt Ha3biBaeTCH BeK-
- -
TopnaH aeJiwmna a, K KOTopofi cTpe'\1HTCH cpe)J.nee ycKopemre acp npn cTpeMJieHim rrpoMemyTKa DpeMeHH L1t K nyJIIO:
- . |
!:!.v dv |
a= hm |
Tt=ctr |
t.t-+0 |
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HJIH, C yqeTOM paBeHCTBa (8), |
(10)
CJie,n:oBaTeJibHo, eeKmop ycKopeHUJl mottKU e iJaHflbtu MOMeflm epe.MeHu paeeH nepeou npou3eoonoi1 orn eeKmopa cKopocmu uAu emopou npou3eoiJflou om paouyca-eeKmopa mottKu no epeMeHu.
Pa3MepnocTb ycKopemm LIP, T. e. )J.mma/(apeMH) 2 ; B KatieCTse e,n:HHHUbi H3MepennH npHMeHHeTcH o6brqno M/c.2 •
lb 4JopMyJibl (10) CJie,n:yeT TaK)Ke, qyo BeKTOp ycKopeHHH TOqKH ~ paaen OTHOlllcHmo 3JieMeHTapnoro npupanrennH BeKTopa CKopocnr dv J{ COOTBeTCTBYIOW,eMy npoMemyTKY BpeMeHll dt.
Hafi,n:eM, KaK pacrroJiaraeTCH BeKTop ano OTHOllleHliiO K TpaeKTOpHH TOtiKH. DpH rrpHV!OJIJ1HeHH01! JJ:BH)KeHHH BeKTOp aHanpaBJieH B)J.OJib rrpRMOH, IIO KOTOpofi JJ;BHmeTCH TO'!Kd.EcJIH_TpaeKTOpHeH TOtiKII HB_:IHeTCH IIJIOCKa5I KpHBaH, TO BeKTOp ycKopeHH5l a, TaK )Ke KaK H BeKTOp acp• JiemHT B ITJIOCIWCTH 3TOH Kpl1BOH H HarrpaBJieii B CTOpOHY ee BOfHYTOCTH. ECJIH TpaeKTOpHH He 5lBJIHeTC5I IIJIOCKOH Kp!IBOH, TO BeKTOp acp HarrpaBJieH B CTOpOHY BOrHyTOCTH TpaeKTOpHII H Jie)KHT B IIJIOCKOCTH, rrpO· XO)J.HW:eii IIepe3 KacaTeJibHY!O K TpaeKTOpHH B TOtiKe M H npHMYIO, napaJIJieJibHYIO KaCaTeJibHOH B COCe)J.HeH ToqKeM1 (pHc. 117). B rrpe}\eJie,
101
IWrJI:a TOtii<a M 1 3Ta rrJio.::KoCTb saHHMaeT no.riomeHHe TaK H33b!Bae~1:0ll conpuKaCatOU{eiicJl nAOCKOCmU, T. e. IT.10CKOCTH, B KOTOpOH npoHCXOJ(HT 6ecKOHetiHO 1\iaJib!H ITOBOpOT KacaTeJibHOH K TpaeKTOpllli npn 3Jie'<1eHTapHoMnepeMern.eHim .D.BIDI<yw.e:ics:r TO'!KH*. CJieJI:oBaTeJib-
Ho, B o6rn.eM cJiyqae 6eKmop ljCKOpeHUJl alleJ.:CUtn 6 conpuKaCaiOU{CUCH. nAocKocmu u nanpa6AeH B cmopoHy 6omymocmu Kpu60ii. orrpeJ(eJieHHH MO.D,YJil'IycKopeHI1H oy.D,eT paCCMOTpeH B § 40
§ 40. OUPEAEJIEHHE CKOPOCTH H YCKOPEHH51 TOI.fKH llPH KOOPJ],HHATHOM CUOCOiiE 3AJ],AHH51 J],BHJKEHH51
HafmeM, KaK BbitiHCJIHIOTCH CI<OpOCTb n ycKopeHne TOtiKH, ec.rm e~ )l.BHmesne sa.naHo ypaBHeHIU!MH (3) Hmi (4). Borrpoc o5 orrpe.neJiemm TpaeKTop!m B 3To:vr cnyqae ObiJI yme pacc:.rorpen B § 37.
<lJop:VlYJibl (8) orrpeJ(e.lJHIOW,He 3Ha'leHHHV II a, COJ(epmaT npoii3BOJ)-Ibie no BpeMeHH oT BeKTopon r H V. B paBeHCTBax, co,'l;epmaIIJ.HX rrpOI13BO)J,Hble OT BeKTOpon, rrepeXO!l; K 33BIICII?-!OCTHM MemJ(y II\ rrpoeK,l!Ul:VIH ocyrn.ecrBJIHeTCH c rroMOUJ.biO cneJI:yiOrn.ei1 Teope:vrbl: npoCKtfUJl npou360ot-wii om eeKmopa rta ocb, Nenoo6UJICHyto 6 oaw-wii cucmeMe Otncttema, pasHa npOU3600flOU Om npoeXI{UU otU/JcpepeHI{Upye.MOZO eeKmopa Ha my JJCe ocb, T. e.
ecJIH |
- dp |
dox |
dp,1 |
rlpz |
( 11) |
q= CiT' TO |
qx= dT' |
q:J= dT' |
qz=Tt • |
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I. |
0 n p e .n e JI e H II e |
c K o p |
o c T II T o q K II. |
BeKTop CKO- |
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pocnr TO'lKIIV=dr)ut. 0TCIOJ.3 Ha OCHOB3Hill! ¢op:.\1YJ1 (11), Y'HlTb!BaH,
'ITOrx=X, ry=lj, rz=Z, H3HJ(eM:
dx |
dq |
dz |
(12) |
Vx=dt' |
VY=dt' Vz=(it |
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W!II
( 12')
r.ne TO'!KaH3.ll. oyKBOH eCTb CHMBOJI .il.HcpcpepeHUIIPOBaH!11'!no Bpe:.\IeHIL TaKHM o6pa3oM, npoetcl{UU cKopocmu mo'tKu Ha ;~oopouHa:nta:te ocu pOfJHbt nepebtJtt npou3600HbtM om coom6erncmeytOw,ux Koop8w-wm rnc<tKU no 6pcMeHu.
3HaH rrpoeKUHH CKOpOCTH, Haft.n;eM ee :\WJI:y.!Jb H narrpaBJieHIIe (T. e.
~' y, KOTOpbie BeKTOp Vo6pa3yeT C IWOp.D,HHaTHblMH OCHMH) ITO <PopMyJiaM
v= Vv;+v~+v;; |
} |
(13) |
|
cos a= vxfv, cos~= r.'ylv, |
cosy= Vz/V. |
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* ,UJJH npocTpaHCTBemwil: KPHBOH (HarrpHMep, n.1H BllHTosoii .'ll!H!!H) s Ka>Knoii TO'!Ke Kpl!BOH 6yneT CBOll COITp!!KacaiOID.aHCll ITJIOClWCTb. Jl:MI n.~OCl{Or'\ KpHBOH C0-
11pHKaCaiOID.allCll llJIOCKOCTb COBrranaeT C llJIOCKOCTbiO 3TO!f Kp!!BOH H HBJIHeTCH o6rn.eii IVIH DCCX ee TO'IeK.
102
2. 0 IT p e J( e JI e H H e y C K 0 p e H H H T 0 q K H. !WpeHHH TOqKH a= dVfdt.
T. e. npoeKt{UU yc1wpeHtL"l mottKU na KoopiJunamrtbte ocu pasnbl nepebw n,JOU360dHbV4 om npoeKt{Uii cKopocmu UAU amopbu.t npou3eodHbU.t om cwmeemcmey10u~ux KoopauHa:n mottKu no apeAtenu . .Mo,J,yJIL H HarrpaBJie-
H:re ycKopeHHH Hai!JJ,yTCH H3 cpop!11YJI
a= Va~-,at -f-a~; |
cos l'r = az/a, |
} |
(15) |
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cos a 1 = axfa, cos ~1 = ayfa, |
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r,J,e ah |
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1, ''hyrJILI,- |
o6pasye:vrhle |
BeKTopoM yCKopeHHH c Koopnu |
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B |
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hc:THbll\iH |
OCH:\1!!. |
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11TaK, eCJI!I J(BH:>KeHlie TO'IKH33,!l,3HO B )l,eKapTOBbiX llpHMOYfOJIHib!X |
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KOOp,:l;!!HaTaX ypaBHeHHHvlil (3) HJIH |
(4), |
TO CKOpOCTb TOtiKH orrpe.'l,eJIH- |
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eTCH no ¢op\JyJia'1 (12) H (13), a ycKopenne- no q)opMyJraM (14) n
(15). |
Dpn 3TOV1 a c.riytiae .'J.Bmxemm, rrponcxoll.HUJ.ero B OJJ.HOH miOCKO- |
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cTI!, |
ao acex cpopMyJiax .ll.O.'DhHa OLITb OT6pomeHa rrpoeKUHH |
Ha ocb z. |
B c.1yqae me npHMO.'IHaelmoroJJ.BIImemm, KoTopoe 3aJJ,aeTc5I O,ll.HHM |
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ypaaHenueM x=f(t), 6y,.J,eT |
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ch: |
(16) |
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Vx = dt ' |
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PaseHCTBa (16) If orrpe,:J;e,TJHIOT snaiieHHH cKopocTH H ycrwpemm TO'IK!i B 3TO.M CJlyqae.
§ 41. PElliEHHE 3A}],AlJ KHHEMATHKH TOlJKff
3aJJ,atiH, pemae:vib!e MeTO,!l,3\IH KHHeMaTHKH TOtiKH, MoryT COCTOHTb B orrpel('.:~.'IeH!III TpaeKTOp!HI, CKOpOCTH HJIH yc;wpeHHH TOtiKH, B 0Tb!CK3HIIH apeMeHH, B Te'leHHeKOTOporo TO'IKarrpOXO,D,HT TOT HJIH H!;fOH ITYTb, H.llf rryTH, rrpOXO,J,mwro 33 TOT II.'IHHHOH npoV1emyTOK BpeMeHU, II T. IT.
DpemJJ,e qe:,r pemaTL ,TJI06yiO H3 TaKoro poJJ,a 3aJJ.ati, HaJI,o ycTaaoBllTL, rro KaKmty saKoHy JI:BIImeTcH TOtiKa . .3ToT 3aKoH MO>KeT OhlTb HerwcpeJJ.CTBeHHO 33;!,3H B yc.10BI15IX 3a,n;aqn (33)J,3'IH 47, 48) HJIH me H3 ycJioBHH 3aJJ,aliii onpeJI:eJieH (3a,1,aq11 49, 50).