7
Dntfnm fm Nmk Hfdkmsfnm Fmtrnhugtnry Kxkrgfsk 7.0
0. Wuppnsk
` p`rtfgok fs dnvfmj wftb gnmst`mt vkongfty v (`onmj tbk `xfs na x). x).
4 vt0 Hfspo`gkdkmt na p`rtfgok fm tfdk t< 4 vt< ∸ Hfspo`gkdkmt na tbk p`rtfgok fm tbk tfdk fmtkrv`o ∁t ( 4 t< ∜ t0) 4 vt< ∜ vt0 4 v ( t< ∜ t0) vkongfty fm tbk tfdk tfdk fmtkrv`o ∸ @vkr`jk vkongfty v ( t< ∜ t0) ∁t ‼ 4 ( t< ∜ t0) 4 v Hfspo`gkdkmt na p`rtfgok fm tfdk t0
Mnw, `s tbk p`rtfgok fs dnvfmj wftb gnmst`mt vkongfty (f.k (f.k., ., wftb gnmst`mt spkkh fm ` jfvkm hfrkgtfnm) fts vkongfty `mh spkkh `t `my fmst`mt wfoo nlvfnusoy lk v.
4 <.5 gd Ú υ r`h s ∜0 75 υ 4 gds ∜0 0;
@s tbk tfp wnuoh lk dnvfmj wftb gnmst`mt spkkh. υ gds ∜0 @vkr`jk spkkh 4 0; Fm 0; s tbk skgnmh b`mh wnuoh rnt`tk tbrnujb 65µ f.k., f.k., tbk hfspo`gkdkmt na fts tfp wfoo lk r <.
∸ Dnhuous na `vkr`jk vkongfty na tbk tfp na skgnmh b`mh fm 0; s.
4 4
r < 0; < < 0;
gds ∜0
8. (`) Vks. Ly gb`mjfmj hfrkgtfnm na dntfnm,
@ms. Xruk. <. @s tbk stnmk wnuoh lk arkk tn `ggkokr`tfnm
umhkr k`rtb‖s jr`vfty ft `ggkokr`tfnm wfoo lk j. j. f.k., 15s tn 7. @ skgnmh b`mh t`iks 0 dfm f.k., gndpoktk nmk rnt`tfnm (f.k., (f.k., rnt`tfnm rnt`tfnm ly `m `mjok na <υ r`h).
∸ @mjuo`r spkkh na skgnmh b`mh <υ r`h 4 15 s
r`h s ∜0 4 υ r` 75
Ofmk`r spkkh na fts tfp 4 r`hfus Ú `mjuo`r spkkh
tbkrk wfoo lk gb`mjk fm vkongfty `mh sn `ggkokr`tfnm. (l) (f) Mn. Fm gurvkh p`tb tbkrk wfoo `ow`ys lk `ggkokr`tfnm. (@s kxpo`fmkh fm tbk prkvfnus `mswkr mn. 7) (ff) Vks. Fm prnckgtfok dntfnm dntfnm tbk p`tb na tbk p`rtfgok fs ` gurvkh nmk wbfok `ggkokr`tfnm na tbk p`rtfgok rkd`fms gnmst`mt. (fff) Vks. Fm gurvkh gurvkh p`tb p`tb tbk `ggkokr`tfnm `ggkokr`tfnm wfoo `ow`ys lk tbkrk. Kvkm fa tbk p`tb fs gfrguo`r wftb gnmst`mt spkkh tbk hfrkgtfnm na tbk `ggkokr`tfnm na tbk p`rtfgok wnuoh kvkry tfdk lk gb`mjfmj.
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Dntfnm fm Nmk Hfdkmsfnm |
;. (`) Xfdk spkkh
4 Gfrgudakrkmgk 4
4 8
Wpkkh <υ Ú 8 gd
4 2 υ 4 <;.07
0gd/s
4
8r <
4<
<υr spkkh
4<
<
υ
v
s
(l) @s p`rtfgok p`rtfgok fs dnvfmj wftb gnmst`mt spkkh na 0 gd/s, fts `vkr`jk spkkh fm `my tfdk fmtkrv`o wfoo lk 0 gd/s. r < | @vkr`jk vkongfty vkongfty|4 X / 8
4
<
υ
0;
spkkh
gds ∜0
4 5.6 gds ∜0 v | @vkr`jk `ggkokr`t `ggkokr`tfnm fnm|4
< X / 8
< < v< υ r
4<
< (0)< υ 8
.<7 gd s ∜< 4 5.<7 1. Hfst`mgk 4 Wpkkh Ú tfdk H0 4 v0t0 H< 4 v< t< H + H< v0t0 + v< t< @vkr`jk spkkh 4 0 4 t0 + t< t0 + t< ( 8 Ú <) + (1 Ú 7) 4 <+7
4 ;.
(wbkrk v 4 spkkh)
Fmtrnhugtnry Kxkrgfsk 7.< 0. @ggkokr`tfnm (huk tn jr`vfty).
v 4 t 7 / 8 hs 4 t7 / 8 ht
8.
0 ` fs pbysfg`ooy gnrrkgt `s ft < jfvks tbk hfspo`gkdkmt na tbk p`rtfgok fm ttb skgnmh (nr `my tfdk umft).
<. st 4 u + `t ∜
∸
s 4
∩ t
s 4
8
st 4 Hfspo`gkdkmt fm t skgnmhs skgnmhss ∜ hfspo`gkdkmt fm ( t ∜ 0) skgnmh 4 ut + 0 `t< ∜ u ( t ∜ 0) + 0 `( t ∜ 0)< < < st 4 st 4 u + `t ‗ `t ‗
<
`
Hfspo`gkdkmt Qkongfty
@ggkokr`tfnm
Xbkrkanrk, tbk jfvkm hfdkmsfnm`ooy fmgnrrkgt.
kqu`tfnm
7 / 8
(jfvkm) …(f) 7
ht 4
+0 t8 7 +0 8
+g
+g : f.k., f.k., s ∔ t : / 8 Hfaakrkmtf`tfmj Kq. (f) w.r.t. tfdk t, 7 h< s 7 8 ∜ 0 4 t ht< 8 ` ∔ t∜0 / 8 ⇜ nr
t : / 8
s) na tbk p`rtfgok ;. Hfspo`gkdkmt ( s) fs
s 4 ( 85 Ú 1) +
0 (∜ 05) 1< <
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01 | Dkgb`mfgs-0
4 85 4 8 s ? 1 s
∸
05
Ft fdpofks tb`t p`rtfgok b`s gndk l`gi `atkr `tt`fmfmj d`xfdud bkfjbt (b (b) jfvkm ly u< b4 < j ( 85)< 4 25 d 25 d < Ú 05
4 ∸
∸ nr
hx
1.
0<; 4 5 ⋋ t +
6.
t 4 <; s <; s 0<; d @vkr`jk vkongfty vkongfty 4 ;s
(hnwmw`rhs) (hnwmw`rhs ) …(f
) hv ht
∸
`
4
@t
t 4 < s
4 ; ∜
` 4 ; ∜ < Ú <
4 0 d/s<
4 ( 85 ∜ 05t) ht x 4 ∩ ( 85 ∜ 05t) ht x 4 85t ∜ ;t< + g
nr
(hnwmw`rhs)
4 <; d/s v 4 05 + ;t ∜ t<
ht
nr
0 < jt <
⇜
05.
H 4 25 + (25 ∜ 15)
4 055 d 055 d v 4 85 ∜ 05t hx 4 85 ∜ 05t
@ms. @ms. Xruk.
Arnd Kq. (f), hx 4 05 + ;t ∜ t< ht
∩ (05 +;;t t ∜ tt ) ht
@s `t t 4 5 tbk v`ouk na x fs x fs zkrn.
∸
x4
g 4 5
nr
x 4 05t +
@s, `t
t 4 5 tbk v`ouk na x fs x fs zkrn
∸
x 4 85t ∜ ;t
<
Anr x tn x tn lk 15 d.
∜ 2 t + 0< 4 5 ∸ t 4 < s nr 1 s Hfspo Hfspo`gk `gkdk dkmt mt fm tfdk tfdk t t vkongfty 4 :. @vkr`jk vkongfty t<
∸
x 4 05t +
Xbus, `t
t 4 7 s
0 < `t < t 0 `t <
ut +
4u+ ∸
↜ 00. u ↜ `
4 v0 + `t `t 4 v< ∜ v0
@vkr`jk vkongfty vkongfty 4
4
Hfspo Hfspo`gk `gkdk dkmt mt fm tfdk tfdk t t
; <
<
t
0 < `t < t 0
∜
7
+g
t7 7
; < (7) <
∜7
4 < ]f d/s 4 (< gns 15µ ]f + < sfm 15µ c]) d/s< 4 (0 ]f +
]
7 c) d/s< y
t v0t +
∜
7
4 75 + <<.; ∜ 6 4 87.; d
v<
2.
<
x 4 (05 Ú 7) +
t
4
<
g 4 5
15 4 85t ∜ ;t< nr
<
` 4 < d/s < 15µ
7
7
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Dntfnm fm Nmk Hfdkmsfnm |
↜ |v| 4 8< + 0< 4 < : d/s ↜ ↜ 0↜ s 4 u t + ` t< < 4 (< ]f) < + 0 (0 ]f + 7 ]c) << <
4 8 ]f + < ]f + < 4 1 ]f + < ↜ |s |4
∸ 0<. U`rt F ↜
U`rt FF Vks. @s kxpo`fmkh lkonw. lkonw. ↜ ] ] v 4 < f +
tbk p`rtfgok fs < f d/s< `mh tbk `ggkokr`tfnm fs < c d/s c d/s< ↜ ∸ s (`t t 4 0 s) 4 (< ]f Ú 0) + 0 (< ]c) 0< <
]
7 c
4 (< ]f + ]c) d
]
7 c 07. x
71 + 0<
4 8 7 d ↜ ] ] v 4 (< f +
…(f)
<
∸
y 4 x <
nr,
x<
4 8y
x 4
∸
hy
∸
ht ↜ v
Xbus,
∩
4
↜
4 < f.k., v x 4 < ]f
y 4 t<
Arnd Kq. (f), ↜ hs 4 (< ]f +
4
(Xbk `lnvk fs tbk kqu`tfnm tn tr`ckgtnry)
hv 4 < c] ht ↜ ] ` 4 < c d/s<
↜ s
↜
4
↜
4 v x + v y 4 (< ]f +
c+ g
< ]
X`ifmj fmftf`o hfspo`gkdkmt tn lk zkrn. ↜ ] ] s (`t t 4 0 s) 4 (< f + c) d
↜ `
4
↜ hv ht
4 < c] d/s<
Fmtrnhugtnry Kxkrgfsk 7.7 0. @t t
4 t0 v
s
ΰ
t <
μ
μ t
t 0 5
0:
β
t 0
t <
v 4 t`m β @s β ? 65µ vt fs + fvk.
ρ
t
@ggkokr`tfnm @ggkokr`tfnm `t t 4 t0 3 `t0
4 t`m ΰ
@s ΰ ? 65µ ` t fs + fvk gnmst`mt.
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02 | Dkgb`mfgs-0 0 dj< t< < < f.k., f.k., IK ∔ t . Zbfok jnfmj up tbk vkongfty wfoo jkt ∜ fvk lut tbk IK wfoo rkd`fm. IK wfoo rkhugk tn zkrn `t tfdk < t wbkm tbk p`rtfgok rk`gbks fts fmftf`o pnsftfnm.
Gnrrkspnmhfmj vkongfty-tfdk wfoo lk
wnuoh lk jt. jt. IK na p`rtfgok wfoo lk
Qkongfty (d/s)
8
2
t (s)
IK
8.
t 4
IK 4
0 dj< t< <
tfdk
b ( t ∜ <)
4 t`m β 4
⇜
< (< ∜ 0) b 4 < ( t ∜ <)
` (d/s<)
4
0
<
β
4 djb
0
7. Wpkkh na l`oo (cust lkanrk d`ifmj afrst
<
β
( t ‗ < ) t
gnoofsfnm wftb aonnr)
4 < jb 4 4 85 d/s 85 d/s
Xfdk t`ikm tn rk`gb jrnumh < Ú 25 4
48s
Wpkkh na l`oo (cust `atkr afrst gnoofsfnm wftb aonnr) 4 85 4 <5 d/s < Xfdk tn `tt`fm d`xfdud bkfjbt ∜ <5 4 < s t 4 ∜05
∸ Xfdk anr tbk rkturm cnurmky tn aonnr 4 < s. Wpkkh (d/s)
8
b
< Ú 05 Ú 25
2
t (s)
U`rtfgok wfoo `tt`fm fts fmftf`o vkongfty f.k., mkt fmgrk`sk fm vkongfty na tbk p`rtfgok wfoo lk zkrn wbkm, `rk` umhkr `-t jr`pb 4 5 (0 + <) Ú < ( ∜ b) ( t ∜ <) <
+
<
45
∜ ( t ∜ <)< 4 5 nr ( t ∜ <)< 4 7 nr t ∜ < 4 ° 7 nr t 4 < ° 7 @ms . @ms . @t tfdk t 4 < + 7 s (t 4 < ∜ 7 mnt pnssflok). nr
7
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Dntfnm fm Nmk Hfdkmsfnm |
s 4 (v @
nr
sfm 75µ 0;5 t 0 755
∜ vL ) t
s
β t
7. Fm afjurk, u 4 spkkh na ln`t
v 4 spkkh na rfvkr aonw L
G @gtu`o p`tb na ln`t
Ln`t s`fofmj hfrkgtfnm
nr
u h 4 855 d
⇜
@ v
+ v<
05 d/s
4
05 d/s
Ú ( 855 d )
4 25 d 8. Okt G lk tbk pnfmt `onmj wbfgb pfont
sbnuoh bk`h tbk po`mk.
` L 4 < d/s<
v L 4 0 d/s
Hrfatfmj huk tn wfmh
∜ 0 4 < d/s Fmftf`o hfspo`gkdkmt na @ w.r.t. @ w.r.t. L L 4 ∜ 05 d
@t tfdk rko`tfvk hfspo`gkdkmt hfspo`gkdkmt na @ na @ w.r.t. w.r.t. L L 0 s 4 ∜ 05 + < t + ( ∜ 0) t< < s4
∜ 05 +
Anr s tn s tn lk dfmfdud hs 45 ht < ∜ (5.; Ú
f.k., f.k., Wpkkh 4 0;5 d/s
∜ 0 d/s<
Qkongfty na @ w.r.t. @ w.r.t. L L 4 7
nr µ 5 7 β
L
` @ 4 0 d/s< ,
nr
L d i 5 5 ;
@
@ggkokr`tfnm na @ w.r.t. @ w.r.t. L L 4 0 ∜ < 4
u < d/s
Mnrtb
05 d
;.
v @ 4 7 d/s,
4 85 s v LG 4 @L
0
<626 s 4 <626 s ;5 dfm 4 ;5 dfm
@G u<
∜ 75µ + β) ;55 Ú 057
;555 0 <<8 4 + 7⋋ 0 7t < 0; < 0; ;555 4 5.;;:: 7 t ;555 t 4 7 Ú 5.;;::
Xfdk tn grnss rfvkr
4 @L 4 LG 4 u v 855 d 4
sfm (025µ
<5t 4 sfm β <5 β 4 sfm ∜0 0 0; 0; sfm (75µ + β) 0 β Mnw 4 755 t ;55 Ú 057 <<8 ;555 nr 4 sfm 75µ gns β + gns 75µ sfm β 7t
t`m β 4 (v (v ‗ v L) @ ‗v
5
4 sfm β 4
06
∸
sdfm
4 ∜ 05 + (< Ú <) ∜ 5.; Ú (<)< 4 ∜ 05 + 8 ∜ <
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<5 | Dkgb`mfgs-0
@FKKK Gnrmkr Wulckgtfvk Pukstfnms (Okvko 0) 0. (`) H 4 v0t0
+ v< t< 4 15 id Ú 0 b + + 25 id Ú 0 b b b < 4 055 id
(l) @vkr`jk spkkh 055 055 id
4
0.; b
4 11.1: id/b. 11.1: id/b.
<. (`) Hfspo`gkdkmt fm afrst twn skgnmhs
4 ( 8) (<) + 0 (1) << <
4 <5 d ∸
@vkr`jk vkongfty 4
<5 d
4 05 d/s
(l) Hfspo`gkdkmt fm afrst anur skgnmhs 4 ( 8) ( 8) + 0 (1) 8< 4 18 d <
∸ Hfspo`gkdkmt fm tbk tfdk fmtkrv`o t 4 < s tn t 4 8 s 4 18 ∜ <5 4 88 d 88 d 88 d ∸ @vkr`jk vkongfty 4 4 << d/s.
7. Okt tbk p`rtfgok t`iks t tfdk tn rk`gb
(l) @vkr`jk vkongfty
4
4
Mkt hfspo`gkdkmt hfspo`gkdkmt Xfdk Xfdk t`ikm t`ikm
∜ 15 d 1.0s
4 ∜ 6.2 d/s 6.2 d/s (hnwmw`rhs) 4 6.2 d/s Χ vt vkongfty 4 8. @vkr`jk vkongfty Χt vt + < vt5 + 7vX <.;v <.; v 4 5 t5 + t5 + X ;t5 + <.;X 4 7 t5 + 7 X ∸ nr X 4 8 t5 ;. (`) @vkr`jk `ggkokr`tfnm
4 4
Afm`o Afm`o vkon vkongf gfty ty ∜ fmftf fmftf`o `o vkon vkongf gfty ty (8
+ 2) s
5∜5 0<
4 5 d/s< v (g) ` 4 t`m β 4 d`x 8
v v d`x
jrnumh. +<5 d/s ‗ 15 d
b
β 8s
2s
0
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Dntfnm fm Nmk Hfdkmsfnm |
(l) @s tbk p`rtfgok p`rtfgok hfh mnt rkturm l`gi hfst`mgk tr`vkookh fm 0< s
1.
4 Hfspo`gkdkmt `t 0< s ∸ @vkr`jk spkkh 4 2 d/s. <0 (`) \` \`hfus ( \) na gfrgok 4 d
:. Unsftfnm vkgtnr `t t 4 5 s ↜ ]
r0 4 (0 f
∸ Gfrgudakrkmgk na gfrgok 4 <υ \ 4 < Ú << Ú <0 d
(`) Hfspo`gkdkmt arnd t 4 5 s tn t 4 8 s ↜ ↜ 4 ( r< ∜ r0 )
<<
4 (1 ]f + 8 ]c) ∜ (0 ]f + < ]c)
4 1d Wpkkh (v) na p`rtfgok
∸
Hfst`mgk dnvkh dnvkh ly p`rtfgok fm < s 4 < d
Xbus, `mjok tbrnujb wbfgb tbk p`rtfgok dnvkh 4 < Ú <υ 4 <υ 4 0<5µ 1 7 D`jmftuhk na @vkr`jk vkongfty hfspo`gkdkmt 4 D`jmftuhk na hfspo`gkdkmt Xfdk ( 4 < s)
(t 4
0<5µ N
↜ vL
4 @L 4 <\ sfm 15µ <
4 \ 7 <
<
∸
@vkr`jk vkongfty 4
]
(; f
+ < ]c) d 8s
4 (0.<; ]f + 5.; ]c) d/s (l) @vkr`jk `ggkokr`tfnm Afm`o fm`o vkon vkongf gfty ty ∜ Fmft Fmftf` f`oo vkon vkongf gfty ty
4 4
@( @(t 4 5 s) L
4 (; ]f + < ]c) d
4 0 d/s
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Dntfnm fm Nmk Hfdkmsfnm |
4 05 ]f ∜ < ]c
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4
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Dntfnm fm Nmk Hfdkmsfnm |
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Dntfnm fm Nmk Hfdkmsfnm |
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