kinematics theory_e
TRANSCRIPT
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YEXWHGW
"lnjhseoulnrpeyshgs.hj" >
LKGENJHGWLkgenjhgs hs tek arnjge `d peyshgs wehge cknbs whte tek gnusk njc kddkgts `d l`th`j `d n
pnrthgbk, rhmhc `afkgts njc ckd`rlnabk a`chks ktg. Lkgenjhgs hs gbnsshdhkc ujckr tw` strknls
jnlkby W t n t h g s njc C y j n l h g s . Cyjnlhgs hs durtekr chvhckc hjt` Ohjklnthgs nj c Ohjkthgs.
> . L@SH@J NJC QKWSL`th`j hs n g`lahjkc pr`pkrty `d tek `afkgt njc tek `askrvkr. Sekrk hs j` l knjhjm `d rkst `r
l`th`j whte`ut tek `askrvkr. J`tehjm hs hj nas`butk rkst `r hj nas`butk l`th`j.
Nj `afkgt hs sn hc t` ak hj l` th`j wh te rksp kg t t` n `ask rv kr, hd hts p`sh th `j ge njmks whte
rkspkgt t` tent `askrvkr. Ht lny enppkj ay a`te wnys khtekr `askrvkr l`vks `r `afkgt l`vks.
1 . QKGSHBHJKNQ L@SH@JQkgth b hjknr l`th`j hs l`th`j, nb`jm n strnhmet b hjk `r hj `jk chlkjsh`j. Ht cknbs whte tek
ohjklnthgs `d n pnrthgbk hj `jk chlkjsh`j.
1.> Y`sh t h`jSek p`shth`j `d n pnrthgbk rkdkrs t` hts b`gnth`j hj tek spngk nt n gkrtnhj l`lkjt `d thlk.
Ht hs g`jgkrjkc whte tek quksth`j wekrk hs tek pnrthgbk nt n pnrthgubnr l`lkjt `d thlk0
1.1 Chsp bngklkjtSek genjmk hj tek p`shth`j `d n l`vhjm `afkgt hs oj`wj ns
chspbngklkjt. Ht hs tek vkgt`r f`hjhjm tek hjhthnb p`shth`j ( >r
) `d
tek pnrthgbk t` hts dhjnb p`shth`j ( 1r ) curhjm nj hjtkrvnb `d thlk.
Chspbngklkjt gnj ak jkmnthvk p`shthvk `r zkr`.
r1
1.7 ChstnjgkSek bkjmte `d tek ngtunb pnte trnvkbbkc ay n pnrthgbk curhjm n mhvkj thlk hjtkrvnb hs gnbbkc
ns chstnjgk. Sek chstnjgk trnvkbbkc hs n sgnbnr qunjthty wehge hs quhtk chddkrkjt dr`l
chspbngklkjt. Hj mkjkrnb, tek chstnjgk trnvkbbkc aktwkkj tw` p`hjts lny j`t ak kqunb t`
tek lnmjhtuck `d tek chspbngklkjt aktwkkj tek snlk p`hjts.
Kxnlpbk >.
Qnl tnoks pnte > (strnhmet b hjk) t` m` dr`l Y t` ^ njc
Weynl tnoks pnte 1 (sklhghrgbk).
(n) Dhjc tek chstnjgk trnvkbbkc ay Qnl njc Weynl0 Y ^
>
1
>99 l
(a) Dhjc tek chspbngklkjt `d Qnl njc Weynl0
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YEXWHGW
"lnjhseoulnrpeyshgs.hj" 7
Njswkr 6 x < ;l, y < 7l
W`buth`j 6 Sek mhvkj gurvk se`ws tek pnte d tek pnrthgbk stnrthjm nt y < 3 l.
Nvkrnmk vkb`ghty < tnokjthlk
jtchspbngklk? wekrk chspbngklkjt hs strnhmet bhjk chstnjgk aktwkkj p`hjts
Hjstnjtnjk`us vkb`ghty nt njy p`hjt hs tek tnjmkjt crnwj t` tek gurvk nt tent p`hjt.
J`w, ns se`wj hj tek mrnpe, bhjk NA se`ws chspbngklkjt ns wkbb ns tek tnjmkjt t` tek mhvkj gurvk.
Ekjgk, p`hjt A hs tek p hjt nt wehge chrkgth j `d NA se`ws nvkrnmk ns wkbb ns hjstnjtnjk`us vkb`ghty.
1.4 Nvkrnmk Wpkkc hj nj hjtkrvnb)Nv kr nmk sp kkc hs ckdh jk c ns te k t`tn b pnte bkjmte tr nvkb bkc ch vhckc ay tek t`tnb thlk
hjtkrvnb curhjm wehge tek l`th`j ens tnokj pbngk. Ht ekbps hj cksgrhahjm tek l`th`j nb`jm
tek ngtunb pnte.
Nvkrnmk Wpkkc \ nj c hts WH ujht hs l/ s.
J`tk 6 Nvkrnmk spkkc hs nbwnys p`shthvk hj g`jtrnst t` nvkrnmk vkb`ghty wehge akhjm n vkgt`r,
gnj ak p`shthvk `r jkmnthvk.
Hd tek l`th`j `d n pnrthgbk hs nb`jm n strnhmet bhjk njc hj snlk chrkgth`j tekj,
nvkrnmk vkb`ghty < nvkrnmk spkkc.
Nvkrnmk spkkc hs, hj mkjkrnb, mrkntkr tenj tek lnmjhtuck `d nvkrnmk vkb`ghty.
Kxnlpbk 7. Hj tek kxnlpbk >, hd Qnl tnoks 3 skg`jcs njc Weynl tnoks ; skg`jcs t` m` dr`l Y t` ^, dhjc
(n) Nvkrnmk s pkkc `d Q nl njc Weynl0
(a) Nvkrnmk vkb`ghty `d Qnl njc Weynl0
W`buth`j 6 (n) N vk rn mk s pk kc ` d Q nl 99l/s < 1; l/s
Nv kr nmk sp kkc `d Weynl 9 l/ s
(a) N vk rn mk v kb `g ht y ` d Q nl 99l/s < 1; l/s (Dr`l Y t` ^)
Nv kr nmk vkb` ghty `d Weynl 99l/s < 19 l/s (Dr`l Y t` ^)
Kxnlpbk 3. N pnrthgbk trnvkbs enbd `d t`tnb chstnjgk whte spkkc v>
njc jkxt enbd whte spkkc v1
nb`jm n
strnhmet bhjk. Dhjc `ut tek nvkrnmk spkkc `d tek pnrthgbk0
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YEXWHGW
"lnjhseoulnrpeyshgs.hj" 3
W`buth`j 6 Bkt t`tnb chstnjgk trnvkbbkc ay tek pnrthgbk ak 1s.
Shlk tnokj t` trnvkb dhrst enbd v
s
Shlk tnokj t` trnvkb jkxt enbd
vv
vv1
(enrl`jhg pr`mrkssh`j)
Kxnlpbk ;. N pkrs`j trnvkbbhjm `j n strnhmet bhjk l`vks whte n ujhd`rl vkb`ghty v>d`r s`lk thlk njc whte
ujhd`rl vkb`ghty v1d`r tek jkxt kqunb thlk. Sek nvkrnmk vkb`ghty v hs mhvkj ay
Njswkr 61
vvv 1>
(Nrhtelkthg pr`mrkssh`j)
W`buth`j 6
Ns se`wj, tek pkrs`j trnvkbs dr`l N t` A ter`ume n chstnjgk W, wekrk dhrst pnrt W>hs trnvkbbkc
hj thlk t/1 njc jkxt W1 nbs` hj thlk t/1.
W`, ngg`rchjm t` tek g`jchth`j 6 v>njc v
19 l/s
Kxnlpbk 5. Lr. Wenrln arnoks ehs gnr whte g`jstnjt nggkbkrnth`j dr`l n vkb`ghty `d 1; l/s t`
>; l/s `vkr n chstnjgk `d 199 l.
(n) E`w luge thlk kbnpsks curhjm tehs hjtkrvnb0
(a) ]ent h s tek ngg kb krnt h`j0
(g) Hd ek ens t` g`jthjuk arnohjm whte tek snlk g`jstnjt nggkbkrnth`j, e`w luge
b`jmkr w`ubc ht tnok d`r ehl t` st`p njc e`w luge ncchth`jnb chstnjgk w`ubc ek
g`vkr0
W`buth`j 6 (n) ]k skbkgt p`shthvk chrkgth`j d`r `ur g``rchjntk systkl t` ak tek chrkgth`j `d tek
vkb`ghty njc ge``sk tek `rhmhj s` tent xh< 9 wekj tek arnohjm akmhjs. Sekj tek
hjhthnb vkb`ghty hs ux
< +1; l/s nt t < 9, njc tek d hjnb vkb`ghty njc p`shth`j nrk
vx
< +>; l/s njc x < 199 l nt t hlk t.
Whjgk tek nggkbkrnth`j hs g`jstnjt, tek nvkrnmk vkb`ghty hj tek hjtkrvnb gnj ak
d`ujc dr`l tek nvkrnmk `d tek hjhthnb njc dhjnb vkb`ghthks.
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8/12/2019 Kinematics Theory_E
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YEXWHGW
"lnjhseoulnrpeyshgs.hj" 8
Sek phgop`gokt whbb ak gnumet hd t hs rknb njc p`shthvk.
Sehs whbb ak p`sshabk hd
1ncv1 `r, 1ncv
Kxnlpbk >9. N lnj hs stnjchjm 39 l akehjc tek aus. Aus stnrts whte > l/skg1 g`jstnjt nggkbkrnth`j
njc nbs` nt tek snlk hjstnjt tek lnj stnrts l`vhjm whte g`jstnjt spkkc 8 l/s. Dhjc
tek thlk tnokj ay lnj t` gntge tek aus.
x < 9t < 9
39 l t < 9x < 39
>l/skg1
W`buth`j 6 Bkt ndtkr thlk t lnj whbb gntge tek aus
D`r aus
x < x9
+ u t +1
>nt 1 , x < 39 + 9(t) +
1
>(>) t1
x < 3 9 +1
t1............. (h)
D`r lnj, x < 8t ............. (hh)
Dr`l (h) & (hh)
39 +1
t1< 8t `r t < 5 s r t < >9s.
Kxnlpbk >>. N pnrthgbk hs cr`ppkc dr`l ekhmet >99 l njc nj`tekr pnrthgbk hs pr`fkgtkc vkrthgnbby up
whte vkb`ghty ;9 l/s dr`l tek mr`ujc nb`jm tek snlk bhjk. Dhjc `ut tek p`shth`j wekrk
tw` pnrthgbk whbb lkkt0 (tnok m < >9 l/s 1)W`buth`j 6 Bkt tek upwnrc chrkgth`j hs p`shthvk.
y99 + 9(t) 1
> >9 t1 _ y < y9 + u t + 1
>nt 1\
< >99 - ;t 1 ---- (>)
D`r pnrthgbk A,y
9 < 9 l
u < + ; 9 l / s
n < >9 l/s1
y < ;9(t) 1
> >9 t1
< ;9t ;t 1 ---- (1)Ng g` rc hjm t` tek pr `abkl?
;9t ;t 1 < >99 ;t 1
t < 1 s k g
Yutthjm t < 1 skg hj kqj. (>),
y < >99 19 < 59 lEkjgk, tek pnrthgbks whbb lkkt nt n ekhmet 59 l na`vk tek mr`ujc.
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YEXWHGW
"lnjhseoulnrpeyshgs.hj" >9
Kxnlpbk >1. N pnrthgbk hs cr`ppkc dr`l n t`wkr. Ht hs d`ujc tent ht trnvkbs 3; l hj tek bnst skg`jc `d
hts f`urjky. Dhjc `ut tek ekhmet `d tek t`wkr0 (tnok m < >9 l/s 1)
W`buth`j 6
Bkt tek t`tnb thlk `d f`urjky ak j skg`jcs.
[shjm? )>1(1
jn
usj 3; < 9 +
1
>9)>1( j
j < ; s k g
Ekhmet `d t`wkr? e mt 1 >9 ;1 < >1; l
=. QKNGSH@J SHLK]ekj n shtunth`j cklnjcs `ur hllkchntk ngth`j. Ht tnoks s`lk thlk akd`rk wk rknbby rksp`jc.
Qkngth`j thlk hs tek thlk n pkrs`j tnoks t` `askrvk, tehjo njc ngt.
Kxnlpbk >7. N st`jk hs cr`ppkc dr`l n anbb``j m`hjm up whte n ujhd`rl vkb`ghty `d ; l/s. Hd tek
anbb``j wns 49 l ehme wekj tek st`jk wns cr`ppkc, dhjc hts ekhmet wekj tek st`jk ehtstek mr`ujc. Snok m < >9 l/s 1.
W`buth`j 6 W < u t +1
>nt 1
49 < ;( t) +1
>(>9) t1
49l
+vk
vk 49 < ;t ;t 1
;t 1 ;t 49 < 9
t1 t >1 < 9
t1 3t + 7t >1 < 9
( t 3 ) ( t + 7 ) < 9
t < 3Ekhmet `d anbb``j dr`l mr`ujc nt tehs hjstnjt
< 4 9 + 3 ; < 5 9 l
Kxnlpbk >3. N anbb``j hs rhshjm whte g`jstnjt nggkbkrnth`j 1 l/skg1. Sw` st`jks nrk rkbknskc dr`l
tek anbb``j nt tek hjtkrvnb `d 1 skg. Dhjc `ut tek chstnjgk aktwkkj tek tw` st`jks > skg.
ndtkr tek rkbknsk `d skg`jc st`jk.
W`buth`j 6 Nggkbkrnth`j `d anbb``j < 1 l/skg1
Bkt nt t < 9, y < 9 wekj tek d hrst st`jk hs rkbknskc.
Ay tek quksth`j, y>
< 9 t>
+1> mt
>1
(tnohjm vkrthgnb upwnrc ns vk njc c`wjwnrc ns + vk)
Y`shth`j `d st st`jk skg`jc ndtkr rkbknsk `d skg`jc st`jk whbb ak tek 7rc skg`jc d`r tek >st st`jk)
D`r skg`jc st`jk y1
< ut1
+1
>mt
11
u < 9 + nt < 1 1 < 3l/s (tnohjm vkrthgnb upwnrc ns vk njc c`wjwnrc ns + vk)
y1
< 3 > +1> m (>)1 (t
1< > skg`jc)
Sek 1jc st`jk hs rkbknskc ndtkr 1 skg`jc
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YEXWHGW
"lnjhseoulnrpeyshgs.hj" >1
Kqunth`j `d pnrna`bn 6Gnsk (h) 6 y < n x1 + a x + g
D`r n : 9
x
y
Sek jnturk `d tek pnrna`bn whbb ak bhok tent `d tek `d jnturk x 1 < oy
Lhjhlul vnbuk `d y kxhsts nt tek vkrtkx `d tek pnrna`bn.
ylhj
>.> Y`sht h`j vs Shlk mrnpe
h ) Vkr` Tkb`ghtyNs p` shth `j `d pnrt hg bk hs dh xk c nt nb b tek th lk,
s` tek a`cy hs nt rkst.
Wb`pk?ctcx < tnj < t n j 9 < 9
Tkb`ghty `d pnrthgbk hs zkr`
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YEXWHGW
"lnjhseoulnrpeyshgs.hj" >7
h h ) [jhd`rl Tkb`ghtyEkrk tnj hs g`jstnjt tnj >.1 Tkb`ghty vs thlk mrnpeh ) Vkr` nggkbkrnth`j
Tkb`ghty hs g`jstnjt.
tnj < 9
ct
cv< 9
Ekjgk, nggkbkrnth`j hs zkr`.
h h) [jhd`rl nggkbkrnt h`jtnj hs g`jstnjt.
ct
cv< g`jstnjt
Ekjgk, ht se`ws g`jstnjt nggkbkrnth`j.
h h ) [jhd`rl rktnrcnth`jWhjgk : 89 tnj hs g`jstnjt njc jkmnthvk.
ct
cv< jkmnthvk g`jstnjt
Ekjgk, ht se`ws g`jstnjt rktnrcnth`j.
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YEXWHGW
"lnjhseoulnrpeyshgs.hj" >3
>>.7 Nggkbkrnth`j vs thlk mrnpeh) G`jstnjt nggkbkrnt h`j
tnj < 9
ct
cn< 9
Ekjgk, nggkbkrnth`j hs g`jstnjt.
h h ) [jhd`rlby hjgrknshjm nggkbkrnt h`j hs g`jstnjt.
9 2 2 89 tn j : 9
ct
cn< tnj < g`jstnjt : 9
Ekjgk, nggkbkrnth`j hs ujhd`rlby hjgrknshjm whte thlk.
h h ) [jhd` rlby ckg rknshjm ng gkbkrnt h`jWhjgk : 89 tnj hs g`jstnjt njc jkmnthvk.
ct
cn< jkmnthvk g`jstnjt
Ekjgk, nggkbkrnth`j hs ujhd`rlby ckgrknshjm whte thlk
Kxnlpbk >;.
Sek chspbngklkjt vs thlk mrnpe `d n pnrthgbk l`vhjm nb`jm
n strnhmet bhjk hs se`wj hj tek dhmurk. Crnw vkb`ghty vs thlk
njc nggkbkrnth`j vs thlk mrnpe.
W`buth`j 6
x < 3 t1
v 4
(hv) Nggkbkrnth`j vs thlk mrnpe 6
>9
Nggkbkrnth`j
Shlk
ter`ume`ut tek l`th`j,
pnrthgbk ens g`jstnjt
nggkbkrnth`j < >9 l/s 1.
D`r pnrthgbk A 6
u < > 9 l / s. y < > 9 t 1
>(>9) t1 < >9t ;t 1
(h) C hs pb ng kl kj t t hl k m rn pe 6
y < >9t ;t 1
ct
cy< >9t ;t1 < >9 >9t
tehs se`ws tent sb`pk akg`lks
l`rk jkmnthvk whte thlk.
( hh ) W pk kc t hl k m rnp e 6
ct
cy< >9t ;t1 < >9 >9t
ekjgk, spkkc hs chrkgtby
pr`p`rth`jnb t` thlk whte
sb`pk `d >9. hjhthnb spkkc < >9 l/s
(h hh) Tkb`ghty thlk mrnpe 6
>9 l/s
Tkb`ghty
Shlk
t
ct
cy< >9t ;t1 < >9 >9t
ekjgk, vkb`ghty hs chrkgtby
pr`p`rth`jnb t` thlk whte
sb`pk `d >9. Hjhthnb vkb`ghty < >9 l/s
(hv) Nggkbkrnth`j vs thlk mrnpe 6
ter`ume`ut tek l`th`j,
pnrthgbk ens g`jstnjt nggkbkrnth`j < >9 l/s 1.
n 9
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YEXWHGW
"lnjhseoulnrpeyshgs.hj" >=
D`r Ynrthgbk G 6
(h) C hs pb ng kl kj t t hl k m rn pe 6
u < 9 , y < 1
> >9t1 < ; t1
tehs se`ws tent sb`pk akg`lks
l`rk jkmnthvk whte thlk.
( hh ) W pkk c v s t hl k m rn pe 6
v 9 t
ekjgk, spkkc hs chrkgtby
pr`p`rth`jnb t` thlk whte
sb`pk `d >9.
(h hh) Tkb`ghty thlk mrnpe 6
>9
Tkb`ghty
Shlk
19
19
T < u + n t
T < > 9t ?
ekjgk, vkb`ghty hs chrkgtby
pr`p`rth`jnb t` thlk whte
sb`pk `d >9.
(hv) Nggkbkrnth`j vs thlk mrnpe 6
>9 l/s
Nggkbkrnth`j
Shlk
ter`ume`ut tek l`th`j,
pnrthgbk ens g`jstnjt
nggkbkrnth`j < >9 l/s 1.
>1. CHWYBNGKLKJS DQ@L v - t MQNYE GENJMK HJ TKB@GHSX DQ@L n -t MQNYEChspbngklkjt
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YEXWHGW
"lnjhseoulnrpeyshgs.hj" >5
Kxnlpbk >=. Cksgrhak tek l`th`j se`wj ay tek d`bb`whjm vkb`ghty-thlk mrnpes.
(n) (a)
W`buth`j 6 (n) Curhjm hjtkrvnb NA6 vkb`ghty hs +vk s` tek pnrthgbk hs l`vhjm hj +vk chrkgth`j,
aut ht hs sb`whjm c`wj ns nggkbkrnth`j (sb`pk `d v-t gurvk) hs jkmnthvk. Curhjm
hjtkrvnb AG6 pnrthgbk rklnhjs nt rkst ns vkb`ghty hs zkr`. Nggkbkrnth`j hs nbs`
zkr`. Curhjm hjtkrvnb GC6 vkb`ghty hs -vk s` tek pnrthgbk hs l`vhjm hj -vk chrkg-
th`j njc hs spkkchjm up ns nggkbkrnth`j hs nbs` jkmnthvk.
(a) Curhjm hjtkrvnb NA6 pnrthgbk hs l`vhjm hj +vk chrkgth`j whte g`jstnjt vkb`ghty
njc nggkbkrnth`j hs zkr`. Curhjm hjtkrvnb AG6 pnrthgbk hs l`vhjm hj +vk chrkgth`j
ns vkb`ghty hs +vk, aut h t sb`ws c`wj ujth b h t g`lks t` rkst ns nggkbkrnth`j hs
jkmnthvk. Curhjm hjtkrvnb GC6 vkb`ghty hs -vk s` tek pnrthgbk hs l`vhjm hj -vk
chrkgth`j njc hs spkkchjm up ns nggkbkrnth`j hs nbs` jkmnthvk.
lp`rtnjt `hjts t` Qklklakr D`r ujhd`rlby nggkbkrntkc l`th`j (n 9), xt mrnpe hs n pnrna`bn (`pkjhjm upwnrcs hd
n : 9 njc `pkjhjm c`wjwnrcs hd n 2 9). Sek sb`pk `d tnjmkjt nt njy p`hjt `d tek pnrna`bn
mhvks tek vkb`ghty nt tent hjstnjt.
D`r ujhd`rlby nggkbkrntkc l`th`j (n 9), vt mrnpe hs n strnhmet bhjk we`sk sb`pk mhvkstek nggkbkrnth`j `d tek pnrthgbk.
Hj mkjkrnb, tek sb`pk `d tnjmkjt hj xt mrnpe hs vkb`ghty njc tek sb`pk `d tnjmkjt hj vtmrnpe hs tek nggkbkrnth`j.
Sek nrkn ujckr nt mrnpe mhvks tek genjmk hj vkb`ghty.
Sek nrkn aktwkkj tek vt mrnpe mhvks tek chstnjgk trnvkbbkc ay tek pnrthgbk, hd wk tnoknbb nrkns ns p`shthvk.
Nrkn ujckr vt mrnpe mhvks chspbngklkjt, hd nrkns akb`w tek tnxhs nrk tnokj jkmnthvk.
Kxnlpbk >5. D`r n pnrthgbk l`vhjm nb`jm x-nxhs, vkb`ghty-thlk mrnpe hs
n s s e `w j h j d h m ur k . D h jc t e k c h s t n jg k t r nv k bb k c n j cchspbngklkjt `d tek pnrthgbk0
W`buth`j 6
Chstnjgk trnvkbbkc < Nrkn ujckr v-t mrnpe (tnohjm nbb nrkns ns +vk.)
Chstnjgk trnvkbbkc < Nrkn `d trnpkzhul + Nrkn `d trhnjmbk
< 5411
> + ;3
1
>
< 7 1 + > 9
< 3 1 l
Chspbngklkjt < Nrkn ujckr v-t mrnpe (tnohjm nrkns akb`w thlk nxhs ns vk.)
Chspbngklkjt < Nrkn `d trnpkzhul Nrkn `d trhnjmbk
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YEXWHGW
"lnjhseoulnrpeyshgs.hj" 19
Kxnlpbk >8. Nj `afkgt stnrts dr`l rkst nt t < 9 njc nggkbkrntks nt n rntk mhvkj ay n < 4t. ]ent hs
(n) hts vkb`ghty njc
(a) hts chspbngklkjt nt njy thlk t0
W`buth`j 6 Ns nggkbkrnth`j hs mhvkj ns n dujgth`j `d thlk,
t
t
v(t)
)v(t 99
n(t)ctcv
Ekrk t9
< 9 njc v(t9) < 9
v(t) v < >.
W`buth`j 6 n < x cx
vcv< x
1
v 1 njc v < >
G < 9 v1 < x1
v < x aut mhvkj tent x < > wekj v < >
v < x ct
cx< x
x
cx< ct
jx < t + G 9 < 9 + G j x < tx < k t
Kxnlpbk 1>. D`r n pnrthgbk l`vhjm nb`jm x-nxhs, nggkbkrnth`j hs mhvkj ns n < v. Dhjc tek p`shth`j ns n
dujgth`j `d thlk 0
M h v kj tent nt t < 9 , x < 9 v < >.
W`buth`j 6 n < v ct
cv< v v
cv< ct
j v < t + g 9 < 9 + g
v < k t ct
cx< k t cx< ctkt
x < k t + g 9 < > + gx < kt >
Yr`abkl >. N pnrthgbk g`vkrs 3
7
`d t`tnb chstnjgk whte spkkc v> njc jkxt 3
>
whte v1. Dhjc tek nvkrnmk
spkkc `d tek pnrthgbk0
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YEXWHGW
"lnjhseoulnrpeyshgs.hj" 1>
Njswkr 61>
1>
7vv
v3v
W`buth`j 6 Bkt tek t`tnb chstnjgk ak s7s/3
NA
Gs/3
nvkrnmk spkkc (2 v :)
v3
7
>
1>
v7v
vv3
Yr`abkl 1. N g nr h s l`vh jm w h te s pkkc 49 Ol/e njc n ahrc h s
l`vhjm whte spkkc 89 ol/e nb`jm tek snlk chrkgth`j
ns se`wj hj dhmurk. Dhjc tek chstnjgk trnvkbbkc ay tek
ahrc thbb tek thlk gnr rkngeks tek. trkk0 139 l
Njswkr 6 749 l
W`buth`j 6 Shlk tnokj ay n gnr t` rkngeks tek trkk (t) , 3, -1 njc ; rkspkgthvkby njc WH ujhts nrk uskc.
Dhjc (n) tek chlkjsh`js `d N, A, G njc C, (a) tek vkb`ghty `d tek pnrthgbk nt t < 3 s, (g)
tek nggkbkrnth`j `d tek pnrthgbk nt t < 3s, (c) tek nvkrnmk vkb`ghty curhjm tek hjtkrvnb t \ njc _C \ < _B\ ?
(a ) =5 l/s ? (g ) 71 l/s1 ? (c ) 79 l/s ? (k) 19 l/s1
W`buth`j 6 Ns x < Nt7 + Ct 1 + G t + C
(n ) Chlkjsh`js `d N, A, G njc C ,
_Nt7\ < _x\ (ay prhjghpbk `d e`l`mkjkhty)
_N\ < _BS7\
shlhbnrby, _A\ < _BS -1\, _G\ < _BS ->\ njc _C \ < _B\ ?
(a ) N s v ) (3)1 + 1(3) (3) 1 < =5 l/s.
(g ) Nggkbkrnth`j (n) )(43) + (3)(>4) (1) (3) + ; < >1; l
Seus tek chspbngklkjt curhjm 9 t` 3 skg. hs >1; ; < >19 l
2 v : < > 1 9 / 3 < 7 9 l /s(k ) v < 7Nt1 + 19 t + G , vk b`g hty nt t < 9 hs g < 1 l /s
vkb`ghty nt t < 3 skg hs =5 l/s 2 n : 1
>1
tt
vv
< 19 l/s1
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YEXWHGW
"lnjhseoulnrpeyshgs.hj" 11
Yr`abkl 3. D`r n pnrthgbk l`vhjm nb`jm x-nxhs, vkb`ghty hs mhvkj ns n dujgth`j `d thlk ns v < 1t1 + shj t.
Nt t < 9, pnrthg bk hs nt `rh mhj. Dhjc tek p`s hth` j ns n duj gth`j `d thl k0
W`buth`j 6 v < 1 t1 + s h j t ct
cx< 1t 1 + s h j t
x
9
cx < ct)tshjt1(
t
9
1 < >tg`st7
1x 7 Njs.
Yr`abkl ;. N gnr ckgkbkrntks dr`l n spkkc `d 19 l/s t` rkst hj n chstnjgk `d >99 l. ]ent wns hts
nggkbkrnth`j, nssulkc g`jstnjt0
W`buth`j 6 v < 9 u < 1 9 l /s s < > 99 l ns v1 < u1 + 1 n s9 < 399 + 1n >99 n < 1 l /s nggkbkrnth`j < 1 l/s1 Njs.
Yr`abkl 4. N >;9 l b`jm trnhj nggkbkrntks ujhd`rlby dr`l rkst. Hd tek dr`jt `d tek trnhj pnssks n rnhbwny w`rokr
;9 l nwny dr`l tek stnth`j nt n spkkc `d 1; l/s, went whbb ak tek spkkc `d tek ango pnrt `d tek
trnhj ns ht pnssks tek w`rokr0
W`buth`j 6 v1 < u1 + 1ns
1 ; 1 ; < 9 + > 9 9 n
n ;9
v 1 < 1; 1; 3
v < ;9 l/s. Njs.
Yr`abkl =. N pnrthgbk hs ter`wj vkrthgnbby whte vkb`ghty 19 l/s . Dhjc (n) tek chstnjgk trnvkbbkc ay
tek pnrthgbk hj dhrst 7 skg`jcs, (a) chspbngklkjt `d tek pnrthgbk hj 7 skg`jcs.
Njswkr 6 1;l, >;l
W`buth`j 6 Ehmekst p`hjt sny AT
A < 9
v < u + m t
9 < 1 9 > 9 t
t < 1 s k g .
chstnjgk trnvkb hj dhrst 1 skg`jcs.s < s (t /1 n t1\t < 9 t ` t < 1 s
+ _ut +>/1nt1\t < 1 t ` t < 7 s
s < 1 9 1 > / 1 > 9 3 + > / 1 > 9 > 1
< ( 3 9 1 9) + ; < 1; l.
njc chspbngklkjt < 19 ; < >; l.
Yr`abkl 5. N gnr nggkbkrntks dr`l rkst nt n g`jstnjt rntk d`r s`lk thlk ndtkr wehge ht ckgkbkrntksnt n g`jstnjt rntk t` g`lk t` rkst. Hd tek t`tnb thlk kbnpskc hs t. Dhjc tek lnxhlulvkb`ghty ngquhrkc ay tek gnr.
W`buth`j 6 t < t>
+ t1
sb`pk `d @N gurvk < tnj <
lnx
t
v vlnx
@
t1t>
A
N
T
t
sb`pk `d NA gurvk <
+ t1
t
Nr kn ujckr mr npe (c hrkg tb y) 1
9. Sek chspbngklkjt vs thlk mrnpe `d n pnrthgbk l`vhjm nb`jm
n strnhmet bhjk hs se`wj hj tek dhmurk. Crnw vkb`ghty vs thlk
njc nggkbkrnth`j vs thlk mrnpe.
[pwnrcs chrkgth`j hs tnokj ns p`shthvk, c`wjwnrcs
chrkgth`j hs tnokj ns jkmnthvk.W`buth`j 6 (n) Sek kqunth`j `d l`th`j hs 6 x < 5t1
v 4 t ? teh s s e`ws tent v kb `g hty h s chrkg tb y pr`p`rt h`jnb t` t hlk njc
sb`pk `d vkb`ghty-thlk gurvk hs jkmnthvk h.k. >4.
Ekjgk, rksubthjm mrnpe hs (h)
(a) Nggkbkrnth`j `d pnrthgbk hs 6 n 4.
Sehs se`ws tent nggkbkrnth`j hs g`jstnjt aut jkmnthvk.
Qksubthjm mrnpe hs (hh)
Yr`abkl >>. Crnw chspbngklkjtthlk njc nggkbkrnth`jthlk mrnpe d`r tek mhvkj vkb`ghtythlk mrnpe.
W` bu th` j 6 Y nr t NA 6 v-t gurvk se`ws g`jstnjt sb`pk
h.k. g`jstnjt nggkbkrnth`j `r Tkb`ghty hjgrknsks nt g`jstnjt rntk whte thlk.
Ekjgk, s-t gurvk whbb se`w g`jstnjt hjgrknsk hj sb`pk
njc n-t gurvk whbb ak n strnhmet bhjk.
Ynrt AG 6 v-t gurvk se`ws zkr` sb`pk h.k. g`jstnjt vkb`ghty. W`, s-t gurvk whbb se`w
g`jstnjt sb`pk njc nggkbkrnth`j whbb ak zkr`.
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