kinematics theory_e

8/12/2019 Kinematics Theory_E

1/24

YEXWHGW

"lnjhseoulnrpeyshgs.hj" >

LKGENJHGWLkgenjhgs hs tek arnjge `d peyshgs wehge cknbs whte tek gnusk njc kddkgts `d l`th`j `d n

pnrthgbk, rhmhc àfkgts 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 àfkgt njc tek àskrvkr. Sekrk hs j` l knjhjm `d rkst `r

l`th`j whteùt tek àskrvkr. J`tehjm hs hj nas`butk rkst `r hj nas`butk l`th`j.

Nj àfkgt hs sn hc t` ak hj l` th`j wh te rksp kg t t` n àsk rv kr, hd hts p`sh th `j ge njmks whte

rkspkgt t` tent àskrvkr. Ht lny enppkj ay a`te wnys khtekr àskrvkr l`vks `r àfkgt 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 àfkgt 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


2/24


3/24

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

Hjstnjtnjkùs 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 hjstnjtnjkùs 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 ùt tek nvkrnmk spkkc `d tek pnrthgbk0


4/24

YEXWHGW


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 terùme 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 wùbc ht tnok d`r ehl t` st`p njc e`w luge ncchth`jnb chstnjgk wùbc ek

g`vkr0

W`buth`j 6 (n) ]k skbkgt p`shthvk chrkgth`j d`r ùr 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

dùjc dr`l tek nvkrnmk `d tek hjhthnb njc dhjnb vkb`ghthks.


8/24


9/24

YEXWHGW


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 mrùjc nb`jm tek snlk bhjk. Dhjc ùt 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 àbkl?

;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 mrùjc.


10/24

YEXWHGW

"lnjhseoulnrpeyshgs.hj" >9

Kxnlpbk >1. N pnrthgbk hs cr`ppkc dr`l n t`wkr. Ht hs dùjc tent ht trnvkbs 3; l hj tek bnst skg`jc `d

hts fùrjky. Dhjc ùt tek ekhmet `d tek t`wkr0 (tnok m < >9 l/s 1)

W`buth`j 6

Bkt tek t`tnb thlk `d fùrjky 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 ùr 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` àskrvk, 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 mrùjc. 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 mrùjc 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 ùt 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


11/24


12/24

YEXWHGW


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`


13/24

YEXWHGW


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.


14/24

YEXWHGW


>>.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


sb`pk `d >9. Hjhthnb vkb`ghty < >9 l/s



pnrthgbk ens g`jstnjt nggkbkrnth`j < >9 l/s 1.

n 9


17/24

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


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


sb`pk `d >9.


>9 l/s

Nggkbkrnth`j

Shlk


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


18/24

YEXWHGW


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


19/24


20/24

YEXWHGW


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


21/24

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


Yràbkl 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.

Yràbkl ;. 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.

Yràbkl 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.

Yràbkl =. 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.

Yràbkl 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)

Yràbkl >>. 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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kinematics theory_e

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