Aa¨vq-3: MwZwe`¨v - WordPress.com · 2017. 3. 1. · Aa¨vq-3: MwZwe`¨v = 2 o25 asin 30 9.8 =...

Aa¨vq-3: MwZwe`¨v cÖkœ1 †Mvjiÿ‡Ki 40 wgUvi mvg‡b †_‡K GKRb dzUej †L‡jvqvo Abyf‚wg‡Ki mv‡_ 30o †Kv‡Y 25ms1 †e‡M ej wKK K‡i|

GKB mg‡q †MvjwKcvi ejwU aivi Rb¨ e‡ji w`‡K 10ms1

mg‡e‡M †`Š‡o hvq| [†eMg eìæb‡bmv miKvwi gwnjv K‡jR; XvKv]

K. Abyf‚wgK cvjøv Kx? 1

L. †bŠKvi ¸Y Uvb‡j wKfv‡e †bŠKv mvg‡bi w`‡K GwM‡q hvq

e¨vL¨v K‡iv| 2

M. wKK Kivi 0.5 †m‡KÛ ci e‡ji †eM KZ? 3

N. ejwU f‚wg‡Z covi Av‡M †MvjwKcvi ejwU ai‡Z cvi‡e

wKbv MvwYwZKfv‡e we‡kølY K‡iv| 4

1 bs cÖ‡kœi DËi

K cÖwÿß e¯‘ ev cÖvm wb‡ÿ‡ci ’̄vb †_‡K m‡e©v”P D”PZvq wM‡q Avevi GKB Abyf‚wgK Z‡j wd‡i Avmvi mg‡q †h Abyf‚wgK `~iZ¡

AwZµg K‡i ZvB Abyf‚wgK cvjøv|

L

aiv hvK, †bŠKvi B we› ỳ‡Z `wo †eu‡a GK e¨w³ BM eivei T ej

cÖ‡qvM K‡i Uvb‡Q| wbDU‡bi Z…Zxq m~Îvbymv‡i †bŠKvI e¨w³i Ici

MB eivei cÖwZwµqv ej T cÖ‡qvM Ki‡e| e¨w³ KZ©„K †bŠKvi Ici

cÖhy³ ej T ỳwU Dcvs‡k wef³ n‡e| GKwU Dcvsk Tcos, hv

†bŠKv‡K mvg‡bi w`‡K wb‡q hv‡e Ges Aci Dcvsk Tsin, hv

†bŠKv‡K K‚‡ji w`‡K wb‡q †h‡Z _vK‡e| wKš‘ gvwS b`xi †mÖvZ‡K

eënvi K‡i êVvi mvnv‡h¨ Gi wecixZ w`‡K GKwU ej Drcbœ

Ki‡e d‡j Tsin AskwU cÖkwgZ n‡e| †mÖvZ KZ©„K †bŠKvi Ici

†cQ‡bi w`‡K cÖhy³ ej n‡j mvg‡bi w`‡K jwä ej n‡e Tcos

| G e‡ji wµqvq wbDU‡bi wØZxq m~Îvbymv‡i †bŠKvq GKwU Z¡iY

m„wó n‡e d‡j †bŠKvi †eM e„w× †c‡Z _vK‡e| †bŠKvi †eM e„w×i

mv‡_ mv‡_ †bŠKvi mv‡c‡ÿ †mÖv‡Zi †eM evo‡Z _vK‡e| G‡Z Gi

gvb evo‡Z _vK‡e| GK mgq Tcos = n‡e, d‡j †bŠKvi Ici

jwä ej k~b¨ n‡e Ges wbDU‡bi cÖ_g m~Îvbymv‡i †bŠKvwU mg‡e‡M

Pj‡Z _vK‡e|

M †Ìqv Av‡Q,

wb‡ÿcY †eM, vo = 25 ms1

wb‡ÿcY †KvY, o = 30o

mgq, t = 0.5s

AwfKl©R Z¡iY, g = 9.8 ms2

0.5s c‡i †eM, v = ?

wb‡ÿcY †e‡Mi Abyf‚wgK Dcvsk, vxo = vo coso

= 25 cos30o

= 21.65 ms1

wb‡ÿcY †e‡Mi Djø¤^ Dcvsk, vyo = vo sino

= 25 sin30o

= 12.5 ms1

0.5s c‡i †e‡Mi Abyf‚wgK Dcvsk, vx = vxo = 21.65 ms1

0.5s c‡i †e‡Mi Djø¤^ Dcvsk, vy = vyo gt

= 12.5 9.8 0.5

= 7.6 ms1

0.5s c‡i †e‡Mi gvb, v = vx2 + vy2

= (21.65)2 + (7.6)2 = 22.95 ms1

GLb, awi, †eM v Abyf‚wg‡Ki mv‡_ †KvY Drcbœ K‡i|

tan = vyvx

= 7.6

21.65

= 19.34o

0.5s ci e‡ji †e‡Mi gvb n‡e 22.95 ms1 Ges Zv Abyf‚wg‡Ki

mv‡_ 19.34 †KvY Drcbœ Ki‡e| (Ans.)

N GLv‡b, ejwUi wb‡ÿcY †eM, vo = 25 ms1



ejwUi cvjøv, R = v2osin 2o

g

= (25)2 sin(2 30o)

9.8 = 55.23 m

ejwUi wePiYKvj, T = 2vosino

g

B A

M

T T

Tcos

Tsin C

†mÖv‡Zi †eM

vy

v

vx

Aa¨vq-3: MwZwe`¨v

= 2 25 sin 30o

9.8 = 2.551s

†Mvj‡cv÷ †_‡K ejwUi cZb we› ỳi ~̀iZ¡, s = (80 55.23)m

= 24.77 m

†MvjiÿK hw` 10 ms1 †e‡M †`Š‡o 2.551s mg‡qi g‡a¨ b~¨bZg

24.77m ~̀iZ¡ AwZµg Ki‡Z cv‡i Z‡eB Zvi c‡ÿ ejwU f‚wg‡Z

covi Av‡MB aiv m¤¢e|

10 ms1 †e‡M †`Šov‡j 2.551s mg‡q †Mvjiÿ‡Ki AwZµvšÍ `~iZ¡,

s = (2.551 10)m

= 25.51m > 24.77 m

myZivs †MvjiÿK ejwU f‚wg‡Z covi Av‡MB ai‡Z cvi‡e|

cÖkœ2 nv‡mg mv‡ne Zvi Mvox wb‡q Awd‡mi c‡_ hvÎv ïiæ Kij, cÖ_‡g †m 10 sec mgZ¡i‡Y Pvjv‡jv| Gici 10 min mg‡e‡M

Pvjv‡bvi ci †eªK †P‡c 5 sec mg‡qi g‡a¨ Mvwo _vwg‡q Awd‡m

cÖ‡ek Kij|

[knx` †eMg †kL dwRjvZzb †bQv gywRe miKvwi gnvwe`¨vjq, nvRvixevM, XvKv]

K. ZvrÿwYK Z¡iY Kv‡K e‡j? 1

L. cÖ‡ÿc‡Ki MwZ wØ-gvwÎK MwZ e¨vL¨v Ki| 2

M. DÏxc‡Ki hvÎv ïiæi 2 sec ci Mvoxi †eM 4ms1 n‡j

nv‡mg mv‡ne 10 sec G KZ ~̀iZ¡ AwZµg K‡iwQj? 3

N. DÏxc‡Ki Av‡jv‡K nv‡mg mv‡n‡ei evmv n‡Z Awd‡mi

`~iZ¡ KZ? †eªK Kivi ci Z¡i‡Yi wKiƒc cwieZ©b n‡qwQj?

4


K mgq eëav‡b k~‡bï KvQvKvwQ n‡j mg‡qi mv‡_ e ‘̄i †e‡Mi cwieZ©‡bi nvi‡K ZvrÿwYK Z¡iY e‡j|

L †Kv‡bv e ‘̄ mgZ¡i‡Y Pj‡Z n‡j Zvi Dci cÖhy³ ej aªæe _vK‡Z n‡e| f‚c„‡ôi KvQvKvwQ Aí D”PZvi eëav‡b e ‘̄i Dci AwfKl©R

ej F = mg aªæe _v‡K †Kbbv, AwfKl©R Z¡iY, g aªæe _v‡K d‡j

e¯‘i Z¡iY I aªæe nq| f‚c„‡ôi mv‡_ Zxh©Kfv‡e wbwÿß †Kvb e ‘̄i

MwZ Ges Abyf‚wgKfv‡e wbwÿß †Kvb e ‘̄i MwZ wØgvwÎK MwZ Ges

MwZc_ GKwU Djø¤̂ Z‡j mxgve× _v‡K Ges Gi Z¡iY aªæe nq|

myZivs ejv hvq ÔcÖv‡mi MwZ mgZ¡i‡Y wØgvwÎK MwZi GKwU DrK…ó

D`vniY|Õ

M †Ìqv Av‡Q, Mvwoi Avw`‡eM, v0 = 0ms1

mgq, t1 = 2 sec

2 sec c‡i †eM, v1 = 4ms1

mgq, t2 = 10 sec

10 sec c‡i AwZµvšÍ ~̀iZ¡, s = ?

awi, mgZ¡iY = a

v1 = v0 + at1

a = v1t1

= 4ms1

2sec = 2ms2

s = vot2 + 12 at2

2

= 0 10 sec + 12 2ms

2 (10 sec)2

= 0 + 100m

= 100m (Ans.)

N †Ìqv Av‡Q, Mvwoi Avw`‡eM, vo = 0ms1

ÔMÕ Ask †_‡K cvB cÖ_g 10 sec-G AwZµvšÍ ~̀iZ¡, s = 100m

GLb, 10 sec ci †eM v n‡j,

v = v0 + a 10 = 0 + 2 10 = 20 ms1

GLb, 20ms1 †eM wb‡q Mvwoi cieZ©x 10 min = 600sec mg‡e‡M

hvq|

G mg‡q Mvwoi AwZµvšÍ `~iZ¡, s1 = 20 600m

= 12000m

cieZ©x 5 sec G MvwowUi †eM 20 ms1 †_‡K n«vm †c‡q 0 ms1 nq|

G mg‡q MvwowUi g›`b, a = 20ms1 0ms1

5 sec

= 4 ms2

G mg‡q AwZµvšÍ ~̀iZ¡, s2 =

20 ms

1 + 0ms1

2 5 sec

= 50 m

nv‡mg mv‡n‡ei evmv †_‡K Awd‡mi ~̀iZ¡, s3 = s + s1 + s2

= (100 + 12000 + 50)m

= 12150m

cÖkœ3 GKRb †L‡jvqvo †Mvj‡cv‡÷i mvg‡b n‡Z GKwU dzUej‡K 10 †Kv‡Y 40ms1 †e‡M wKK K‡i| ejwU 1 sec ci

†Mvj‡cv‡÷i Abyf‚wgK ev‡i AvNvZ K‡i| [ivRkvnx K‡jR, ivRkvnx]

K. cwigv‡ci cig ÎæwU Kv‡K e‡j? 1

L. cÖv‡mi MwZ wØgvwÎK n‡jI Zvi Z¡iY GKgvwÎK e¨vL¨v Ki| 2

M. †Mvj‡cv÷ n‡Z KZ ~̀‡i ejwU‡K wKK Kiv n‡q‡Q? 3

Aa¨vq-3: MwZwe`¨v N. †MvjiÿK ejwU ai‡Z bv cvi‡j †Mvj n‡e

wKbvMvwYwZKfv‡e hvPvB Ki| 4


K †Kv‡bv GKwU ivwki cÖK…Z gvb I cwigvcK…Z gv‡bi cv_©K¨‡K cig ÎæwU e‡j|

L cÖv‡mi g‡a¨ GKB mv‡_ †e‡Mi Abyf‚wgK I Djø¤^ Dcvsk _v‡K| wKš‘ Gi Z¡iY ïay Dj¤^ w`‡K KvR K‡i Ges Abyf‚wgK eivei Z¡iY

k~b¨ nq| ZvB cÖv‡mi †eM wØgvwÎK n‡jI Z¡iY GKgvwÎK|

M †`Iqv Av‡Q,

wb‡ÿcY †KvY, o = 10

wb‡ÿcY †eM, v0 = 40ms1

mgq, t =1s

†Mvj †cv‡÷i ~̀iZ¡, x = ?

Avgiv Rvwb, x = vxot + 12axt

2

= vx.t [ ax = 0]

= (vocoso)t

= (40 cos10) 1m

x = 39.39m (Ans.)

N GLv‡b,


wb‡ÿcY †eM, vo = 40ms1

mgq, t = 1s

t mgq ci †e‡Mi Abyf‚wgK Dcvsk, vx = v0cos0

= 40 cos 10

= 39.39 ms–1

t mgq ci †e‡Mi Djø¤^ Dcvsk, vy = vy0 – gt

= v0 sin0 – 9.8 1

= 40 sin 10 – 9.8

= 2.85 ms1 [wb¤œgyLx]

jwä †eM v Gi w`K n‡e wb‡Pi w`‡K|

A_©vr †Mvj‡cv÷ AvNvZ Kivi mgq ejwU wb‡Pi w`‡K MwZkxj

wQ‡jv| myZivs †MvjiÿK ejwU ai‡Z bv cvi‡j †Mvj nIqvi

m¤¢vebv _vK‡e|

cÖkœ4 GKRb cvwL wkKvix e‡bi g‡a¨ GKwU evwoi 14.0m DuPz cÖvPx‡ii †fZ‡i GKwU Mv‡Q GKwU mv`v eK emv †`L‡jb| wZwb

†`qvj n‡Z 30m ̀ ~‡i Ae ’̄vb K‡i 30 †Kv‡Y 40ms1 †e‡M eKwU‡K

jÿ¨ K‡i u̧wj Qyuo‡jb| [mvZKvwbqv miKvwi K‡jR, PÆMÖvg]

K. bvj †f±i wK? 1

L. `ywU †f±i KLb j¤̂ I mgvšÍivj nq? 2

M. wkKvixi †Quvov ¸wjwU m‡e©v”P KZ D”PZvq DV‡e? 3

N. ¸wjwU cÖvPxi UcKv‡Z cvi‡e wKbv we‡kølY Ki| 4


K †h †f±‡ii gvb k~b¨ Zv‡K k~b¨ †f±i ev bvj †f±i e‡j|

L ỳwU †f±‡ii WU ¸Ydj hw` k~b¨ nq Z‡e †f±iØq j¤̂ n‡e|

KviY, A .B = ABcos = ABcos90 = 0 [ cos 90 = 0]

Avevi, ỳwU †f±‡ii µm ¸Ydj hw` k~b¨ nq Z‡e †f±iØq

mgvšÍivj n‡e| KviY, A

B =

0

ev, ABsin^n =

0

ev, sin = 0 [ A, B I ^n 0]

ev, sin = sin0 Ges sin 180

= 0 Ges 180

M †`Iqv Av‡Q, wb‡ÿcY †KvY, o = 30


g = 9.8ms2

m‡e©v”P D”PZv, H = ?

Avgiv Rvwb, H = vo2sin2o

2g

ev, H = (40)2 (sin30)2

2 9.8

H = 20.41m

wkKvixi †Quvov ¸wjwU m‡e©v”P 20.41m D”PZvq DV‡e| (Ans.)

N GLv‡b, †`qv‡ji ~̀iZ¡, x = 30m

wb‡ÿcY †eM, v0 = 40ms1 Ges wb‡ÿcY †KvY, 0 = 30 vy

v

vx

Aa¨vq-3: MwZwe`¨v cÖv‡mi MwZc‡_i mgxKiY †_‡K Avgiv Rvwb,

y = x(tano) gx2

2(vocoso)2

= 30 tan30 9.8 (30)2

2(40cos30)2

= 17.32 88202400

= 17.32 3.675

y = 13.645

A_©vr 30m ~̀i‡Z¡ ¸wjwU 13.645m D”PZvq †cuŠQv‡e| wKš‘ cÖvPx‡ii

D”PZv 14.0m|

myZivs ¸wjwU cÖvPxi UcKv‡Z cvi‡e bv|

cÖkœ5 eb¨v ~̀M©Z GjvKvq GKwU wegvb †_‡K ïKbv Lvev‡ii c¨v‡KU †djvi Rb¨ GKwU wegvb iIbv n‡jv wKQyÿY c‡i wegv‡bi

cvBjU A‰_ cvwbi g‡a¨ 3km Abyf‚wgK ~̀i‡Z¡ GKwU DuPz f‚wg‡Z

500m e¨vmv‡a©i GKwU ~̀M©Z AÂj †`L‡Z †c‡q 1000kg f‡ii

GKwU Lvev‡ii c¨v‡KU †d‡j w`j| G mgq wegvbwU 1km D”PZv

w`‡q f‚wgi mgvšÍiv‡j 220ms1 †e‡M MwZkxj wQj|

[miKvwi gwnjv K‡jR, cvebv]

K. mij Qw›`Z ¯ú›`b Kv‡K e‡j? 1

L. gv_vq fvix e ‘̄ wb‡q Abyf‚wgKfv‡e wKQy`~i hvIqvi ciI

AwfKl© ej Øviv K…ZKvR k~b¨ nq †Kb? 2

M. DÏxc‡Ki Lvev‡ii c¨v‡KUwUi 5sec c‡i †eM KZ wQ‡jv?

3

N. DÏxc‡Ki Lvev‡ii c¨v‡KUwU ~̀M©Z AÂ‡j co‡e wK? Zv

MvwYwZKfv‡e e¨vL¨v Ki| 4


K hw` †Kvb e ‘̄i Z¡iY GKwU wbw`©ó we›`y †_‡K Gi mi‡Yi mgvbycvwZK Ges me©`v H we›`y AwfgyLx nq, Zvn‡j e ‘̄i GB MwZ‡K

mij Qw›`Z ¯ú›`b e‡j|

L gv_vq fvix e ‘̄ wb‡q Abyf‚wgKfv‡e wKQy`~i †M‡j, †m‡ÿ‡Î AwfKl©R ej Lvov wb‡Pi w`‡K wµqv K‡i, Avi miY nq Abyf‚wgK

w`K eivei| A_©vr AwfKl©R ej Ges mi‡Yi ga¨eZ©x †KvY 90|

Avgiv Rvwb,

KvR, W = Fscos

ev, W = Fscos90

W = 0

myZivs G‡ÿ‡Î, AwfKl©R ej Øviv K…ZKvR k~b¨|

M g‡b Kwi, 5sec ci wegv‡bi †eM v Ges Gi Abyf‚wgK I Djø¤̂ Dcvsk h_vµ‡g vx I vy

Abyf‚wgK Dcvs‡ki †ÿ‡Î,

vx = vocoso + axt

= 220 cos0 + 0

= 220 ms1

Djø¤^ Dcvs‡ki †ÿ‡Î,

vy = vo + ayt

= 0 + ( 9.8) 5

= 49 ms1

GLv‡b,

wegv‡bi †eM, vo = 220ms1


mgq, t = 5sec

AwfKl©R Z¡iY, g = 9.8ms2

Djø¤^ w`K eivei Z¡iY, ay = g

[ wb¤œgyLx)

Abyf‚wgK w`K eivei Z¡iY, ax = 0

5 †m‡KÛ ci †eM, v = ?

5 †m‡KÛ ci †eM, v = vx2 + vy2

= (220)2 + ( 49)2 = 225.4ms1 (Ans.)

g‡b Kwi, v, Abyf‚wg‡Ki mv‡_ †KvY ˆZwi K‡i|

tan = vyvx

= 49

200

= 12.56 (Ans.)

N aiv hvK, †h we› ỳ †_‡K e ‘̄ †Q‡o †`qv nq †mwU g~jwe›`y (x0, y0) Ges Lvov Dc‡ii w`K y Aÿ abvZ¥K|

GLv‡b, x0 = y0 = 0

Dj¤^ miY, y y0 = 1000m

Dj¤^ Avw`‡eM, vyo= 0

Dj¤^ Z¡iY, ay = g = 9.8ms2

f‚c„‡ô †cuŠQ‡Z mgq = t sec

Abyf‚wgK Avw` †eM, vxo = 220ms1

Abyf‚wgK Z¡iY, ax = 0

vy

v

vx

Aa¨vq-3: MwZwe`¨v Abyf‚wgK ~̀iZ¡, (x x0) = d = ?

Avgiv Rvwb,

Djø¤^ MwZi †ÿ‡Î,

y = y0 + vyot + 12ayt

2

ev, (y y0) = vyot + 12 ayt

2

ev, 1000 = 0 + 12 (9.8) t

2

ev, t2 = 1000 2

9.8

ev, t = 14.285 sec

myZivs, c¨v‡KUwU 14.285 sec ci f‚c„‡ô †cuŠQ‡e|

x = x0 + vx0t + 12 axt

2

ev, (x xo) = vxot + 12axt

2

ev, d = 220 14.286m

d = 3142.92m

cÖkœg‡Z, c¨v‡KUwU Qvovi wVK AvM gyn~‡Z© wegvbwU Abyf‚wgKfv‡e

D³ AÂj †_‡K 3km ev 3000m `~‡i wQj Ges AÂjwUi e¨vmva©

500m|

Avi MvwYwZKfv‡e Avgiv †`Ljvg †h, c¨v‡KUwU f‚wg ¯úk© Kivi

AvM gyn~‡Z© Abyf‚wgKfv‡e 3142.92 m ~̀iZ¡ AwZµg K‡i|

myZivs Lvev‡ii c¨v‡KUwU ~̀M©Z AÂ‡ji †fZ‡iB ci‡e|

cÖkœ6 cv‡ki wPÎvbyhvqx A I B we› ỳ †_‡K ỳwU e¯‘‡K GKB mg‡q wb‡ÿc Kiv n‡jv| A we› ỳwU 1.5m D”PZvq Aew ’̄Z|

[KzwoMÖvg miKvwi gwnjv K‡jR]

K. PµMwZi e¨vmva© wK? 1

L. UK© ïay cÖhy³ e‡ji Dci bq, j¤^ ~̀i‡Z¡i DciI wbf©ikxj

e¨vL¨v Ki| 2

M. A we›`y †_‡K e¯‘wU wbwÿß nIqvi 1s ci Gi †eM KZ n‡e

†ei Ki| 3

N. DÏxc‡Ki e¯‘Ø‡qi g‡a¨ †Kvb e ‘̄wU cÖ_‡g gvwU‡Z co‡e MvwYwZK we‡kølY K‡i †`LvI| 4


K hw` †Kv‡bv „̀p e ‘̄i GKwU wbw ©̀ó we›`y †hLv‡b e ‘̄wUi mg Í̄ fi †K›`ªxf‚Z Av‡Q aiv nq Ges N~Y©b Aÿ mv‡c‡ÿ H we› ỳ‡Z RoZvi

åvgK mgMÖ e¯‘wUi RoZvi åvg‡Ki mgvb nq, Z‡e Aÿ n‡Z H

we›`yi ~̀iZ¡‡K PµMwZi e¨vmva© e‡j|

L U‡K©i msÁv n‡Z Avgiv Rvwb UK©, =

r

F

GLv‡b, r Ae ’̄vb †f±i Ges

F n‡”Q cÖhy³ ej|

U‡K©i gvb n‡e, = Frsin

GLv‡b rsin n‡”Q N~Y©b‡K›`ª n‡Z e‡ji wµqv‡iLvi j¤̂ ̀ ~iZ¡| A_©vr

UK© e¯‘i Dci cÖhy³ ej Ges j¤̂ ~̀i‡Z¡i Dci wbf©ikxj|

M †`Iqv Av‡Q, A e ‘̄wUi Avw`‡eM, vo = 20ms1

A e¯‘wUi wb‡ÿcY †KvY, o = 30

mgq, t = 1sec

awi, Avw`‡e‡Mi Abyf‚wgK I Dj¤̂ Dcvsk h_vµ‡g, vxo I vyo

1 sec ci †eM, vx = vocoso + axt

= 20cos30 1 + 0

= 17.32 ms1

Ges vy = vyo + ayt

= vosino 9.8t

= 20sin30 9.8 1

= 0.2 ms1

†eM, v = vx2 + vy2 = (17.32)2 + (0.2)2

v = 17.33ms1 (Ans.)

jwä †eM Abyf‚wg‡Ki mv‡_ †KvY ˆZwi Ki‡j

tan = vyvx

= 0.2

17.32

= 0.6615 (Ans.)

A

20 ms1

30

10 ms1

60

B

1.5

Aa¨vq-3: MwZwe`¨v N A e ‘̄wUi †ÿ‡Î,

cÖv‡mi m~Î n‡Z cvB,

y yo = vyot + 12 ayt

2

ev, 1.5m = v0sinot 12gt

2

ev, 1.5 = 20sin30t 12 9.8t

2

ev, 4.9t2 10t + 1.5 = 0

t = 1.88sec

B e ‘̄wUi †ÿ‡Î, T = 2vosin0

g = 2 10sin60

9.8

T = 1.76 sec

Dc‡iv³ MvwYwZK we‡kølY n‡Z †`Lv hvq †h, e ‘̄Øq GKB mv‡_

wbt‡ÿc Ki‡j B e ‘̄wU cÖ_g gvwU‡Z co‡e| KviY B e ‘̄wUi

DÇq‡bi mgqKvj A e ‘̄wUi Zzjbvq Kg|

cÖkœ7

†Kvb GKwU e ‘̄‡K 70 ms1 †e‡M k~‡b¨ wb‡ÿc Kiv nj| GKwU

wbw`©ó mgq c‡i e ‘̄wU 117.6m D”PZvq DV‡e|

[mybvgMÄ miKvwi gwnjv K‡jR]

K. cÖv‡mi MwZ Kq gvwÎK? 1

L. Avgv‡ì ˆ`bw›`b Rxe‡b e„ËvKvi MwZ Acwinvh© e¨vL¨v Ki| 2

M. e¯‘wU KLb 117.6m D”PZvq DV‡e? 3

N. Avbyf‚wgK cvjøv I me©vwaK D”PZv mgvb n‡Z n‡j e ‘̄wUi

Dci Kx kZ© Av‡ivc Ki‡Z n‡e? 4


K cÖv‡mi MwZ wØgvwÎK|

L e„ËvKvi MwZ Avgv‡ì Rxe‡bi mv‡_ IZ‡cÖvZfv‡e RwoZ| ê`y¨wZK cvLvi MwZ, Nwoi KuvUvi MwZ, Gme e„ËvKvi MwZi

D`vniY| KjKviLvbvi cÖ‡Z¨K h‡š¿ nvRv‡iv hš¿vs‡k e„ËvKvi MwZ

Kv‡R jv‡M| e„ËvKvi MwZ‡K Kv‡R jvwM‡q wekvj D‡ovRvnvR I

†nwjKÞvi k~‡b¨ Do‡Q| Mvwoi PvKvi e„ËvKvi MwZ Mvwo‡K †`Š‡o

wb‡q hv‡”Q| Gfv‡e e„ËvKvi MwZ ˆ`bw›`b Rxe‡b Acwinvh©|

M †Ìqv Av‡Q,


wb‡ÿc †KvY, = 44.427

D”PZv, y = 117.6m

mgq, t = ?

g = 9.8ms2

Avgiv Rvwb, cÖv‡mi †ÿ‡Î, y = vosinot 12gt

2

ev, 117.6 = 70 sin44.427 t 12 9.8 t

2

ev, 117.6 = 40.0t 4.9t2

ev, 24 = 10t t2 [4.9 Øviv fvM K‡i cvB]

ev, t2 10t + 24 = 0

ev, t2 4t 6t + 24 = 0

ev, t(t 4) 6(t 4) = 0

ev, (t 4) (t 6) = 0

t = 4s A_ev, t = 6s

k~‡b¨ wbwÿß e ‘̄ GKB D”PZv ỳBevi AwZµg K‡i| GLv‡b, 4s

DVvi mgq I 6s bvgvi mgq|

4s ci 117.6 m G DV‡e| (Ans.)

N †Ìqv Av‡Q,



awi, me©vwaK D”PZv = H

Abyf‚wgK cvjøv = R

Ges wb‡ÿcY, †KvY =

hLb R = H, o = ?

Avgiv Rvwb, R = vo2

g sin2o

Ges H = vo2

2g sin2o

R = H

vog sin2o =

vo2g sin

2o

Vo

= 44.427

Aa¨vq-3: MwZwe`¨v

ev, sin2o = 12sin

2o

ev, 2sinocoso = 12 sin

2o

ev, 4sinocoso = sin2o

ev, 4sinocoso sin2o = 0

ev, sino (4coso sin0) = 0

nq, sino = 0 A_ev, 4coso sino = 0

ev, 0 = 0 ev, sin0 = 4coso

ev, sinocoso

= 4

ev, tano= 4

ev, o = tan1(4)

= 75.964

GLv‡b, 0 = 0 cÖ‡ÿc‡Ki cÖviw¤¢K we›`y wb‡`©k K‡i|

. = 75.964 Gi †ÿ‡Î R = H n‡e|

myZivs Abyf‚wgK cvjøv I me©vwaK D”PZv mgvb n‡Z n‡j e ‘̄wU‡K

Abyf‚wg‡Ki mv‡_ 75.964 †Kv‡Y wb‡ÿc Ki‡Z n‡e|

cÖkœ8 60m D”PZv wewkó GKwU cvnv‡oi P‚ov n‡Z GKwU Kvgv‡bi ¸wj 25 ms1 †e‡M Abyf‚wg‡Ki mv‡_ 53 †Kv‡Y †Quvov

n‡”Q| [K`gZjv c~e© evmv‡ev ¯‹zj GÛ K‡jR]

K. w¯úÖs aªæeK Kv‡K e‡j? 1

L. GKwU eo e„wói †duvUv †f‡½ A‡bK¸‡jv †QvU †duvUvq

cwiYZ Ki‡j ZvcgvÎvi Kx cwieZ©b n‡e e¨vL¨v Ki| 2

M. Kvgv‡bi ¸wjwU f‚wg n‡Z m‡e©v”P KZ D”PZvq DV‡e? 3

N. cvnv‡oi P‚ov n‡Z DÏxc‡K ewY©Z ¸wji Abyiƒc GKwU

Kvgv‡bi ¸wj GKB mgq GKB †e‡M Abyf‚wgK eivei

wb‡ÿc Kiv n‡j, †KvbwU gvwU‡Z AvNvZ Ki‡e? MvwYwZK

we‡kølY Ki| 4


K †Kv‡bv w¯úÖs‡K Gi mvg¨ve ’̄v n‡Z 1m cÖmvwiZ ev msKzwPZ Ki‡Z †h cwigvY ej cÖ‡qvM Ki‡Z nq, Zv‡K w¯úÖs aªæeK e‡j|

L GKwU eo e„wói †duvUv †f‡½ A‡bK¸‡jv †QvU †duvUvq cwiYZ Ki‡j me©‡gvU †ÿÎdj e„w× cvq| c„ôkw³i `iæY G‡ÿ‡Î A‡bK

kw³i `iKvi nq| e„nr cvwbi †duvUv n‡Z G kw³ †kvlY Kiv nq

weavq G‡ÿ‡Î ZvcgvÎvi n«vm NU‡e|

M DÏxcK †_‡K cvB,


wb‡ÿcY †KvY, 0 = 53

Avgiv Rvwb, m‡e©v”P D”PZv, H = v02sin20

2g

= (25)2 (sin53)2

2 9.8 m = 20.34 m

†h‡nZz ¸wjwU 60m DuPz Qv` †_‡K †Quvov n‡qwQj, ZvB ¸wjwU f‚wg

†_‡K m‡e©v”P (60 + 20.34) ev 80.34m D”PZvq DV‡e| (Ans.)

N Avgiv Rvwb, h = v0 sin0t1 12gt1

2

ev, 60 = 25 sin53 t1 12 9.8t1

2

ev, 4.9t12 19.97t1 60 = 0

t1 = 6.08sec A_ev, t1 = 2.011 sec

mg‡qi FYvZ¥K gvb MÖnY‡hvM¨ bq| A_©vr t1 = 6.08 sec A_©vr 1g

†ÿ‡Î ¸wjwUi gvwU‡Z AvNvZ Ki‡Z 6.08 sec mgq jvM‡e|

Avevi Abyf‚wgK eivei wbwÿß ¸wji †ÿ‡Î,

y = 12gt

22

ev, t2 = 2yg

ev, t2 = 2 60

9.8

t2 = 3.5 sec

G‡ÿ‡Î ¸wjwUi gvwU‡Z AvNvZ Ki‡Z 3.5sec mgq jvM‡e|

GLv‡b, t2 < t1

60 m

53

V0 = 25ms1

Aa¨vq-3: MwZwe`¨v A_©vr Abyf‚wgK eivei wbwÿß ¸wjwU Av‡M gvwU‡Z AvNvZ Ki‡e|

cÖkœ9 evsjv‡`k ebvg Bsj¨vÐ †U÷ g¨v‡P Zvwgg BKevj †evjvi †eb †÷vKm Gi †Qvov e‡j 40 †Kv‡Y 30ms1 AvNvZ K‡ib| [wc‡ivRcyi miKvwi gwnjv K‡jR]

K. UK© Kv‡K e‡j? 1

L. mylg †e‡M N~Y©vqgvb e„ËvKvi MwZ‡Z Z¡iY _v‡K †Kb? 2

M. Zvwgg BKev‡ji AvNvZK…Z ejwU KZÿY fvmgvb _vK‡e?

3

N. Zvwgg BKev‡ji cÖvšÍ †_‡K evDÛvwii ~̀iZ¡ 90 wgUvi n‡j

†¯‹vi 4 bv 6 n‡e? 4


K hv †Kvb AN~Y©bkxj e¯‘‡Z NyY©b m„wó K‡i ev NyY©vqgvb e ‘̄i †KŠwYK †e‡Mi cwieZ©b K‡i Zv‡K UK© e‡j|

L Avgiv Rvwb, †e‡Mi cwieZ©b N‡U ïay Gi gvb ev w`K ev Df‡qi cwieZ©‡bi Øviv| myZivs, †Kv‡bv e ‘̄i †e‡Mi gv‡bi ( ª̀æwZ)

cwieZ©b bv NU‡j I Gi w`‡Ki cwieZ©b NU‡j †e‡Mi cwieZ©b

N‡U| †e‡Mi cwieZ©b (v ) Ak~b¨ n‡j Z¡i‡Yi msÁvbymv‡i

a = v

t Z¡i‡Yi Ak~b¨ gvb _v‡K| ZvB mg`ªæwZ‡Z e„ËvKvi c‡_

Pjgvb e ‘̄i Z¡iY _v‡K| GwU Ab¨fv‡eI e¨vL¨v Kiv hvq, e„Ëc‡_

N~Y©iZ †Kv‡bv e ‘̄i Ici e„‡Ëi †K‡›`ªi w`‡K me©`v †K› ª̀gyLx ej

wµqv K‡i| D³ e‡ji `iæb e ‘̄wU‡Z Z¡iY N‡U _v‡K|

M GLv‡b, AvNvZK…Z †eM, v0 = 30ms1

AvNvZK…Z †e‡Mi †KvY, = 40


†ei Ki‡Z n‡e,

AvNvZK…Z ejwUi fvmgvb _vKvi mgq, T = ?

Avgiv Rvwb,

T = 2v0 sin

g

= 2 30 sin 40

9.8

= 3.94 s (Ans.)

N (M) Ask †_‡K Avgiv AvNvZK…Z ejwUi fvmgvb mgqKvj T †c‡qwQ|

Avgiv Rvwb, T mg‡q ejwU Abyf‚wgK w`‡K †h ~̀iZ¡ AwZµg K‡i

†mUvB ejwUi cvjøv|

AZGe Avgiv cvB, R = v0cos T

ev, R = (30 cos40 3.94)m

R = 90.5 m

†h‡nZz R > 90m| myZivs ejwU fvmgvb _vKv Ae ’̄vq evDÛvwi

AwZµg Ki‡e| A_©vr †¯‹vi n‡e 6|

cÖkœ10 evsjv‡`k wR¤^vey‡qi ga¨Kvi wgicyi †U‡÷ mvwKe GKwU ej‡K e¨v‡Ui mvnv‡h¨ AvNvZ Kivq ejwU 45 †Kv‡Y Ges 20ms1

†e‡M †evjv‡ii Dci w`‡q gv‡Vi evB‡i †h‡Z ïiæ K‡i| ga¨ gvV

†_‡K GKRb wdìvi † ùŠov‡Z ïiæ Ki‡jb| wdìviwU e‡ji jvB‡b

†cuŠQv‡bvi Av‡MB †mwU Q°v‡Z cwiYZ nq| gv‡Vi †fZi ejwUi

AwZµvšÍ `~iZ¡ 35m, XvKvq g = 9.8ms1|[†bvqvLvjx miKvwi gwnjv K‡jR]

K. w¯’wZ¯’vcKZv Kv‡K e‡j? 1

L. Lvov Dc‡i wbwÿß e ‘̄i Abyf‚wgK ~̀iZ¡ k~Y¨ nq †Kb e¨vL¨v Ki| 2

M. DÏxc‡Ki ejwU me©vwaK KZ D”PZvq DV‡e? 3

N. DÏxc‡Ki wdìvi E‡aŸ© jvd w`‡q 3m D”PZvq ej ai‡Z

cv‡ib| wZwb hw` mgq g‡Zv e‡ji jvB‡b †cuŠQ‡Z cvi‡Zb

Zvn‡j wZwb ejwU K¨vP wb‡Z mg_© n‡Zb wK? DË‡ii

mc‡ÿ MvwYwZK we‡kølY `vI| 4


K ej cÖ‡qv‡M †Kv‡bv e ‘̄i ˆ`N©¨, AvKvi ev AvqZ‡bi cwieZ©b NUv‡bv n‡j ej AcmviY Kiv gvÎB e¯‘wU c~e©ve ’̄vq wd‡i Avmvi

ag©‡K w¯’wZ¯’vcKZv e‡j|

L Lvov Dc‡i wbwÿß e ‘̄i †ÿ‡Î Abyf‚wgK w`‡K wb‡ÿcY †e‡Mi Dcvsk k~b¨| ZvB wbwÿß e ‘̄i Abyf‚wgK ~̀iZ¡I k~b¨ nq|

M DÏxcK †_‡K mvwK‡ei †ÿ‡Î cvB,

wb‡ÿcY †KvY, = 45

wb‡ÿcY †eM, u = 20ms1

me©vwaK D”PZv, H = ?

Avgiv Rvwb, me©vwaK D”PZv, H = u2sin2

2g m

= (20)2 (sin45)2

2 9.8

= 10.20m (Ans.)

N DÏxcK †_‡K cvB,

wb‡ÿcY †KvY, = 45

wb‡ÿcY †eM, u = 20ms1

ejwUi AwZµvšÍ `~iZ¡, x = 35m

Avgiv Rvwb, y = x tan gx2

2u2cos2

Aa¨vq-3: MwZwe`¨v

= 35 tan(45) 9.8 (35)2

2 (20)2 (cos 45)2 m

= 4.99m

ejwU 35m `~‡i gvwU †_‡K 4.99 m D”PZvq _vK‡e| DÏxcK

†_‡K Rvb‡Z cvwi, wdìvi E‡aŸ© jvd w`‡q 3m D”PZvq ej ai‡Z

cv‡ib|

myZivs wdìvi mgqg‡Zv e‡ji jvB‡b †cuŠQv‡Z cvi‡jI wZwb K¨vP

wb‡Z mg_© n‡Zb bv|

cÖkœ11 45o †Kv‡Y 30m/s MwZ‡e‡M GKwU e¯‘ f‚-c„ô n‡Z k~‡b¨ wb‡ÿc Kiv n‡jv| [¸iæ`qvj miKvwi K‡jR, wK‡kviMÄ]

K. UK© wK? 1

L. ¸wj Qyo‡j e›`yK †cQ‡bi w`‡K av°v †`q †Kb? 2

M. e¯‘wU m‡e©v”P KZ D”PZvq D‡V Ges KZ ̀ ~‡i f‚wg‡K AvNvZ

Ki‡e? 3

N. †`LvI †h, e ‘̄wU wb‡ÿc †e‡M f‚-c„ô‡K AvNvZ K‡i| 4


K †Kv‡bv wbw ©̀ó A‡ÿi Pviw`‡K N~Y©vqgvb †Kv‡bv e ‘̄‡Z Z¡iY m„wói Rb¨ cÖhy³ Ø‡›Øi åvgKB UK©|

L e›`yK †_‡K ¸wj †Qvov n‡j e› ỳKwU wcQ‡bi w`‡K GKwU av°v †`q| wbDU‡bi MwZi Z…Zxq m~Î †_‡K Gi GKwU e¨vL¨v †`qv hvq|

e›`yK †jvW Kivi mgq GKwU w¯úÖs‡K msKzwPZ Kiv nq| wUªMvi Pvcv

n‡j w¯úÖswU ¸wj‡K m‡Rv‡i AvNvZ K‡i, G‡Z ¸wji wcQ‡b _vKv

eviæ‡ì we‡ùviY N‡U| G we‡ùvi‡Y m„ó M¨vm ¸wji Ici cÖPÐ ej

cÖ‡qvM K‡i Ges ¸wjwU cÖPÐ †e‡M †ewo‡q Av‡m| wbDU‡bi Z…Zxq

m~Îvbymv‡i hZÿY ¸wji Ici ej wµqvkxj wQj ZZÿY ¸wjI

e›`y‡Ki Ici cðvr w`‡K cÖwZwµqv ej cÖ‡qvM K‡i‡Q| GRb¨B

¸wj †Quvovi mgq e› ỳK wcQ‡bi w`‡K GKwU av°v †`q|

M †Ìqv Av‡Q,



m‡e©v”P D”PZv, H = ?

f‚wg‡Z AvNvZ Kivi ~̀iZ¡ ev Abyf‚wgK cvjøv, R = ?


GLb, H = v20sin

2o

2g

= (30)2 (sin45o)2

2 9.8 m

= 22.959 m (Ans.)

Ges, R = v20sin2o

g = (30)2 sin(2 45o)

9.8

= 91.84 m (Ans.)

N †Ìqv Av‡Q,



g‡b Kwi, e ‘̄wUi wePiYKvj = T = 2vosino

g

myZivs T mgq c‡i e ‘̄wUi †eMB n‡e f‚-c„ô‡K AvNvZ Kivi †eM|

g‡bKwi, f‚c„ô‡K AvNvZ Kivi †eM = v

vo Gi Abyf‚wgK Dcvsk, vxo = vo coso = 30 cos45o

= 30

2 ms1

vo Gi Djø¤^ Dcvsk, vyo = vo sino = 30 sin45o = 30

2 ms1

v Gi Abyf‚wgK Dcvsk, vx = vxo = 30

2 ms1

v Gi Djø¤^ Dcvsk, vy = vyo gT

= 30

2 g

2vo sinog

= 30

2 2 30

1

2

= 30

2 ms1

v = vx2 + vy2 =

30

2

2

+

30

2

2

= 30 ms1 = vo

GLb, g‡bKwi, v, Abyf‚wg‡Ki mv‡_ †KvY ˆZwi K‡i|

tan = vyvx

=

30

2

30

2

= 1.

vy

v

vx

Aa¨vq-3: MwZwe`¨v = 45o = o

myZivs †`Lv hv‡”Q †h, e ‘̄wU wb‡ÿcY †e‡MB f‚-c„ô‡K AvNvZ

Ki‡e|

cÖkœ12 20.20m DuPz GKwU LywUi Dci evbi e‡m wQj| LywUi †Mvov n‡Z 35m ỳ‡i f‚wg n‡Z 30 †Kv‡Y I 45ms1 †e‡M ¸wj

†Qvov n‡jv| GKB mg‡q evbiwU Lvov wb‡Pi w`‡K jvd w`j|

[†g‡nicyi miKvwi K‡jR, †g‡nicyi]

K. Ae¯’vb †f±i Kv‡K e‡j? 1

L. †f±‡ii WU ¸Yb I µm ¸Y‡bi g‡a¨ cv_©K¨ wjL| 2

M. DÏxc‡Ki Av‡jv‡K ¸wjwUi wePiYKvj wbY©q Ki| 3

N. ¸wjwU evb‡ii Mv‡q jvM‡e wK? DË‡ii mc‡ÿ hyw³ `vI|

4


K cÖm½ KvVv‡gvi g~j we›`yi mv‡c‡ÿ Ab¨ †Kv‡bv we›`yi Ae ’̄vb †h †f±i Øviv cÖKvk Kiv nq Zv‡KB H we›`yi Ae¯’vb †f±i e‡j|

L †f±‡ii WU ¸Yb Ges µm ¸Y‡bi g‡a¨ cv_©K¨ wb¤œiƒc:

WU ¸Yb µm ¸Yb

1. WU ¸Yb GKwU †¯‹jvi

ivwk| Gi †Kv‡bv w`K †bB|

1. µm ¸Yb GKwU †f±i

ivwk| WvbnvwZ ¯µz wbqg

†_‡K Gi w`K wbY©q Kiv

hvq|

2. WU ¸Y‡bi gvb ivwkØ‡qi

gv‡bi Ges AšÍfz©³ ÿz`ªZi

†Kv‡Yi cosine-Gi ¸Yd‡ji

mgvb|

2. µm ¸Yd‡ji gvb

ivwkØ‡qi gv‡bi Ges

AšÍfz©³ ÿz`ªZi †Kv‡Yi sine-

Gi ¸Yd‡ji mgvb|

3. WU ¸Yb wewbgq m~Î

†g‡b P‡j|

3. µm ¸Yb wewbgq m~Î

†g‡b P‡j bv|

4. †f±i ỳwU ci¯úi j¤̂

n‡j WU ¸Yb k~b¨ nq|

4. †f±i ỳwU ci¯úi

mgvšÍivj n‡j µm ¸Yb k~b¨

nq|

M †`Iqv Av‡Q, ¸wji wb‡ÿcY †eM, v0 = 45 ms1

Ges wb‡ÿcY †KvY, 0 = 30

Avgiv Rvwb, AwfKl©R Z¡iY, g = 9.8 ms2

¸wjwUi wePiY Kvj, T = 2v0 sin0

g

= 2 45 ms1 sin 30

9.8 ms2

= 4.591 sec (Ans.)

N †`Iqv Av‡Q, ¸wjwUi wb‡ÿcY †eM, v0 = 45 ms1

wb‡ÿcY †KvY, 0 = 30

LywUi †Mvov n‡Z wb‡ÿcY we›`yi ~̀iZ¡, x = 35 m

Avgiv Rvwb, AwfKl©R Z¡iY, g = 9.8 ms2

GLb, ¸wjwUi Abyf‚wgK cvjøv, R = v02 sin 20

g

= (45 ms1)2 sin(2 30)

9.8 ms2

= 178.949 m 35 m

GLb, g‡b Kwi, x Ae ’̄v‡b ¸wji y Aÿ eivei miY y.

y = x tan0 gx2

2v02 cos20

=

35 tan 30

9.8 (35)2

2 (45)2 (cos 30)2 m

= 16.25 m

GLb, x = 35 m Abyf‚wgK ~̀iZ¡ AwZµ‡g ¸wjwUi t mgq jvM‡j,

x = v0 cos0 t

t = x

v0 cos 0

= 35 m

45 ms1 cos 30

= 0.898 sec

GLb, GB t = 0.898 sec mg‡q evbiwUi AwZµvšÍ ~̀iZ¡, h1 n‡j,

h1 = 12

gt2

= 12

9.8 (0.898)2 m

= 3.95 m

t mg‡q evbwUi Abyf‚wgK †_‡K D”PZv, h = (20.20 3.95)

= 16.25 m

†`Lv hv‡”Q, y = h = 16.25 m

myZivs ¸wjwU evb‡ii Mv‡q jvM‡e|

cÖkœ13 GK e¨w³ 40m DuPz‡Z GKwU evbi‡K †`‡L Kjv Qy‡o gvij, hv Avbyf‚wg‡Ki mv‡_ 18o †Kv‡Y 32ms1 †e‡M †Mj d‡j

evbi Zv aivi Rb¨ jvwd‡q coj GKwU Wv‡j hvi D”PZv f‚wg †_‡K

13m. [h‡kvi miKvwi wmwU K‡jR, h‡kvi]

Aa¨vq-3: MwZwe`¨v K. ZvrÿwYK †eM Kv‡K e‡j? 1

L. v t †jL †_‡K wKfv‡e Z¡iY cvIqv hvq? 2

M. wb‡ÿ‡ci 1s c‡i Kjvi †eM KZ? 3

N. evb‡ii c‡ÿ Kjv aiv m¤¢e wKbv MvwYwZK e¨vL¨v `vI| 4


K †Kvb MwZkxj e ‘̄i †Kvb GKwU we‡kl gyn~‡Z©i †eM‡K ZvrÿwYK †eM e‡j|

L v t †jL †_‡K e ‘̄i †h †Kvb gyn~‡Z©i Z¡iY wbY©q Kiv hvq| †Kvb eµ‡iLvi †Kvb we› ỳ‡Z AswKZ ¯úk©‡Ki Xvj‡KB H we› ỳ‡Z

eµ‡iLvi Xvj wn‡m‡e we‡ePbv Kiv nq| t ebvg v †jLwP‡Î t Gi

mv‡c‡ÿ v Gi e„w×i nvi dvdt Øviv GB Xvj cÖKvk Kiv nq| †h‡nZz,

a = dvdt| ZvB †Kvb we‡kl gyn~‡Z© t ebvg v †jLwP‡Îi Xvj Øviv H

gyn~‡Z©i Z¡iY a cvIqv hvq|

Dc‡ii v t †jLwP‡Îi P we› ỳ‡Z Aw¼Z ¯úk©K APB Gi Xvj Øviv

H gyn~‡Z©i Z¡iY a cvIqv hvq,

a = BCAC

M †Ìqv Av‡Q,

Kjvi wb‡ÿcY †KvY, o = 18o


1s ci Kjvi †e‡Mi Djø¤^ Dcvsk, vy = vo sino gt

= 32 sin18o 9.8 1

= 0.0885 ms1

Ges Abyf‚wgK Dcvsk, vx = vo coso

= 32 cos 18o

= 30.43 ms1

1s ci Kjvi †eM, v = v2x + v2y

= (30.43)2 + (0.0885)2

= 30.43 ms1 (Ans.)

N †Ìqv Av‡Q, f‚wg †_‡K Wv‡ji D”PZv, h = 13m

awi, 13m D”PZvq DV‡Z Kjvi cÖ‡qvRbxq mgq = t

Zvn‡j, h = vo sino t 12 gt

2

ev, 13 = 32 sin 18o t 12 9.8 t

2

ev, 4.9t2 9.88t + 13 = 0

ev, t = 9.88 (9.88)2 4 4.9 13

2 4.9

(+) gvb wb‡q cvB, t = 2.287 s

Avevi, awi, evbiwU 40m D”PZv †_‡K Wv‡j †b‡g Avm‡Z

cÖ‡qvRbxq mgq = t

Zvn‡j, (40 13) = 12 gt

2

ev, 27 = 12 gt

2

ev, t = 549.8 = 2.347 s

†h‡nZz, evbiwUi Wv‡j †b‡g Avm‡Z mgq KjvwU Wv‡ji D”PZvq

DVvi mgq A‡cÿv †ewk ZvB evb‡ii c‡ÿ Kjv aiv m¤¢e bq|

cÖkœ14 60 kmh–1 MwZ‡eM m¤úbœ GKwU †Uªb 328 m e¨vmva© wewkó †ijjvBb euvK †bqvi mgq jvBbPz¨Z n‡q ewMmn D‡ë hvq|

`yN©Ubv ’̄‡j jvB‡bi cvZØ‡qi gaëZ©x ~̀iZ¡ 1m Ges †fZ‡ii cvZ

A‡cÿv evB‡ii cvZwU 7 cm DuPz wQj|

[miKvwi mvi`v my›ìx gwnjv K‡jR, dwi`cyi; cÖkœ-2]

K. nvi‡gvwbK Kv‡K e‡j? 1

L. AvKvk †gNjv _vK‡j wkwki c‡o bv †Kb? 2

M. DÏxc‡K ewY©Z `yN©Ubv¯’‡j †UªbwU wbivc‡` m‡e©v”P KZ

†Kv‡Y AvbZ n‡Z cvi‡e? 3

N. MvwYwZK we‡køl‡Yi gva¨‡g DÏxc‡K D‡jøwLZ †ij ỳN©Ubvi

KviY D`NvUb Ki| 4


K Dcmyi¸‡jvi K¤úv¼ hw` g~j my‡ii K¤úv‡¼i mij ¸wYZK nq, Zvn‡j †mB mKj Dcmyi‡K nvi‡gvwbK e‡j|

L w`‡bi †ejvq m~‡h©i Zv‡c f‚-c„ô msjMœ evZvm Mig _v‡K Ges Rjxq ev®ú Øviv Am¤ú„³ _v‡K| †gNnxb iwÎ‡Z f‚-c„ô Zvc wewKiY

B

C

t t O

A

P

v

Aa¨vq-3: MwZwe`¨v K‡i VvÛv n‡Z _v‡K Ges cwi‡k‡l Ggb GKwU ZvcgvÎvq DcbxZ

nq hLb evZvm Rjxq ev®ú Øviv m¤ú„³ nq Ges Rjxq ev®ú Nbxf‚Z

n‡q wkwki R‡g|

wKš‘ AvKvk †gNv”Qbœ _vK‡j f‚-c„ô Zvc wewKiY K‡i VvÛv n‡Z

cv‡i bv| KviY †gN Zvc‡ivax c`v_© e‡j f‚-c„ô n‡Z wewKiYRwbZ

Kvi‡Y Zvc cwievwnZ n‡Z cv‡i bv| d‡j f‚-c„ô VvÛv nq bv Ges

wkwki R‡g bv|

M †Ìqv Av‡Q,

`yN©Ubv ’̄‡j jvB‡bi cvZØ‡qi gaëZ©x ~̀iZ¡, x = 1m

cvZ ỳwUi gaëZ©x D”PZv, h = 7 cm = 7 10–2 m

wbivc` e¨vswKs †KvY, = ?

Avgiv Rvwb,

tan = hx

ev, tan = 7 10–2m

1m

ev, tan = 7 10–2

ev, tan1 (7 102)

= 4

myZivs ỳN©Ubv ’̄‡j †UªbwU m‡e©v”P 4 †Kv‡Y wbivc‡` AvbZ n‡Z

cvi‡e| (Ans.)

N †Ìqv Av‡Q,

`yN©Ubv ’̄‡j †UªbwUi MwZ‡eM, v = 60 kmh–1

= 60 1000

3600 ms–1 = 16.67 ms–1

euv‡Ki e¨vmva©, r = 328m

Avgiv Rvwb, AwfKl©R Z¡iY, g = 9.8 ms–2

GLb, ỳN©Ubv¯’‡j †UªbwU Djø‡¤î mv‡_ †Kv‡Y AvbZ _vK‡j,

tan = v2

rg

ev, tan = (16.67 ms–1)2

328m 9.8 ms–2

ev, tan = 0.0864

= 4.94

wKš‘ ÔMÕ Ask †_‡K cvB wbivc` e¨vswKs †KvY, = 4

Avevi, wbivc` e¨vswKs †KvY Gi Rb¨ wbivc` v n‡j,

tan = v2

rg

ev, v = tan4 9.8 328 ms–1

v = 15ms–1 < 16.67 ms–1

myZivs †`Lv hv‡”Q †UªbwU wbivc` †e‡Mi †P‡q †ewk †e‡M Pj‡Z

wM‡q Djø‡¤î mv‡_ wbivc` e¨vswKs †Kv‡Yi †P‡q †ewk †Kv‡Y AvbZ

n‡q‡Q|

GKvi‡Y, †K› ª̀gyLx e‡ji †P‡q †K›`ªvwegyLx ej †ewk nIqvq †UªbwU

`yN©Ubvq cwZZ n‡q‡Q|

cÖkœ15 AešÍx 43m DuPz Qv` n‡Z GKwU ej‡K Lvov wb‡P †d‡j w`j| Zvi evÜex wcÖqv GKB iK‡gi Ab¨ GKwU ej‡K 4ms–1 †e‡M

Dc‡ii w`‡K wb‡ÿc Ki‡jv| [†g‡nicyi miKvwi K‡jR, †g‡nicyi; cÖkœ-

3]

K. f‚-w¯’i DcMÖn Kv‡K e‡j? 1

L. msiÿYkxj ej I AmsiÿYkxj e‡ji g‡a¨ cv_©K¨ wjL|

2

M. f‚wg n‡Z KZ D”PZvq AešÍxi e‡ji w ’̄wZ kw³ MwZkw³i

A‡a©K n‡e? 3

N. f‚wg n‡Z Dc‡ii †Kv‡bv we›`y‡Z wK wcÖqvi e‡ji MwZkw³

w¯’wZkw³i mgvb n‡e? hyw³mn DËi `vI| 4


K †h K…wÎg DcMÖ‡ni AveZ©bKvj 24 N›Uv Ges †eM c~e© AwfgyLx Zv‡K f‚-w ’̄i DcMÖn e‡j|

L msiÿYkxj ej I AmsiÿYkxj e‡ji cv_©K¨:

msiÿYkxj ej AmsiÿYkxj ej

1. †Kvb KYv GKwU c~Y© Pµ

m¤úbœ K‡i Zvi Avw` Ae ’̄v‡b

wd‡i Avm‡j msiÿYkxj ej

Øviv K…Z Kv‡Ri cwigvY k~b¨

nq|

1. †Kvb KYv GKwU c~Y© Pµ

m¤úbœ K‡i Zvi Avw` Ae ’̄v‡b

wd‡i Avm‡j AmsiÿYkxj ej

Øviv K…Z Kv‡Ri cwigvY k~b¨

nq bv|

2. msiÿYkxj ej Øviv †Kvb

KYvi Ici K…Z KvR KYvwUi

MwZc‡_i Ici wbf©i K‡i bv,

†Kej KYvi Avw` Ae ’̄vb I

2. AmsiÿYkxj ej Øviv †Kvb

KYvi Ici K…Z KvR KYvwUi

Avw` Ae ’̄vb I †kl Ae ’̄v‡bi

h

x

Aa¨vq-3: MwZwe`¨v †kl Ae¯’v‡bi Ici wbf©i

K‡i|

cvkvcvwk KYvwUi MwZ c‡_i

IciI wbf©i K‡i|

3. msiÿYkxj ej Øviv K…Z

KvR m¤ú~Y©iƒ‡c cybiæ×vi Kiv

m¤¢e|

3. AmsiÿYkxj ej Øviv K…Z

KvR m¤ú~Y©iƒ‡c cybiæ×vi Kiv

m¤¢e bq|

4. msiÿYkxj e‡ji wµqvi

†ÿ‡Î hvwš¿K kw³i wbZ¨Zv m~Î

Lv‡U|

4. AmsiÿYkxj e‡ji wµqvi

†ÿ‡Î hvwš¿K kw³i wbZ¨Zv m~Î

Lv‡U bv|

M †Ìqv Av‡Q,

Qv‡ì D”PZv, h = 43 m

Avgiv Rvwb, AwfKl©R Z¡iY, g = 9.8 ms–2

g‡b Kwi, f‚wg †_‡K x D”PZvq AešÍxi e‡ji w ’̄wZ kw³,

MwZkw³i A‡a©K n‡e Ges e‡ji fi m.

x D”PZvq w¯’wZkw³ = mgx

Ges x D”PZvq MwZkw³ = mgh – mgx = mg(h – x)

kZ©g‡Z,

mgx = 12 mg(h – x)

ev, x = 12 h –

12x

ev, x + 12x =

12h

ev, 3x2 =

h2

x = h3

= 433 m

= 14.33m (Ans.)

N

†Ìqv Av‡Q, f‚wg n‡Z Qv‡ì D”PZv, h = 43 m

ejwUi Avw`‡eM, v0 = 4 ms–1 (DaŸ©gyLx)

Rvbv Av‡Q, AwfKl©R Z¡iY, g = 9.8 ms–2

g‡b Kwi, ejwU Qv‡ì Dc‡i h D”PZvq D‡V †eM k~b¨ nq|

h = v02

2g = (4 ms–1)2

2 9.8 ms–2 = 0.8163 m

ejwUi f‚wg †_‡K m‡e©v”P D”PZv, h = (43 + h)m

= (43 + 0.8163) m

= 43.8163 m

Ges m‡e©v”P D”PZvq †eM, v = 0 ms–1

GLb, g‡b Kwi, f‚wg †_‡K x D”PZvq wcÖqvi e‡ji MwZkw³

w¯’wZkw³i mgvb n‡e|

GLb, x D”PZvq w¯’wZkw³ = mgx

Ges MwZkw³ = mgh – mgx

= mg(h – x)

kZ©g‡Z, mg(h – x) = mgx

ev, h – x = x

ev, 2x = h

ev, x = h2

x = 43.8163

2 = 21.91 m

myZivs, f‚wg †_‡K 21.91 m D”PZvq wcÖqvi e‡ji MwZkw³

w¯’wZkw³i mgvb n‡e|

h

h h – x

v = 0ms–1

43 m x

vo = 4ms–1

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