Aa¨vq-3: MwZwe`¨v - WordPress.com · 2017. 3. 1. · Aa¨vq-3: MwZwe`¨v = 2 o25 asin 30 9.8 =...
Transcript of Aa¨vq-3: MwZwe`¨v - WordPress.com · 2017. 3. 1. · Aa¨vq-3: MwZwe`¨v = 2 o25 asin 30 9.8 =...
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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 e`iæb‡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
e¨envi K‡i ˆeVvi 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 †`Iqv 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
wb‡ÿcY †KvY, o = 30o
AwfKl©R Z¡iY, g = 9.8 ms2
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
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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
2 bs cÖ‡kœi DËi
K mgq e¨eav‡b k~‡b¨i 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 e¨eav‡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 †`Iqv 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 †`Iqv 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
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Aa¨vq-3: MwZwe`¨v N. †MvjiÿK ejwU ai‡Z bv cvi‡j †Mvj n‡e
wKbvMvwYwZKfv‡e hvPvB Ki| 4
3 bs cÖ‡kœi DËi
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 †KvY, o = 10
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
4 bs cÖ‡kœi DËi
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
wb‡ÿcY †eM, v0 = 40ms1
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
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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 A‡j co‡e wK? Zv
MvwYwZKfv‡e e¨vL¨v Ki| 4
5 bs cÖ‡kœi DËi
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
wb‡ÿcY †KvY, o = 0
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
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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 A‡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¯‘؇qi g‡a¨ †Kvb e ‘̄wU cÖ_‡g gvwU‡Z co‡e MvwYwZK we‡kølY K‡i †`LvI| 4
6 bs cÖ‡kœi DËi
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‡`i 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
7 bs cÖ‡kœi DËi
K cÖv‡mi MwZ wØgvwÎK|
L e„ËvKvi MwZ Avgv‡`i Rxe‡bi mv‡_ IZ‡cÖvZfv‡e RwoZ| ˆe`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 †`Iqv Av‡Q,
wb‡ÿcY †eM, vo = 70ms1
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 †`Iqv Av‡Q,
wb‡ÿcY †eM, vo = 70ms1
AwfKl©R Z¡iY, g = 9.8ms2
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
8 bs cÖ‡kœi DËi
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 †eM, v0 = 25ms1
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
9 bs cÖ‡kœi DËi
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
AwfKl©R Z¡iY, g = 9.8ms2
†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? Dˇii
mc‡ÿ MvwYwZK we‡kølY `vI| 4
10 bs cÖ‡kœi DËi
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
11 bs cÖ‡kœi DËi
K †Kv‡bv wbw ©̀ó A‡ÿi Pviw`‡K N~Y©vqgvb †Kv‡bv e ‘̄‡Z Z¡iY m„wói Rb¨ cÖhy³ ؇›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‡`i 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 †`Iqv Av‡Q,
wb‡ÿcY †eM, vo = 30 ms1
wb‡ÿcY †KvY, o = 45o
m‡e©v”P D”PZv, H = ?
f‚wg‡Z AvNvZ Kivi ~̀iZ¡ ev Abyf‚wgK cvjøv, R = ?
AwfKl©R Z¡iY, g = 9.8 ms2
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 †`Iqv Av‡Q,
wb‡ÿcY †eM, vo = 30 ms1
wb‡ÿcY †KvY, o = 45o
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? Dˇii mc‡ÿ hyw³ `vI|
4
12 bs cÖ‡kœi DËi
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 ivwk؇qi
gv‡bi Ges AšÍfz©³ ÿz`ªZi
†Kv‡Yi cosine-Gi ¸Yd‡ji
mgvb|
2. µm ¸Yd‡ji gvb
ivwk؇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
13 bs cÖ‡kœi DËi
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 †`Iqv Av‡Q,
Kjvi wb‡ÿcY †KvY, o = 18o
wb‡ÿcY †eM, vo = 32 ms1
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 †`Iqv 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 cvZ؇qi ga¨eZ©x ~̀iZ¡ 1m Ges †fZ‡ii cvZ
A‡cÿv evB‡ii cvZwU 7 cm DuPz wQj|
[miKvwi mvi`v my›`ix 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
14 bs cÖ‡kœi DËi
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 iw·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 †`Iqv Av‡Q,
`yN©Ubv ’̄‡j jvB‡bi cvZ؇qi ga¨eZ©x ~̀iZ¡, x = 1m
cvZ ỳwUi ga¨eZ©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 †`Iqv 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‡¤^i 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‡¤^i 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
15 bs cÖ‡kœi DËi
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 †`Iqv Av‡Q,
Qv‡`i 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
†`Iqv Av‡Q, f‚wg n‡Z Qv‡`i 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‡`i 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
-
Aa¨vq-3: MwZwe`¨v