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c_Ö g c‡Îi As‡Ki mgvavb ‡gv: kvn Rvgvj
mnKvix Aa¨vcK (c`v_©weÁvb wefvM )
First Paper Mathematics Solution we G Gd kvnxb K‡jR ‡ZRMuvI, XvKv
3| MwZwe`¨v (Dynamics) ‡dvb: +8801670856105, +88029125630, +88029115369
e-mail: [email protected]
1| GKwU e›`‡y Ki ¸wj †Kvb †`Iqv‡ji g‡a¨ 0.04m cÖ‡ek Kivi 3| 20ms-1 †e‡M MwZkxj GKwU e¯‘i †eM cwÖ Z †m‡K‡Û 3ms-1 nv‡i
ci A‡a©K †eM nvivq | MywjwU †`Iqv‡ji
g‡a¨ Avi KZUKz z cÖ‡ek Ki‡e? n«vm cvq| †_‡g hvIqvi Av‡M e¯‘wU KZ `i~ Z¡ AwZµg Ki‡e?
Avgiv Rvwb, v2 = u2 – 2as GLv‡b,
ev, 0 = 202 – 2(3)s Avw`‡eM, u = 20 ms-1
g‡b Kwi, ev, 6s = 400 g›`b, a = 3 ms-2
jÿ¨¯‡’ j c‡Ö e‡ki gyn~‡Z© ¸wji Avw`‡eM = u ev, s 400 ‡kl‡eM, v = 0
Ges ¸wjwU AviI x wgUvi `i~ Z¡ c‡Ö ek Ki‡e| 6 _vgvi Av‡M e¯‘wU KZ…K
0.04 m cÖ‡ek Kivi ci †eM n‡e = u s 66.7 m (Ans.) AwZµvšÍ `i~ Z,¡ s = ?
2
4| Dc‡ii w`‡K wbwÿß GKwU ej †Uwj‡dvb Zvi‡K 0.70ms-1 `ªæwZ‡Z
Ges †kl †eM n‡e 0 (k~b¨)|
Avgiv Rvwb, cÖ_g As‡ki Rb¨ AvNvr K‡i| †Qvovi ¯’vb †_‡K ZviwUi D”PZv 5.1m n‡j ejwUi Avw`
u 2 u2 2a(0.04) `ªæwZ KZ wQj?
2 Avgiv Rvwb,
v2 u2 2gh GLv‡b,
0.08a u 2 u 2 (0.7)2 u 2 2 9.8 5.1
4 u 2 (0.7)2 2 9.8 5.1 D”PZv, h =5.1m
u 2 0.49 99.96 g = 9.8 ms-2
a 3u 2 3u 2 ..... .......... (1) ‡kl †eM, v =0.70ms-1
4 0.08 0.32 Avw` `ªæwZ, u =?
wØZxq As‡ki Rb¨,
0 u 2 2ax u2 100.45
2
u 100.45 10.02 ms-1 (Ans.)
ev, 0 u 2 2 3u 2 x 5| GKwU †Ubª 3ms-2 mgZi¡ ‡b Pj‡Q Ges Avw`‡eM 10m/s †UªbwU hLb
2 0.32 60m c_ AwZµg Ki‡e ZLb Gi †eM KZ n‡e|
ev, 6u 2x u 2 Avgiv Rvwb, GLv‡b,
0.32 4
v2 u2 2as Zi¡ Y, a = 3ms-2
x 0.32 0.0133 m (Ans.) v2 102 2 3 60 Avw`‡eM, u = 10 ms-1
64 v2 100 360 miY, s = 60m
v2 460
‡kl‡eM, v = ?
2| 50 wgUvi DPyu †_‡K GKwU e¯‘ fw~ g‡Z cwZZ nq| v 460 21.447 21.45ms1 (Ans)
(K) fywg‡Z †cŠQu ‡Z Gi KZ mgq jvM‡e?
(L) fwy g‡Z †cŠQu evi ce~ © gny ‡~ Z© Gi †eM KZ n‡e? 6| GKwU e¯‡‘ K 98 ms-1 †e‡M Lvov Dc‡ii w`‡K wb‡ÿc Kiv n‡j
(K) h ut 1 gt 2 GLv‡b, †`LvI †h, 3 Sec I 17 Sec mg‡q e¯i‘ †eMØq mgvb wKš‘ w`K
2 D”PZv, h =50m
Avw`‡eM, u = 0 wecixZ gLy x| GLv‡b,
50 0 1 9.8 t 2 g = 9.8 ms-2 Avgiv Rvwb, Avw`‡eM, u = 98 ms-1
2 (K) mgq, t = KZ? 3 †mt c‡i †eM
(L) ‡kl †eM, v = KZ ? mgq, t1 = 3S
50 4.9t 2 v1 = u gt1 mgq, t2 = 17S
t 2 50 ev, v1 = 989.8×3 †kl‡eM, v1 =?
4.9 ev, v1 = 9829.4 †kl‡eM, v2 =?
t 50 v1 = 68.6ms-1
4.9
Avevi, 17 †mt c‡i †eM
t = 3.19 s (Ans.)
Avevi, v2 = u gt2
(L) v = u + gt ev, v2 = 98 9.8×17
⇒ v = 0 + 9.8 × 3.19 ev, v2 = 98 166.6
v = 31.26 ms-1 (Ans.) v2 = 68.6 ms-1
3 †mt I 17 †mt c‡i †eR Øq mgvb I wecixZ (cÖgvwYZ)
3| MwZwe`¨v (Dynamics) 2
7| s 1 t3 3t mÎ~ vbmy v‡i GKwU e¯‘ mij †iLvq Pj‡Q| t 2 2 1.414s
3 ‡k‡li 1m `i~ Z¡ AwZµg Ki‡Z mgq
2 †m‡KÛ ci Gi †eM KZ n‡e? jv‡M, t t2 t1 (1.414 1) s 0.414s (Ans.)
Avgiv Rvwb,
v ds GLv‡b, 10| GKwU ‡Uªb w¯’i Ae¯v’ b n‡Z 10ms-2 Z¡i‡Y Pj‡Z Avi¤¢ Kij| GKB
dt mgq, t = 2 Sec mgq GKwU Mvwo 100ms-1 mg‡e‡M †U‡ª bi mgvšiÍ v‡j Pjv ïiæ Kij| †Uªb
‡eM, v =? MvwowU‡K KLb wcQ‡b †dj‡e?
v d 1 t 3 3t
dt 3 g‡b Kwi, t mgq ci †Ubª MvwowU‡K GLv‡b,
wcQ‡b †d‡j P‡j hv‡e, Mvwoi mg‡eM, V = 100ms-1
v 1 3t 2 3 t mgq †Ub KZK… AwZµvšÍ `i~ Z¡, ‡U‡ª bi Z¡iY, a = 10ms-2
3
x 0 1 at 2 mgq, t = ?
v t2 3 2
v 22 3 [t Gi gvb ewm‡q] x 1 10 t 2
2
v 7 GKK („Ans.)
8| 54 kmh1 †e‡M PjšÍ GKwU †ij Mvwo‡Z †÷mb †_‡K wKQy `y‡i x 5t 2... ... ... (1)
0.75ms-2 g›`b m„wóKvix †eKª ‡`Iqvq MvwowU †÷m‡b G‡m †_‡g †Mj| t mg‡q Mvwo KZK… AwZµvšÍ `i~ Z,¡ x Vt
x 100t ... ... ... (2)
†÷mb ‡_‡K KZ `~‡i †eªK †`Iqv n‡qwQj Ges MvwowU _vg‡Z KZ
mgq †j‡MwQj? kZ©g‡Z †Uªb hLb MvwowU‡K AwZµg Ki‡e ZLb x x n‡e|
Avgiv Rvwb, 5t 2 100t
v2 = u22as GLv‡b, t 100 t 20s (Ans.)
0 152 2 0.75 s Avw`‡eM, u = 54 kmh-1 5
s 1515 m
541000 ms1 11| w¯’ive¯v’ †_‡K Pj‡Z Avi¤¢ K‡i 625m `i~ Z¡ AwZµg Ki‡j GKwU
2 0.75 3600
s 150 m (Ans.) e¯‘i †eM 125ms-1 nj| Z¡iY wbYq© Ki|
=15ms-1
Avevi, v = u at g›`b, a = 0.75ms-2 Avgiv Rvwb, GLv‡b,
Avw`‡eM, u = 0
0 15 0.75 t †kl‡eM, v = 0 v2 u 2 2as AwZµvšÍ `~iZ¡ , s = 625m
1252 0 2 a 625
t 15 s mgq, t = ?
0.75 miY, s=? a 1252 ms 2 ‡kl †eM, v = 125 ms-1
2 625 Zi¡ Y, a =?
t 20s (Ans.) a 12.5ms 2 (Ans.)
9| GKwU e¯‘ w¯i’ Ae¯’vb n‡Z hvÎv ïiæ K‡i c_Ö g †m‡K‡Û 1m 12| 64m DuPz `vjv‡bi Qv` †_‡K 5kg f‡ii GKwU cv_i †Q‡o w`‡j
`i~ Z¡ AwZµg K‡i| cieZ©x 1m `~iZ¡ AwZµg Ki‡Z KZ mgq f~wg‡Z †cuŠQv‡Z Gi KZ mgq jvM‡e?
jvM‡e| GLv‡b, Avgiv Rvwb,
Avgiv Rvwb, GLv‡b,
s1 ut1 1 at12 Avw`‡eM, u = 0 h ut 1 gt 2 Avw`‡eM, u = 0
2 2 AwZµvšÍ `i~ Z¡ , h = 64m
mgq, t1 = 1s fi, m= 5kg
1 0 1 a(1)2 miY, s1 =1m 64 0 1 9.8 t 2 mgq, t =?
2 Z¡iY, a=? 2
1 a 64 4.9t 2
2
t 64 t 3.61s. (Ans.)
a 2ms2 4.9
GLb cÖ_g †_‡K s2 = (1m+1) =2m `~iZ¡ AwZµg Ki‡Z mgq 13| w¯’i Ae¯’vb n‡Z hvÎv Avi¤¢ K‡i GKwU e¯‘ c_Ö g †m‡K‡Û 2m `~iZ¡
jv‡M = t2
AwZµg K‡i| cieZx© 2m `~iZ¡ AwZµg Ki‡Z e¯w‘ Ui KZ mgq
s2 ut 2 1 at 2 jvM‡e| GLv‡b,
2 2 Avgiv Rvwb,
Avw`‡eM, u = 0
2 0 1 2 t 2 s1 ut1 1 at 2 mgq, t1 = 1s
2 2 2 1 miY, s1 =1m
Z¡iY, a =?
t 2 2 2 0 1 a(1) 2
2 2
3| MwZwe`¨v (Dynamics) 3
2 a 15| GKwU cÖv‡mi AbyfywgK cvjvø 96m Ges Avw`‡eM 66 ms-1| wb‡ÿc
2
†KvY KZ?
a 4ms2
Avgiv Rvwb, GLv‡b,
GLb c_Ö g †_‡K s2 = (2m+2m) = 4m `~iZ¡ AwZµg Ki‡Z
mgq jv‡M = t2 R vo2Sin 2o Abyfwy gK cvjøv, R = 96 m
g
Avw`‡eM, vo = 66ms-1
AwfKlR© Zi¡ Y, g = 9.8ms-2
s2 ut 2 1 at 2 ev , Sin2θo Rg wb‡ÿc †KvY, = ?
2 2
v 2
0
4 0 1 4 t 2 ev, Sin2o 96 9.8
2 2 662
t 2 2 ev, 2θo Sin 1(0.2159)
2
t 2 2 1.414s ev, 2θo 12.47
‡k‡li 2m `~iZ¡ AwZµg Ki‡Z mgq θo 6.24 (Ans.)
jv‡M, t t 2 t1 (1.414 1) s 0.414s (Ans.)
14| GKwU e¯‘ c_Ö g `yB †m‡K‡Û 30m I cieZ©x Pvi †m‡K‡Û 16| GKwU e¯‡‘ K 40ms-1 †e‡M Abf~ ywg‡Ki mv‡_ 60° ‡Kv‡Y wb‡ÿc Kiv
nj| mev© waK D”PZv Ges Abyf~wgK cvjvø wbYq© Ki|
150m ‡Mj| Zi¡ Y AcwiewZ©Z _vK‡j e¯‘wU Gi ci GK †m‡K‡Û Avgiv Rvwb,
KZUv c_ AwZµg Ki‡e? GLv‡b, (v o Sin θ o ) 2
Avgiv Rvwb, H
`~iZ¡, s1 = 30m
s1 ut1 1 at 2 mgq, t1 = 2s 2g GLv‡b,
2 1 miY, s2 = (30+150) Avw`‡eM, v0 = 40ms-1
40 Sin 602
30 u 2 1 a 22 =180m wb‡ÿc †KvY 60º
2 miY, s7 =? H AwfKlR© Zi¡ Y,
2 9.8 g = 9.8ms-2
u a 15..........(1) mev© waK D”PZv, H = ?
H 40 Sin 60 2
cÖ_g †_‡K t2= (2+4)= 6 †m‡K‡Û e¯w‘ U hvq Abfy wy gK cvjøv, R = ?
2 9.8
s2=(30+150)m=180m
40 0.86602 2
s2 ut2 1 at22
2 H
2 9.8
180 u 6 1 a 62
2 34.6408 2
u 3a 30..........(2) H
2 9.8
u a 15..........(1)
1199 .9850
we‡qvM K‡i, 2a= 15 a 7.5ms2 H
GLb (1) bs mgxKi‡Y a Gi gvb ewm‡q,
2 9.8
u 7.5 15 u 7.5ms1
6 ‡m‡K‡Ûi c‡ii †m‡KÛ A_v© r 7g †m‡K‡Û AwZµvšÍ `~iZ¡, H 61.22 m (Ans .)
Avevi,
R v 2 Sin 2θo
0
g
st u a (2t 1) R 402Sin (2 60)
2 9.8
s7 7.5 7.5 (2 7 1) R 402 Sin ( 2 60)
2 9.8
s7 7.5 7.5 13 R 1600 Sin 120)
2 9.8
s7 7.5 48.75 R 1600 0.86602
9.8
s7 56.25m (Ans)
R 1385.632
9.8
R 141.39 m (Ans.)
3| MwZwe`¨v (Dynamics) 4
17| nvB‡Wvª ‡Rb cigvbiy g‡W‡ji GKwU B‡jKUbª GKwU †cÖvU‡bi y 28.86751346 6.533333345
Pviw`‡K 5.2 ×10 -11 m e¨vmv‡a©i GKwU e„ËvKvi c‡_ 2.18 ×106 y 22.33 m (Ans.)
ms-1 †e‡M c`Ö wÿb K‡i| B‡jKU‡ª bi fi 9.1 ×10-31 kg n‡j
†K›`gª yLx ej KZ? 21| GKwU cÖv‡mi Abfy wy gK cvjvø 79.53 m Ges wePiYKvj 5.3 s n‡j
Avgiv Rvwb, wb‡ÿc †KvY I wb‡ÿc †eM KZ?
F mv 2 Avgiv Rvwb, GLv‡b,
r R vo2Sin 2o Abyfwy gK cvjvø , R = 79.53 m
wePiYKvj, T=5.3s
F 9.1 10 -31 (2.18106)2 g wb‡¶c ‡eM, vo = ?
5.2 10-11 79.53 v2oSin 2o wb‡¶c †KvY, = ?
F 8.316108 N (Ans.) 9.8
18| 0.250kg f‡ii GKwU cv_i LÛ‡K 0.75m j¤^v GKwU myZvi vo2Sin2o 779.394.. ... (1)
GK cÖv‡šÍ †eu‡a eË„ vKvi c‡_ cÖwZ wgwb‡U 90 evi Niy v‡j myZvi Dci Aevi, T 2voSino
g
KZ Uvb co‡e| GLv‡b,
Avgiv Rvwb, 5.3 2voSino
fi, m = 0.250 kg 9.8
F m2r e¨vmva,© r = 0.75 m
F m 2πn 2 r mgq t = 1 min. 2voSino 51.94 ... ... ... (2)
(1) bs mgxKiY‡K (2) bs mgxKiY Øviv fvM K‡i cvB,
t = 60s.
cvKmsL¨v, n = 90 cvK| vo2Sin 2o 779.394
F 0.25 23.1416902 0.75 Uvb, F = ? 2voSino 51.94
60
F 16.65 N (Ans.) v 2 2Sin oCoso 779.394
o
19| 9.2 ms-1 †e‡M GKwU ÿz`ª e¯‘‡K Lvov Dc‡ii w`‡K wb‡ÿc
Kiv nj| GwU KZ mgq c‡i f-~ c„‡ô wd‡i Avm‡e? 2voSino 51.94
v0Coso 15 ... ... .. (3)
Avgiv Rvwb, GLv‡b, (2) bs mgxKiY‡K (3) bs mgxKiY Øviv fvM K‡i cvB,
Avw`‡eM, vo = 9.2 ms-1
T 2voSino 2voSino 51.94
g wb‡¶c †KvY, 0º v0Coso 15
AwfKlR© Zi¡ Y,
T 2 9.2 Sin90 g = 9.8ms-2 tan o 3.463
9.8 DÌvb cZ‡bi †gvU 2
T 2 9.2 1 mgq T =? tan o 1.732
9.8
o tan1 1.732
T 18.4
9.8 o 60(Ans.)
T 1.877s (Ans.) (3) bs mgxKi‡Y 0 Gi gvb ewm‡q cvB,
v0Cos60 15
20| Abyfywg‡Ki mv‡_ 30°†KvY f~-cô„ †_‡K 50ms-1 †e‡M GKwU v0 1 15
2
e‡y jU †Qvov nj| e‡y jUwU 50m `~‡i Aew¯’Z GKwU †`Iqvj‡K KZ
D”PZvq AvNvZ Ki‡e| v0 30ms 1 (Ans.)
Avgiv Rvwb, GLv‡b,
y (tan 0 )x 2(v0 g x2 Avw`‡eM, vo = 50 ms-1 22| GKwU ej‡K fw~ gi mv‡_ 30°†KvY K‡i Dc‡ii w`‡K wb‡ÿc Kiv
cos 0 )2 n‡j GwU 20m `‡~ i GKwU `vjv‡bi Qv‡` wM‡q coj| wb‡ÿc we›`y †_‡K
wb‡¶c †KvY, º
AwfKl©R Zi¡ Y,
g = 9.8ms-2 Qv‡`i D”PZv 5m n‡j ejwU KZ †e‡M †Qvov n‡qwQj|
y (tan30 )x g (50)2 Abfy ywgK `i~ Z,¡ x=50m Avgiv Rvwb,
cos30 Dj¤^ `~iZ,¡ y=?
2(v0 )2 g x2 GLv‡b,
y (tan 0 )x cos 0 )2
2(v0 wb‡¶c †KvY, º
y (tan30)50 g (50)2 AwfKlR© Zi¡ Y,
2(v0 cos30 )2 g = 9.8ms-2
Abyfwy gK `i~ Z,¡ x=20m
y 0.57735026950 9.8 (50)2 5 tan3020 9.8 (20)2
2(50 0.86602540)3 2 2v02 cos2 30 Dj¤^ `~iZ,¡ y=5m
Avw`‡eM, vo =?
3| MwZwe`¨v (Dynamics) 5
9.8 400 24| GKwU MvÖ ‡gv‡dvb †iKW© cwÖ Z wgwb‡U 45 evi N‡y i| Gi †K›`ª †_‡K
5 0.57735026920
9cm `‡~ i †Kvb we›`iy `æª wZ KZ?
2 v02 0.75
v r
3920 GLv‡b,
5 11.54 v02 1.5
v 2n r mgq, t = 1m =60s
3920 6.54 t cvKmsL¨v, n=45
e¨vmva,© r =9cm=0.09m
v 2 1.5 v 2 3.14 45 0.09 `ªæwZ, v =?
0 60
v 2 3920 v0 20ms1(Ans.) v 0.42ms1 (Ans.)
0 6.54 1.5
23| GKRb †jvK 48 ms-1 †e‡M GKwU ej Lvov Dc‡ii w`‡K
wb‡ÿc K‡i| ejwU KZ mgq k‡~ Y¨ _vK‡e Ges m‡ev© ”P KZ Dc‡i
DV‡e?
Avgiv Rvwb, GLv‡b,
T 2voSinθo †eM, v0 = 48 ms-1
g wb‡¶c †KvY, =º
T 2 48Sin90 DÌvb cZ‡bi †gvU mgq,T =?
9.8 D”PZv, H =?
T 2 481
9.8
T 9.795 s. ( Ans.)
Avevi,
H vo2Sin 2o
2g
H 482 (Sin90)2
2 9.8
H 117.5m (Ans.)
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