math solution 1st 03 dynamics (PDF)

cÖ_g c‡Îi As‡Ki mgvavb
First Paper Mathematics Solution
3| MwZwe`¨v (Dynamics)
1| GKwU e›`y‡Ki ¸wj †Kvb †`Iqv‡ji g‡a¨ 0.04m cÖ‡ek Kivi
ci A‡a©K †eM nvivq | MywjwU †`Iqv‡ji
g‡a¨ Avi KZUzKz cÖ‡ek Ki‡e?

u
2

24

Ges †kl †eM n‡e 0 (k~b¨)|
Avgiv Rvwb, cÖ_g As‡ki Rb¨

4| Dc‡ii w`‡K wbwÿß GKwU ej †Uwj‡dvb Zvi‡K 0.70ms-1 `ªæwZ‡Z
AvNvr K‡i| †Qvovi ¯’vb †_‡K ZviwUi D”PZv 5.1m n‡j ejwUi Avw`
`ªæwZ KZ wQj?
Avgiv Rvwb,
GLv‡b,
v 2  u 2  2gh
D”PZv, h =5.1m
 (0.7) 2  u 2  2  9.8  5.1 g = 9.8 ms-2
-1
 u 2  (0.7) 2  2  9.8  5.1 ‡kl †eM, v =0.70ms
Avw` `ªæwZ, u =?
 u 2  0.49  99.96
 u 2  100.45
u  100.45  10.02 ms-1 (Ans.)

2

al

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bd

u
2
   u  2a(0.04)
2
u2
 0.08a  u 2 
4
3u 2
3u 2
a

..... .......... (1)
4  0.08 0.32

m

wØZxq As‡ki Rb¨,
2

Ja

u
0     2ax
 2
2

Sh

M

d

0.32
 0.0133 m (Ans.)
6 4

C

op
yr

ig

ht

@

2| 50 wgUvi DPuy †_‡K GKwU e¯‘ f~wg‡Z cwZZ nq|
(K) fywg‡Z †cŠuQ‡Z Gi KZ mgq jvM‡e?
(L) fywg‡Z †cŠuQevi c~e© gyn~‡Z© Gi †eM KZ n‡e?

1
(K) h  ut  gt 2
2
1
 50  0  9.8  t 2
2
 50  4.9 t 2
50
 t2 
4. 9
50
t
4.9

t = 3.19 s (Ans.)
Avevi,
(L) v = u + gt
⇒ v = 0 + 9.8 × 3.19
 v = 31.26 ms-1 (Ans.)

5| GKwU †Uªb 3ms-2 mgZ¡i‡b Pj‡Q Ges Avw`‡eM 10m/s †UªbwU hLb
60m c_ AwZµg Ki‡e ZLb Gi †eM KZ n‡e|
Avgiv Rvwb,
GLv‡b,
Z¡iY, a = 3ms-2
2
2
v  u  2as
Avw`‡eM, u = 10 ms-1
2
2
 v  10  2  3  60
miY, s = 60m
 v 2  100  360
‡kl‡eM, v = ?
2
 v  460
 v  460  21.447  21.45ms 1 ( Ans )

ah

3u 2
u
ev , 0     2 
x
0.32
2
6u 2 x u 2
ev ,

0.32
4
 x

3| 20ms-1 †e‡M MwZkxj GKwU e¯‘i †eM cÖwZ †m‡K‡Û 3ms-1 nv‡i
n«vm cvq| †_‡g hvIqvi Av‡M e¯‘wU KZ `~iZ¡ AwZµg Ki‡e?
Avgiv Rvwb, v2 = u2 – 2as
GLv‡b,
ev, 0 = 202 – 2(3)s
Avw`‡eM, u = 20 ms-1
ev, 6s = 400
g›`b, a = 3 ms-2
400
‡kl‡eM, v = 0
ev, s 
_vgvi Av‡M e¯‘wU KZ…K
6
 s  66.7 m (Ans.) AwZµvšÍ `~iZ,¡ s = ?

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o

jÿ¨¯’‡j cÖ‡e‡ki gyn~‡Z© ¸wji Avw`‡eM = u
Ges ¸wjwU AviI x wgUvi `~iZ¡ cÖ‡ek Ki‡e|

 0.04 m cÖ‡ek Kivi ci †eM n‡e =

‡dvb: +8801670856105, +88029125630, +88029115369
e-mail: sjamal59@gmail.com

m

g‡b Kwi,

‡gv: kvn Rvgvj
mnKvix Aa¨vcK (c`v_©weÁvb wefvM )
we G Gd kvnxb K‡jR ‡ZRMuvI, XvKv

GLv‡b,
D”PZv, h =50m
Avw`‡eM, u = 0
g = 9.8 ms-2
(K) mgq, t = KZ?
(L) ‡kl †eM, v = KZ ?

6| GKwU e¯‘‡K 98 ms-1 †e‡M Lvov Dc‡ii w`‡K wb‡ÿc Kiv n‡j
†`LvI †h, 3 Sec I 17 Sec mg‡q e¯‘i †eMØq mgvb wKš‘ w`K
wecixZ gyLx|
GLv‡b,
Avgiv Rvwb,
Avw`‡eM, u = 98 ms-1
3 †mt c‡i †eM
mgq, t1 = 3S
v1 = u gt1
mgq, t2 = 17S
ev, v1 = 989.8×3
†kl‡eM, v1 =?
†kl‡eM, v2 =?
ev, v1 = 9829.4
 v1 = 68.6ms-1
Avevi, 17 †mt c‡i †eM
v2 = u gt2
ev, v2 = 98 9.8×17
ev, v2 = 98 166.6
 v2 = 68.6 ms-1
 3 †mt I 17 †mt c‡i †eR Øq mgvb I wecixZ (cÖgvwYZ)

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3| MwZwe`¨v (Dynamics)

1
7| s  t 3  3t m~Îvbymv‡i GKwU e¯‘ mij †iLvq Pj‡Q|
3

 t 2  2  1.414s
‡k‡li 1m `~iZ¡ AwZµg Ki‡Z mgq
jv‡M, t  t 2  t1  (1.414  1) s  0.414s (Ans.)

2 †m‡KÛ ci Gi †eM KZ n‡e?
Avgiv Rvwb,

ds
dt

8| 54 kmh1 †e‡M PjšÍ GKwU †ij Mvwo‡Z †÷mb †_‡K wKQy `y‡i
0.75ms-2 g›`b m„wóKvix †eªK ‡`Iqvq MvwowU †÷m‡b G‡m †_‡g †Mj|
†÷mb ‡_‡K KZ `~‡i †eªK †`Iqv n‡qwQj Ges MvwowU _vg‡Z KZ
mgq †j‡MwQj?
Avgiv Rvwb,
v2 = u22as
GLv‡b,

 0  152  2  0.75  s
15 15
s
m
2  0.75
 s  150 m (Ans.)

54  1000 1
ms
3600

Avevi, v = u  at

24

5t 2  100 t
100
t
 t  20s (Ans.)
5
11| w¯’ive¯’v †_‡K Pj‡Z Avi¤¢ K‡i 625m `~iZ¡ AwZµg Ki‡j GKwU
e¯‘i †eM 125ms-1 nj| Z¡iY wbY©q Ki|
Avgiv Rvwb,

v 2  u 2  2as
 125 2  0  2  a  625
125 2
a
ms 2
2  625
 a  12.5ms 2 (Ans.)

GLv‡b,
Avw`‡eM, u = 0
AwZµvšÍ `~iZ¡ , s = 625m
‡kl †eM, v = 125 ms-1
Z¡iY, a =?

d

Sh

 0  15  0.75  t
15
t
s
0.75
 t  20s (Ans.)

kZ©g‡Z †Uªb hLb MvwowU‡K AwZµg Ki‡e ZLb x  x  n‡e|

Ja

=15ms-1
g›`b, a = 0.75ms-2
†kl‡eM, v = 0
mgq, t = ?
miY, s=?

 x   100 t ... ... ... (2)

ah



t mg‡q Mvwo KZ…K AwZµvšÍ `~iZ¡, x   Vt

m

Avw`‡eM, u = 54 kmh-1

2
1
 x   10  t 2
2
 x  5t 2 ... ... ... (1)

bd

1
 3t 2  3
3
 v  t2  3
 v  2 2  3 [t Gi gvb ewm‡q]
 v  7 GKK („Ans.)
v

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o

m

d 1 3

 t  3t 
dt  3


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v

10| GKwU ‡Uªb w¯’i Ae¯’vb n‡Z 10ms-2 Z¡i‡Y Pj‡Z Avi¤¢ Kij| GKB
mgq GKwU Mvwo 100ms-1 mg‡e‡M †Uª‡bi mgvšÍiv‡j Pjv ïiæ Kij| †Uªb
MvwowU‡K KLb wcQ‡b †dj‡e?
g‡b Kwi, t mgq ci †Uªb MvwowU‡K
GLv‡b,
wcQ‡b †d‡j P‡j hv‡e,
Mvwoi mg‡eM, V = 100ms 1
t mgq †Ub KZ…K AwZµvšÍ `~iZ¡,
‡Uª‡bi Z¡iY, a = 10ms 2
1
mgq, t = ?
x  0  at 2

GLv‡b,
mgq, t = 2 Sec
‡eM, v =?

al

v

2

2

ig

ht

@

M

9| GKwU e¯‘ w¯’i Ae¯’vb n‡Z hvÎv ïiæ K‡i cÖ_g †m‡K‡Û 1m
`~iZ¡ AwZµg K‡i| cieZ©x 1m `~iZ¡ AwZµg Ki‡Z KZ mgq
jvM‡e|
GLv‡b,
Avgiv Rvwb,
Avw`‡eM, u = 0
1
s1  ut 1  at12
mgq, t1 = 1s
2
miY, s1 =1m
1
2
Z¡iY, a=?
 1  0  a (1)

op
yr

a
2
 a  2ms 2
1

C

GLb cÖ_g †_‡K s2 = (1m+1) =2m `~iZ¡ AwZµg Ki‡Z mgq
jv‡M = t2

1 2
at 2
2
1
 2  0   2  t 22
2
2
 t2  2
s 2  ut 2 

12| 64m DuPz `vjv‡bi Qv` †_‡K 5kg f‡ii GKwU cv_i †Q‡o w`‡j
f~wg‡Z †cuŠQv‡Z Gi KZ mgq jvM‡e?
Avgiv Rvwb,
GLv‡b,

1
h  ut  gt 2
2
1
 64  0   9.8  t 2
2
 64  4.9t 2
64
t
 t  3.61 s.
4.9

Avw`‡eM, u = 0
AwZµvšÍ `~iZ¡ , h = 64m
fi, m= 5kg
mgq, t =?

(Ans.)

13| w¯’i Ae¯’vb n‡Z hvÎv Avi¤¢ K‡i GKwU e¯‘ cÖ_g †m‡K‡Û 2m `~iZ¡
AwZµg K‡i| cieZ©x 2m `~iZ¡ AwZµg Ki‡Z e¯‘wUi KZ mgq
jvM‡e|
GLv‡b,
Avgiv Rvwb,
Avw`‡eM, u = 0

1
s1  ut1  at 12
2

 2  0

1
a(1) 2
2

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mgq, t1 = 1s
miY, s1 =1m
Z¡iY, a =?

3| MwZwe`¨v (Dynamics)
3
15| GKwU cÖv‡mi AbyfywgK cvjøv 96m Ges Avw`‡eM 66 ms-1| wb‡ÿc
†KvY KZ?
GLv‡b,
Avgiv Rvwb,
2
AbyfywgK cvjøv, R = 96 m
v Sin 2 o
GLb cÖ_g †_‡K s2 = (2m+2m) = 4m `~iZ¡ AwZµg Ki‡Z
R o
Avw`‡eM, vo = 66ms-1
g
mgq jv‡M = t2
AwfKl©R Z¡iY, g = 9.8ms-2
1
R g
wb‡ÿc †KvY,  = ?
s  ut  at 2
ev , Sin2θ 

a
2
2
 a  4ms 2

o

2

 θ o  6.24 (Ans.)

1
 30  u  2  a  2 2
2
u  a  15..........(1)

=180m

miY, s7 =?
†m‡K‡Û

Sh

@

we‡qvM K‡i, 2a= 15  a  7.5ms 2
GLb (1) bs mgxKi‡Y a Gi gvb ewm‡q,

ht

ig

op
yr

C

bd

40 Sin

H

 H 

 H 

2
60  

2  9.8

40 Sin

 H 

2
60  

2  9.8

40  0.86602 2
2  9.8
34 .6408 2

2  9.8
1199 .9850

2  9 .8

 H  61.22 m (Ans .)
Avevi,
R

6 ‡m‡K‡Ûi c‡ii †m‡KÛ A_©vr 7g †m‡K‡Û AwZµvšÍ `~iZ¡,

 s7  56.25m ( Ans)

2g

2

u  7.5  15 u  7.5ms 1

a
st  u  (2t  1)
2
7.5
 s7  7.5 
( 2  7  1)
2
7.5
 s7  7.5 
13
2
 s7  7.5  48.75

hvq

H 

 H 

d

1
 s2  ut2  at 22
2
1
 180  u  6  a  6 2
2
u  3a  30..........(2)
u  a  15..........(1)

e¯‘wU

(v o Sin θ o ) 2

ah

6

M

cÖ_g †_‡K t2= (2+4)=
s2=(30+150)m=180m

|e

`~iZ¡, s1 = 30m
mgq, t1 = 2s
miY, s2 = (30+150)

1
s1  ut1  at 12
2

16| GKwU e¯‘‡K 40ms-1 †e‡M Ab~fywg‡Ki mv‡_ 60° ‡Kv‡Y wb‡ÿc Kiv
nj| me©vwaK D”PZv Ges Abyf~wgK cvjøv wbY©q Ki|
Avgiv Rvwb,

al

14| GKwU e¯‘ cÖ_g `yB †m‡K‡Û 30m I cieZ©x Pvi †m‡K‡Û
150m ‡Mj| Z¡iY AcwiewZ©Z _vK‡j e¯‘wU Gi ci GK †m‡K‡Û
KZUv c_ AwZµg Ki‡e?
GLv‡b,
Avgiv Rvwb,

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o

‡k‡li 2m `~iZ¡ AwZµg Ki‡Z mgq
jv‡M, t  t 2  t 1  (1.414  1) s  0.414s (Ans.)

24

 t 2  2  1.414s

m

v 20
96  9.8
ev, Sin2 o 
66 2
ev , 2θ o  Sin 1 (0.2159)
ev, 2θ o  12.47

2
1
 4  0   4  t 22
2
2
 t2  2

m

2

Ja

2

v 0 Sin 2θ o
g

40 2 Sin ( 2  60)
9.8
40 2 Sin ( 2  60 )
 R
9.8
R

 R 

 R

 R

1600 Sin 120 )
9.8
1600  0.86602

9.8
1385 .632

9.8
 R  141.39 m (Ans.)

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GLv‡b,
Avw`‡eM, v0 = 40ms-1
wb‡ÿc †KvY 60º
AwfKl©R Z¡iY,
g = 9.8ms-2
me©vwaK D”PZv, H = ?
AbyfywgK cvjøv, R = ?

m

3| MwZwe`¨v (Dynamics)
4
17| nvB‡Wªv‡Rb cigvbyi g‡W‡ji GKwU B‡jKUªb 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 cÖ`wÿb K‡i| B‡jKUª‡bi fi 9.1 ×10-31 kg n‡j
†K›`ªgyLx ej KZ?
21| GKwU cÖv‡mi AbyfywgK cvjøv 79.53 m Ges wePiYKvj 5.3 s n‡j
Avgiv Rvwb,
wb‡ÿc †KvY I wb‡ÿc †eM KZ?
Avgiv Rvwb,
mv 2
GLv‡b,
F 
v o2Sin 2 o
r
AbyfywgK cvjøv, R = 79.53 m
R
9.1 10 -31  (2.18106 )2
g
wePiYKvj, T=5.3s
F
2
wb‡¶c ‡eM, vo = ?
5.2 10-11
v oSin 2 o
8

79
.
53

 F  8. 316  10 N (Ans.)
wb‡¶c †KvY,  = ?

60

2 v oSin  o
g
2v Sin  o
 5.3  o
9. 8
 2v oSin  o  51.94 ... ... ... (2)

bd

|e

(1) bs mgxKiY‡K (2) bs mgxKiY Øviv fvM K‡i cvB,

v o2Sin 2 o 779.394

2v oSin  o
51.94



 F  16.65 N (Ans.)

v o2 2Sin  o Coso 779.394

2 v oSin  o
51.94
 v 0 Cos o  15 ... ... .. (3)

Ja



(2) bs mgxKiY‡K (3) bs mgxKiY Øviv fvM K‡i cvB,

18.4
9. 8
 T  1.877s (Ans.)

M

d

Sh

ah

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?
Avgiv Rvwb,
GLv‡b,
2 v Sin  o
T o
Avw`‡eM, vo = 9.2 ms-1
g
wb‡¶c †KvY, 0º
2  9.2  Sin 90
AwfKl©R Z¡iY,
T
9.8
g = 9.8ms-2
DÌvb cZ‡bi †gvU
2  9. 2  1
T
mgq T =?
9.8

24

Aevi, T 

al



 v Sin 2 o  779.394.. ... (1)

m

18| 0.250kg f‡ii GKwU cv_i LÛ‡K 0.75m j¤^v GKwU myZvi
GK cÖv‡šÍ †eu‡a e„ËvKvi c‡_ cÖwZ wgwb‡U 90 evi Nyiv‡j myZvi Dci
KZ Uvb co‡e|
GLv‡b,
Avgiv Rvwb,
fi, m = 0.250 kg
2
F  m r
e¨vmva©, r = 0.75 m
2
 2πn 
mgq t = 1 min.
 F m
 r
= 60s.
 t 
2
cvKmsL¨v, n = 90 cvK|
 23.141690
 F  0.25
  0.75 Uvb, F = ?

.c
o

9.8

2
o

@

T

ht

ig

op
yr

Avw`‡eM, vo = 50 ms-1
wb‡¶c †KvY, º
AwfKl©R Z¡iY,
g = 9.8ms-2
AbyfywgK `~iZ¡, x=50m
(50)2 Dj¤^ `~iZ¡, y=?
2

C

g
y  (tan 0 ) x 
x2
2
2( v 0 cos  0 )

g
2(v0 cos30 )
g
 y  (tan30)50
(50)2
2
2(v0 cos30 )
 y  0.577350269  50 

 o  tan 1 1.732

  o  60(Ans.)
(3) bs mgxKi‡Y 0 Gi gvb ewm‡q cvB,

20| Abyfywg‡Ki mv‡_ 30°†KvY f~-c„ô †_‡K 50ms-1 †e‡M GKwU
ey‡jU †Qvov nj| ey‡jUwU 50m `~‡i Aew¯’Z GKwU †`Iqvj‡K KZ
D”PZvq AvNvZ Ki‡e|
GLv‡b,
Avgiv Rvwb,

 y  (tan30 )x 

2v oSin  o 51.94

v 0 Cos o
15
3.463
 tan  o 
2
 tan  o  1.732

9.8
 (50) 2
2
2(50 0.866025403
)

v 0 Cos60  15
1
 v 0   15
2
 v 0  30ms 1 (Ans.)
22| GKwU ej‡K f~wgi mv‡_ 30°†KvY K‡i Dc‡ii w`‡K wb‡ÿc Kiv
n‡j GwU 20m `~‡i GKwU `vjv‡bi Qv‡` wM‡q coj| wb‡ÿc we›`y †_‡K
Qv‡`i D”PZv 5m n‡j ejwU KZ †e‡M †Qvov n‡qwQj|
Avgiv Rvwb,

y  (tan 0 ) x 

g
x2
2
2( v 0 cos  0 )

 5  tan30 20

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9.8
(20)2
2
2v cos 30
2
0

GLv‡b,
wb‡¶c †KvY, º
AwfKl©R Z¡iY,
g = 9.8ms-2
AbyfywgK `~iZ¡, x=20m
Dj¤^ `~iZ¡, y=5m
Avw`‡eM, vo =?

3| MwZwe`¨v (Dynamics)
24| GKwU MÖv‡gv‡dvb †iKW© cÖwZ wgwb‡U 45 evi Ny‡i| Gi †K›`ª †_‡K
9cm `~‡i †Kvb we›`yi `ªæwZ KZ?
v  r
GLv‡b,
2n
mgq, t = 1m =60s
v
r
cvKmsL¨v, n=45
t
e¨vmva©,r =9cm=0.09m
2  3.14  45  0.09
v
`ªæwZ, v =?
60

9.8  400
 5  0.577350269  20 
2  v02  0.75
3920
 5  11.54  2
v0 1.5

3920
 6.54
v 02  1.5
3920
 v 02 
 v 0  20ms 1 (Ans.)
6.54  1.5


m

 v  0.42ms 1 (Ans.)

24
|e

bd

†eM, v0 = 48 ms-1
wb‡¶c †KvY, =º
DÌvb cZ‡bi †gvU mgq,T =?
D”PZv, H =?

al

2 v oSinθ o

g
2  48  Sin 90
T
9.8
2  48  1
T
9. 8
 T  9.795 s. ( Ans.) 
T

.c
o

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‡e©v”P KZ Dc‡i
DV‡e?
Avgiv Rvwb,
GLv‡b,

H

Ja

v o2Sin 2  o
2g
482 (Sin 90) 2 
2  9.8

ah

H

m

Avevi,

C

op
yr

ig

ht

@

M

d

Sh

 H  117.5m (Ans.) 

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5