03 dynamics .pdf

03| MwZwe`¨v

‡gv: kvn& Rvgvj
mnKvix Aa¨vcK (c`v_©weÁvb wefvM )

(Dynamics)

we G Gd kvnxb K‡jR XvKv
‡dvb: 01670 856105, 9125630, 9115369

E-mail: sjamal59@gmail.com

Mo‡eM (Average Velocity):
msÁv: †h †Kvb mgq e¨eav‡b †Kvb e¯‘i M‡o cÖwZ GKK mg‡q †h miY nq Zv‡K e¯‘wUi Mo †eM e‡j|


 Δr
e¨L¨v: t mgq e¨eav‡b †Kvb e¯‘i miY r n‡j Mo †eM v 
n‡e|
t
‡eM (Velocity):

msÁv: mgq e¨eavb k~‡b¨i KvQvKvwQ n‡j mg‡qi mv‡_ e¯‘i mi‡Yi nvi‡K †eM e‡j|








Δr
Δr
e¨L¨v: t mgq e¨eav‡b †Kvb e¯‘i miY r n‡j †eM v  lim
wKš‘
n‡”Q Mo †eM v | myZivs v  lim v
t 0 t
t  0
t

ah

Ja

m
al

A_©vr mgq e¨eavb k~‡b¨i KvQvKvwQ n‡j Mo †e‡Mi mxgvwšÍK gvb‡KB †eM e‡j|
mg‡eM ev mylg †eM (Uniform Velocity) :
hw` †Kvb e¯‘i MwZKv‡j Zvi †e‡Mi gvb I w`K AcwiewZ©Z _v‡K Zvn‡j †mB e¯‘i †eM‡K mg‡eM e‡j| A_©vr †Kvb e¯‘
hw` wbw`©ó w`‡K mgvb mg‡q mgvb c_ AwZµg K‡i Zvn‡j e¯‘i
†eM‡K mg‡eM e‡j| k‡ãi †eM, Av‡jvi †eM, cÖf„wZ mg‡e‡Mi cÖK…ó
cÖvK…wZK D`vniY|
Amg‡eM (Variable Velocity) t
‡Kvb e¯‘i MwZKv‡j hw` Zvi †e‡Mi gvb ev w`K ev DfqB cwiewZ©Z nq Zvn‡j †mB †eM‡K Amg‡eM e‡j| Avgiv
mPvivPi †h MwZkxj e¯‘ †`wL Zv‡`i †eM Amg‡eM|

ht

©

Sh

ZvrÿwYK †eM (Instantaneus Velocity) :
GKwU e¯‘ mij ev eµ c‡_ Amg‡e‡M Pj‡j cÖwZwbqZ Gi †e‡Mi cwieZ©b nq| Gfv‡e Amg‡e‡M PjšÍ †Kvb e¯‘i †h
†Kvb gyû‡Z©i †eM‡K H e¯‘i ZvrÿwYK †eM e‡j| ZvrÿwYK †e‡Mi w`K e¯‘wUi H gyû‡Z©i Ae¯’v‡b AswKZ MwZc‡_i
¯úk©K eivei|
Z¡iY (Acceleration) :

C
op

yr
ig


 v

mg‡qi mv‡_ †eM e„w×i nvi‡K Z¡iY e‡j| t mgq e¨eav‡b e¯‘i †e‡Mi cwieZ©b v n‡j Z¡iY a 
n‡e| Ab¨fv‡e
t
ejv hvq mgq e¨eavb k~‡b¨i KvQvKvwQ n‡j mg‡qi mv‡_ e¯‘i †eM e„w×i nvi‡K Z¡iY e‡j| t mgq e¨eav‡b e¯‘i †e‡Mi



v
cwieZ©b v n‡j Z¡iY a  lim
n‡e| Gi GKK ms-2
t  0  t
mgZ¡iY ev mylg Z¡iY (Uniform Acceleration):

GKB w`‡K GKB mgq e¨eav‡b †e‡Mi e„w×i nvi mgvb n‡j Zv‡K mgZ¡iY ev mylg Z¡iY e‡j| AwfK‡l©i Uv‡b gy³fv‡e
cošÍ e¯‘i †eM e„w×i nvi‡K AwfKl©R Z¡iY e‡j| AwfKl©R Z¡iY, mgZ¡iY wewkó MwZi GKwU cÖKó… D`vniY| mgZ¡i‡Y,
Z¡i‡Yi gvb I w`K mg‡qi mv‡_ AcwiewZ©Z _v‡K| mgZ¡i‡Y MwZkxj
e¯‘‡Z mgej wµqvK‡i e¯‘i cici †m‡K‡Ûi †e‡Mi AšÍiB mgZ¡iY|
wP‡Î GKwU mij‡iLv eivei cici †m‡K‡Ûi †eM †`wL‡q Gi Z¡i‡bi cÖK…wZ wb‡`©k Kiv n‡q‡Q| GLv‡b mgZ¡i‡Yi gvb
2ms2 |

AmgZ¡iY (Variable Acceleration):
GKB mgq e¨eav‡b †e‡Mi e„w×i nvi mgvb bv n‡j Zv‡K AmgZ¡iY e‡j| evm †Uªb, †gvUiMvwo BZ¨vw`i Z¡iY Amg Z¡i‡Yi
D`vniY| wP‡Î GKwU mij‡iLv eivei cici †m‡K‡Ûi †eM †`wL‡q Gi Z¡i‡bi cÖK…wZ wb‡`©k Kiv n‡q‡Q| GLv‡b Z¡i‡Yi
gvb AmgZ¡iY|

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03| MwZwe`¨v (Dynamics)

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ZvrÿwYK Z¡iY (Instantaneous acceleration): †Kvb GKwU MwZkxj e¯‘i mg‡qi e¨eavb k~‡b¨i KvQvKvwQ n‡j mg‡qi

mv‡_ e¯‘i †eM cwieZ©‡bi nvi‡K ZvrÿwYK Z¡iY e‡j| t mgq e¨eav‡b e¯‘i †e‡Mi cwieZ©b v n‡j Z¡iY


v
a  lim
n‡e|
t  0  t

miY (Displacement)t wbw`©ó w`‡K e¯‘i Ae¯’v‡bi cwieZ©b‡K miY e‡j| miY‡K s ev d Øviv cÖKvk Kiv nq| Gi GKK
wgUvi| wbw`©ó w`‡K †Kvb e¯‘ t mgq a‡i v †e‡M Pj‡j, miY s = v t n‡e| miY GKwU †f±i ivwk|
Av‡cwÿK †eM (Relative
Av‡cwÿK †eM e‡j|

(Mean velocity):

`ywU MwZkxj e¯‘i GKwUi Zzjbvq (mv‡c‡ÿ) AciwUi Ae¯’v‡bi cwieZ©‡bi nvi‡K

†Kvb GKwU MwZkxj e¯‘i cÖ_g Ges †kl †eM Gi AwfgyL GKB n‡j Zv‡`i †hvM d‡ji

m
al

ga¨ †eM

velocity):

A‡a©K‡K ga¨ †eM e‡j| ‡Kvb wbw`©ó w`‡K †Kvb e¯‘i Avw`‡eM vi I †kl †eM vf n‡j ga¨‡eM =

vi  v f
2

n‡e|

ah

Ja

MwZi msµvšÍ mgxKiY mgvKj‡bi mvnv‡h¨ Dc¯’vcb:
(K) vx  vxo  ax t cÖwZcv`b|
g‡bKwi, X Aÿ eivei GKwU e¯‘ mylg Z¡i‡Y MwZkxj| Av‡iv awi, GB MwZi cÖviw¤¢K kZ©vw` nj mgq Mbbvi ïiæ‡Z
A_©vr hLb t = 0 ZLb Avw` Ae¯’vb x = 0 Ges Avw`‡eM vx = vxo | Av‡iv awi, t mgq ci e¯‘wUi Ae¯’vb x = x Ges
Ges †eM vx = vx| ‡h‡nZz †h †Kvb gyn‡~ Z©i mg‡qi mv‡c‡ÿ †Kvb KYvi †eM e„w×i nvi‡K Z¡iY e‡j|
dv x
dt
 dv x  ax dt

Sh

myZivs, ax 

vx

©

hLb t = 0 ZLb vx = vxo Ges x = xo Avevi, hLb t = t ZLb vx = vx Ges x = x
GB mxgvi g‡a¨ mgxKiY‡K mgvKjb K‡i cvB,
v xo

0

 ax t 

t

yr
ig

 v x 

vx

 ax  aª æeK 

ht

t

 dvx  ax  dt
v x0

o

C
op

 v x  v xo  a x t  0 
 v x  v xo  a x t mgxKiYwU cÖwZcv`b Kiv nj|

1
2

(L) x  xo  vxo t  ax t 2

cÖwZcv`b:

g‡bKwi, X Aÿ eivei GKwU e¯‘ ax mylg Z¡i‡Y MwZkxj| Av‡iv awi, GB MwZi cÖviw¤¢K kZ©vw` nj mgq Mbbvi ïiæ‡Z
A_©vr hLb t = 0 ZLb Avw` Ae¯’vb x = xo Ges Avw`‡eM vx = vxo | Av‡iv awi, t mgq ci e¯‘wUi Ae¯’vb x = x Ges
Ges †eM vx = vx| ‡h‡nZz †h †Kvb gyn‡~ Z©i mg‡qi mv‡c‡ÿ †Kvb KYvi †eM e„w×i nvi‡K Z¡iY e‡j|
dvx
dt
 dvx  ax dt hLb t = 0 ZLb vx = vxo Ges x = xo Avevi, hLb t = t ZLb vx = vx Ges x = x GB

myZivs, ax 

mxgvi g‡a¨ mgxKiY‡K mgvKjb K‡i cvB,
vx

t

 ax  aª æeK 

 dvx  ax  dt

v xo

 v

0



vx
x v xo

 ax t o
t

 vx  vxo  ax t  0

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03| MwZwe`¨v (Dynamics)
 vx  vxo  ax t ... ... ... ... ... (1)

(1) mgxKi‡Y vx 

3

†h †Kvb gyû‡Z© e¯‘i miY e„w×i nvi‡K †eM e‡j| D³

dx
 v xo  a x t
dt
 dx  vxo dt  a x t dt
x
t
t
  dx  v xo  dt  a x  t dt
xo
o
o

dx
ewm‡q cvB,
dt

t

t2 
 x   v xo t   ax  
 2 o
1
 x  xo  vxo t  0   a x t 2  0 
2
1
 x  xo  vxot  a xt 2 mgxKiYwU cÖwZcv`b Kiv nj|
2
1 2
1 2
ev,  x  xo  vxot  a xt  S  v xot  a xt mgxKiYwUI cÖwZcv`b Kiv nj| w¯’i
2
2
1
2
Ae¯’vb †_‡K mgZ¡i‡Y Pjgvb e¯‘i †ÿ‡Î, v xo  0 Ges a x  aª æeK ,d‡j S  0   aª æeK  t  S  aª æeK t 2
2
2
 S t Kv‡RB, w¯’i Ae¯’vb †_‡K mgZ¡i‡Y Pjgvb e¯‘i AwZµvšÍ `yiZ¡ mg‡qi e‡M©i mgvbycvwZK|
t
o

Sh

ah

Ja

m
al

x
xo

©

(M) v 2x  v 2xo  2ax ( x  x o ) cÖwZcv`b :
awi, GKwU e¯‘ X Aÿ eivei ax mylg Z¡i‡Y MwZkxj| GB MwZi cÖviw¤¢K kZ©vw` nj hLb mgq Mbbvi ïiæ‡Z hLb t = 0
ZLb Avw` Ae¯’vb x = xo Ges Avw`‡eM vx = vxo Avevi, t mgq ci KYvwUi Ae¯’vb x Ges †eM vx | †h‡nZz ‡h †Kvb
gyn‡~ Z© mg‡qi mv‡c‡ÿ e¯‘i †eM e„w×i nvi‡K Z¡iY e‡j|

ht

dv x
dt
dv
dx
 ax  x 
dx dt
dv
 dx

 ax  x  v x

 vx 

dx
 dt

 v x dv x  a x dx
hLb x = xo ZLb vx = vxo Ges hLb x = x ZLb vx = vx GB mxgvi g‡a¨ Dc‡iv³ mgxKiY‡K mgvKjb K‡i cvB,

C
op

yr
ig

myZivs Z¡ iY a x 

vx

x

 v x dv x  ax  dx

v xo

xo

2
x

vx

v 
    a xx
x x
 2  vxo
o
v 2x  v 2xo
 ax ( x  x o )
2
 v 2x  v 2xo  2a x ( x  x o )


mgxKiYwU cÖwZcv`b Kiv nj|

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03| MwZwe`¨v (Dynamics)

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cošÍ e¯‘i m~Î eY©bv (Laws of falling bodies) :
evavnxb fv‡e cošÍ e¯‘ wb‡¤§v³ wZbwU m~Î †g‡b P‡j| 1589 wLª÷v‡ã weÁvbx M¨vwjwjI m~Î wZbwU Avwe®‹vi K‡ib t
1g m~Ît e¯‘ mgvb mg‡q mgvb c_ AwZµg K‡i|
2q m~Ît wbw`©ó mg‡q e¯‘ †h †eM jvf K‡i Zv H mg‡qi mgvbycvwZK| t mg‡q v †eM jvf Ki‡j, m~Îvbyhvqx †eM n‡e,
vt

ah

djvdjt evqyk~b¨ ¯’v‡b mKj e¯‘ mgvb mg‡q mgvb c_ AwZµg K‡i|

Ja

m
al

3q m~Ît wbw`©ó mg‡q e¯‘ KZ…K AwZµvšÍ `~iZ¡ H mg‡qi e‡M©i mgvbycvwZK| t mg‡q AwZµvšÍ `yiZ¡ h n‡j, m~Îvbyhvqx
D”PZv n‡e, ht 2
¯^b©gy`ªv I cvjK cixÿv:
hš¿cvwZt (K) j¤^v GKwU k³, †gvUv I duvcv `yBgyL †Lvjv KvPbj B| (L) GKwU Uzwc C
(M) GKwU ÷c KK© S (N) GKwU cvjK |
cixÿvi weeiY: KvPb‡ji GKcÖv‡šÍGKwU Uzwc C Ges Aci cÖv‡šÍGKwU ÷c KK© S _v‡K|
Uzwc Ly‡j GKwU ¯^b©gy`ªv G Ges GKwU cvjK F b‡ji g‡a¨ XyKv‡bv nq| ócK‡K©i Pvwe Ly‡j
cv‡¤úi mvnv‡h¨ bjwU‡K evqyc~b© ev evqyk~b¨ Kiv hvq| bjwU‡K nVvr Dwë‡q gy`ªv I cvjK‡K
wb‡Piw`‡K co‡Z †`Iqv nq| cixÿvq †`Lv hvq †h (1) evqyc~b© Ae¯’vq gy`ªvwU cvj‡Ki Av‡M
wb‡Pi cÖv‡šÍc‡o| (2) evqyk~b¨ Ae¯’vq gy`ªv I cvjK GKB mv‡_ wb‡Pi cÖv‡šÍ c‡o|

C
op

yr
ig

ht

©

Sh

(K) (vt) MÖv‡di mvnv‡h¨ v = v0+at cÖgvY:
mgZ¡i‡Y MwZkxj †Kvb e¯‘i †ÿ‡Î X A‡ÿi w`‡K mgq t Ges Y A‡ÿi w`‡K †eM v wb‡q v ebvg t ‡jL wPÎ AsKb
Kiv nj| GB ‡jLwPÎ †_‡K t mg‡q e¯‘i AwZµvšÍ `~iZ¡ s wbb©q Kiv hvq| AB ‡iLvi Dci †h †Kvb we›`y P †bqv nq| P
†_‡K X A‡ÿi Dci PQ j¤^ Uvbv nq| Zvn‡j OQ = t mg‡q AwZµvšÍ `~iZ¡ s n‡e AOQP ‡ÿ‡Îi †ÿÎdj|
Avw`‡eM,
v0 = OA=RQ.... ... .... (1)
‡kl †eM,
v = PQ .... .... ..... .... (2)
wKš‘, PQ = PR+RQ ... ... ..(3)
myZivs, v = PR+RQ
ev, v = PR+ v0 ... ... (4)
Avgiv Rvwb,Z¡iY a = AB ‡iLvi Xvj|
PR
a
AR
wKš‘, AR = OQ= t



a

PR
t

AZGe, PR= at
(4) bs mgxKi‡Y PR Gi gvb ewm‡q,
v = at+ v0
v = v0 + at

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03| MwZwe`¨v (Dynamics)

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1
2

(L) mgZ¡iY MwZi †ÿ‡Î †eM ebvg mgq (v  t)†jLwPÎ AsKb Ges †jLwPÎ n‡Z s  v o t  at 2 mgxKiYwU cÖwZcv`b:
mgZ¡i‡Y MwZkxj †Kvb e¯‘i †ÿ‡Î X A‡ÿi w`‡K mgq t Ges Y A‡ÿi w`‡K †eM v wb‡q v ebvg t ‡jL wPÎ AsKb
Kiv nj| GwU Y Aÿ‡K †Q`Kvix GKwU mij †iLv nq hv, v = vo+at mgxKiY †g‡b P‡j| GB ‡jLwPÎ †_‡K t mg‡q e¯‘i
AwZµvšÍ `~iZ¡ s wbb©q Kiv hvq| AB ‡iLvi Dci †h †Kvb we›`y P †bqv nq| P †_‡K X A‡ÿi Dci PQ j¤^ Uvbv nq|
Zvn‡j OQ = t mg‡q AwZµvšÍ `~iZ¡ s n‡e AOQP ‡ÿ‡Îi
†ÿÎdj| aiv hvK, KYvwUi mgZ¡iY a
Ges Avw`‡eM, vo = AO
AwZµvšÍ mgq, t = OQ
Ges t mg‡q AwZµvšÍ `~iZ¡, s = AOQP ‡ÿ‡Îi †ÿÎdj|
= AOQR ‡ÿ‡Îi †ÿÎdj  ARP ‡ÿ‡Îi †ÿÎdj|
×AR×PR

©

Sh

ah

s = AO×OQ + 12 ×OQ×PR [∵ AR = OQ ]
wKš‘ AB ‡iLvi Xvj n‡”Q KYvwUi Z¡iY a,
PR
a 
AR
PR = a×AR
= a×OQ
s = AO×OQ + 12 ×OQ×a×OQ
 s = AO×OQ + 12 ×a×OQ2
1
 s  v o t  at 2 mgxKiYwU cÖwZcv`b Kiv nj| 
2

m
al

1
2

Ja

= AO×OQ +

(M) mgZ¡iY MwZi †ÿ‡Î †eM ebvg mgq (v  t)†jLwPÎ AsKb Ges †jLwPÎ n‡Z

(mgvšÍivj `yB evûi † hvMdj)  DPPZv
2

s

C
op

s 

yr
ig

ht

cÖwZcv`b:
wPÎ †_‡K †`L‡Z cvB, `~iZ¡ (s) AwZµg Ki‡Z e¯‘wUi mgq jv‡M = t
s = UªvwcwRq‡gi †ÿÎdj AOQP

(OA  QP )  OQ

2
(v

v)

t
s 
... ... ... (6)
2
vv
vv
0
0
Avgiv Rvwb, a 
 t
t
a
0

t Gi

gvb (6) bs mgxKi‡Y ewm‡q cvB,

 s  (v  v)(v  v 0 )
2a
 2as  v2  v02
0

 v 2  v 02  2as

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2

v v

2
0

 2as

mgxKiYwU

03| MwZwe`¨v (Dynamics)

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cÖkœt cÖvm Kv‡K e‡j?
DËit †Kvb e¯‘‡K Abyfywg‡Ki mv‡_ wZh©Kfv‡e †Kvb ¯’v‡b wb‡ÿc Kiv n‡j Zv‡K cÖvm e‡j| wZh©Kfv‡e wbwÿß
wXj, ey‡j‡Ui MwZ BZ¨vw` cÖvm MwZi D`vniY|
cÖÖkœt Abyfywg‡Ki mv‡_ wZh©Kfv‡e wbwÿß cÖv‡mi MwZc‡_i mgxKiY wbb©q Ki Ges †`LvI †h, GB MwZc_ Awae„ËvKvi|

Ja

m
al

DËit g‡bKwi, evqyga¨w¯’Z O we›`y n‡Z GKwU cÖvm‡K wb‡ÿc Kiv nj|
wb‡ÿc †eM ev Avw`‡eM = vo
wb‡ÿc ‡KvY = 
g wb‡Pi w`‡K wµqvkxj| AZGe ay = -g; ax = 0;
wb‡ÿc we›`y I g~j we›`y GKB nIqvq xo = yo = 0
 Avw`‡e‡Mi Abyf~wgK Dcvsk = voCoso
Ges Avw`‡e‡Mi Dj¤^ Dcvsk = voSino
X Aÿ eivei MwZi cwieZ©b D³ Aÿ eivei Z¡i‡Yi Dci wbf©ikxj| Y Aÿ eivei MwZi cwieZ©b D³ Aÿ
eivei Z¡i‡Yi Dci wbf©ikxj| G `ywU Aÿ eivei MwZi cwieZ©b Awbf©ikxj|
awi t mg‡q cÖvmwU P(x,y) Ae¯’v‡b _v‡K| ZLb Gi †eM = v
Abyf~wg‡Ki w`‡K Z¡iY, ax= 0
Abyf~wg‡Ki w`‡K miY = x

Sh

ah

x = voCoso t + 12 axt2
ev, x = voCoso t + 0
[ax= 0]
ev, x = voCoso t
x
t 
.......................(1)
vo Cos o

©

Dj¤^ w`‡K Z¡iY ay=g;
Dj¤^ w`‡K miY y; Abyiƒcfv‡e

o

o



x

v Cos 
o


x

g 
 v Cos 
2
 o
1

yr
ig

ev, y  v Sin 

ht

y=voSinot 12 gt2

o

o






2

[t Gi gvb ewm‡q]

2

 y  bx  cx

C
op


 2
g
 x
ev, y  tan  o x   2
2
 2vo Cos  o 



g
awi, aª æeK tan θo  b Ges 2 2  c 
2voCos θo



Dc‡iv³ mgxKiYwU GKwU Awae„‡Ëi mgxKiY|  cÖv‡mi MwZc_ GKwU Awae„Ë (c¨viv‡evjv)|
cÖkœt cÖgvY Ki, evqynxb Ae¯’vq f~wg n‡Z D”PZvq Aew¯’Z †h †Kvb Ae¯’vb n‡Z Abyf~wgK Awfgy‡L wbwÿß e¯‘i MwZc_
GKwU Awae„Ë|
g‡bKwi, k~‡b¨ Aew¯’Z O we›`y n‡Z vo †e‡M f~wgi mgvšÍiv‡j GKwU e¯‘KYv wbwÿß nj| e¯‘ KYvwU g Gi cÖfv‡e
bx‡P co‡e| awi cÖ‡ÿcb Z‡j Abyf~wgK OX †iLv X Aÿ Ges OY †iLv Y Aÿ| awi t mgq c‡i e¯‘ KYvwU MwZ c‡_i
P(x,y) we›`y‡Z gyn‡~ Zi Rb¨ Ae¯’vb Ki‡e| g bx‡Pi w`‡K wµqvkxj|
AZGe ay = g; ax= 0 ;
Avw`‡e‡Mi Abyf~wgK Dcvsk = vo
Ges Avw`‡e‡Mi Dj¤^ Dcvsk = 0

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03| MwZwe`¨v (Dynamics)

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tmg‡q AwfKl©RZ¡iYnxb Abyf~wgK miY x = vot
 x 2  v o2 t 2 ...

...

... ...

(1 )

... ...

tmg‡q Dj¤^ miY y = 0.t + 12 gt2
y=

1
2

gt2...

.... .... .... .... .... ....

(2)

(1) ‡K (2) Øviv fvM K‡i cvB
x2
v 2t 2
 1o 2
y
2 gt

x 2 2 v 2o

y
g

 2v 2 
 x 2   o  y
 g 


2vo2
awi
,
 4a  aª æeK 

g



 x 2  4ay

m
al



Ja

Dc‡iv³ mgxKiYwU GKwU Awae„‡Ëi mgxKiY| ZvB wbwÿß e¯‘i MwZc_ GKwU Awae„Ë (c¨viv‡evjv)|

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cÖkœt Abyfywg‡Ki mv‡_ wZh©K fv‡e wbwÿß e¯‘i ‡ÿ‡Î (K) m‡e©v”P D”PZvq †cŠQ‡Z mgq (L) m‡e©v”P D”PZv (M) wePiY
Kvj (N) cvjøv (O) me©vwaK cvjøv wbb©q Ki|
g‡b Kwi, evqyga¨w¯’Z O we›`y n‡Z GKwU cÖvm‡K vo †e‡M o †Kv‡Y wZh©Kfv‡e wb‡ÿc Kiv nj| cÖvmwU t mg‡q m‡e©v”P
D”PZv P(x,y) G Ae¯’vb Ki‡e Ges ZLb Gi †eM n‡e v|
(K) m‡e©v”P D”PZvq †cŠQ‡Z mgqt vo †e‡Mi Dj¤^ Dcvsk voSino
t mgq c‡i P we›`y‡Z †eM, vy = voSino gt.................(1)
P we›`yMvgx m‡e©v”P D”PZvq vy= 0..................................... (2)
(1) bs mgxKi‡Y vy= 0 ewm‡q cvB

C
op

(L) m‡e©v”P D”PZvt
g‡bKwi, m‡e©v”P D”PZv = H

yr
ig

0 = voSino gt
v Sin o
 t o
..................................(3)
g

 H = voSinot  12 gt2

v Sin o 1  vo Sin o 

 H  vo Sin o  o
 2 g 
g
 g 

H 

vo Sin o 2  vo Sin o 2
g

2g

2

 (3)

bs n‡ Z t Gi gvb ewm‡ q  



v o2Sin 2 o
... ... ... ... ... ... ... (4) 
2g
(M) DÇqb (wePiY) Kvj (Time of Flight) t
g‡b Kwi wePiY Kvj T A_©vr T mg‡q cÖvmwU mgZ‡j wd‡i Av‡m|
H 

 t mg‡q Dj¤^ w`‡K miY y = voSinot  12 gt2GB mgxKi‡Y mgq t = T Ges miY y = 0 ewm‡q cvB,
0 = voSinoT  12 gT2

ev, 12 gT2 = voSinoT

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03| MwZwe`¨v (Dynamics)

8

2vo Sinθo
... ... ... ... ... ... (5) 
g
(N) cvjøv (Range)t
g‡b Kwi cvjøv R A_©vr T mg‡q cÖvmwU Abyfywg‡Ki w`‡K †h `~iZ¡ AwZµg K‡i ZvBB cvjøv R
 R = ( voCoso ) × T
2v Sin o
 R  voCos o  o
[(5) bs n‡Z T Gi gvb ewm‡q]
g
T 

R

vo2 2Sin o Cos o
g

Ja

m
al

vo2 Sin 2 o
R 
...........................(6)
g
(O) me©vwaK cvjøv (Maximum Range) t
g‡bKwi me©vwaK cvjøv Rmax| wbw`©ó vo Gi Rb¨, Sin20 Gi gvb me©vwaK n‡j cvjøv n‡e me©vwaK| Sin20 Gi
me©vwaK gvb = 1
A_©vr Sin20 = 1
ev, Sin20 = Sin900
ev, 20 = 900

0 = 450 myZivs wb‡ÿc †KvY0 = 450 n‡j cvjøv me©vwaK
v 2 Sin 2  45o
 me©vwaK cvjøv Rmax  o
g
2
v Sin 90 o
 Rmax  o
g
v2 1
 Rmax  o
g
2
v
 Rmax  o ... ... ... ... ... (7)
g

yr
ig

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

  
cªkœt ˆiwLK †eM I †KŠwbK †e‡Mi msÁv `vI Ges G‡`i g‡a¨ m¤úK© ¯’vcb Ki| ev, v  r ev , v    r cÖgvb Ki |

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op

  
ev, v    r cÖgvY Ki |

‰iwLK †eM (Linear Velocity)t wbw`©ó w`‡K ˆiwLK c‡_ †Kvb e¯‘ GKK mg‡q †h `yiZ¡ AwZµg K‡i Zv‡K H e¯‘i ‰iwLK
†eM e‡j| ˆiwLK †eM‡K v Øviv cÖKvk Kiv nq| wbw`©ó w`‡K e¯‘ t mg‡q d `~iZ¡ AwZµg Ki‡j †eM v 

d
n‡e| †eM
t

GKwU †f±i ivwk| ˆiwLK †e‡Mi GKK ms-1
‡KŠwYK †eM (Angular Velocity) t mgq e¨eavb k~‡b¨i KvQvKvwQ n‡j †Kvb we›`y ev Aÿ‡K †K›`ª K‡i e„ËvKvi c‡_
Pjgvb †Kvb e¯‘i mg‡qi mv‡_ †KŠwbK mi‡Yi nvi‡K †KŠwbK †eM e‡j| Ab¨ K_vq e„ËvKvi c‡_ †Kvb e¯‘ GKK mg‡q
†h †KŠwbK `~iZ¡ AwZµg K‡i Zv‡K H e¯‘i †KŠwbK †eM e‡j| †KŠwbK †eM‡K  Øviv cÖKvk Kiv nq| wbw`©ó w`‡K e¯‘ t

n‡e| †KŠwbK †e‡Mi GKK rad s-1
t
† KvY
Pvc
L


 T-1
Gi gvÎv n‡”Q 
mgq
e¨vmva©  mgq L  T

mg‡q  ‡KvY Drcbœ Ki‡j †KŠwbK †eM  

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03| MwZwe`¨v (Dynamics)

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m¤úK© (Relation) t
g‡bKwi GKwU e¯‘KYv OC= OB = r e¨vmva© wewkó GKwU e„‡Ëi
cwiwa eivei‡KŠwbK †e‡M Nyi‡Q| hw` T †m‡K‡Û e¯‘ KYvwU e„‡Ëi cwiwa
eivei GKevi Ny‡i Av‡m Z‡e †KŠwbK `~iZ¡  =  †iwWqvb n‡e|
 ‡KŠwbK †eM, ω 

ev, T 

2π
T

2π
... ... ... ... ...(1)
ω

GLb e¯‘ KYvwU hw` e„ËvKvi c‡_ bv Ny‡i H GKB mg‡q mij †iLv eivei PjZ Z‡e T mg‡q e¯‘KYvwU e„ËwUi

T 

2πr
T

2πr
... ... ... ... ...( 2)
v

m
al

cwiwai mgvb c_ r `~iZ¡ AwZµg KiZ|  ˆiwLK †eM v 

yr
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Ja

(1) bs I (2) mgxKiYØq n‡Z cvB
2 2 r


v
1 r
 
 v
 v = r A_©vr ‰iwLK †eM = †KŠwbK †eM × e„‡Ëi e¨vmva©|
v = r mgxKi‡Yi ‡f±i iƒc:
  
g‡b Kwi, u    r ... ... ... (3) 
 

u ‡f±‡ii gvb u  r sin 90  r [  r ]
 


µm ¸Y‡bi wbqg Abymv‡i,   r ev , u †f±‡ii AwfgyL Ges v †f±‡ii AwfgyL Awfbœ| Avevi v = r| †`Lv hv‡”Q †h,
 
gvb I w`K we‡ePbvq u I v ‡f±i Awfbœ|
 
 u  v ... ... ... (4)
  
(3) I (4) n‡Z v    r (cÖgvwYZ)

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‡K›`ªgyLx ej (Centripetal Force): hLb †Kvb e¯‘ e„ËvKvi c‡_ Nyi‡Z _v‡K ZLb †h ej e¯‘i Dci H e„‡Ëi †K›`ª
Awfgy‡L wµqv K‡i e¯‘wU‡K e„ËvKvi c‡_ MwZkxj iv‡L Zv‡K †K›`ªgyLx ej e‡j| m f‡ii e¯‘ r e¨vmva© wewkó e„ËvKvic‡_
v mg`ªæwZ‡Z Nyi‡Z _vK‡j Zvi †K›`ªgyLx ej  m

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†K›`ªwegyLx ej (Centrifugal Force): hLb †Kvb e¯‘ e„ËvKvi c‡_ Nyi‡Z _v‡K ZLb †h ej H e„‡Ëi †K‡›`ªi wecixZ
w`‡K cÖ‡qvM K‡i Zv‡K †K›`ªwegyLx ej e‡j| m f‡ii e¯‘ r e¨vmva© wewkó e„ËvKvic‡_ v mg`ªæwZ‡Z Nyi‡Z _vK‡j Zvi
†K›`ªwegyLx ej  m

v2
|
r

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