Intermediate physics 1st paper

঩দাথ঱বফজ্ঞান প্রশ্ন ঳ৃজন঱ীর ফা তত্ত্বীয় যমভনই য঴াক না যকন, ঩দাথ঱বফজ্ঞাননয যভৌবরক বফলয়঳ভূ঴ ঳ফ঳ভয় একই থানক । এই
যভৌবরক বফলয়঳ভূ঴ ঳ম্পনক঱ আ঩নায ঩বযস্কায ধাযণা থাকনর ,আ঩বন অফ঱যই যম যকান উদ্দী঩নকয জ্ঞান, অনুধাফন, প্রনয়াগ ঑ উচ্চতয দক্ষতায
প্রশ্ন঳ভূন঴য উত্তয খুফ ঳ুন্দয কনয ঳াবজনয় বরনখ আ঳নত ঩াযনফন।এই ই-ফুনক ঩দাথ঱বফজ্ঞাননয যভৌবরক বফলয়঳ভূ঴ ঳ম্পনক঱ খুফ ঳঴নজ ধাযনা ঑
ফুঝায জনয ঩দাথ঱বফজ্ঞাননয গুযত্ব঩ূন঱ বফলয়঳ভূন঴য প্রনয়াজনীয় উদা঴যণ ঑ বিত্র঳঴ ফযাখযাভূরক ঳ভাধান কনয যদ঑য়া ঴নয়নে ।
ন োটঃ ‚ভূর ফই তথা ঩াঠ্য ফইনয়য বফকল্প বকেু নাই ।‛ ঩দাথ঱বফজ্ঞান বানরা বানফ ফুঝায জনয আ঩নানক অফ঱যই বফববন্ন যরখনকয ফই ঩ড়নত ঴নফ
অথ঱াৎ একই টব঩ক্স বফববন্ন ফইনত ঩ড়নত ঴নফ। যম টাকা বদনয় যকাবিিং কযনফন য঳ই টাকা বদনয় বফববন্ন যরখনকয ফই বকনুন এফিং তা একফায কনয
঴নর঑ ঩ড়ু ন , আ঱া কবয যকাবিিং এ ঩ড়ায যিনয় বানরা পরাপর ঩ানফন ।
ব঱ক্ষাথ঱ী কানে একান্ত অনুনযাধ , ঩দাথ঱বফজ্ঞান না ফুনঝ ভুখস্ত কযায বফ঱ার বু রবট বু নর঑ কযনফন না।ন঳টা বননজয ঩ানয় বননজই কু ড়ার ভাযায
঳ভাথ঱ক।কাযন এনত ঳ভয় ঑ যভধা দুনটাই অ঩িয় ঴য় , এফিং এনত বফলয়টায উ঩য একটা অভূরক অস্ববস্ত ঑ বীবত িনর আন঳।
অনটানভবটক স্ক্রনরয ভাধযনভ ই-ফুক ঩ড়া / বযনড়য জনযঃ
আ঩নায ই−ফুক ফা pdf বযডানযয Menu Bar এয View অ঩঱নবট যত বিক কনয Auto /Automatically Scroll অ঩঱নবট
ব঳নরক্ট করুন (অথফা ঳যা঳বয যমনত  Ctrl + Shift + H )। এবার ↑ up Arrow ফা ↓ down Arrow যত বিক কনয
আ঩নায ঩ড়ায ঳ুবফধা অনু঳ানয স্ক্রর স্পীড বঠ্ক কনয বনন।
঳যা঳বয যমনত অধযানয়য নানভয উ঩য বিক করুনঃ
1. যবক্টয (Vector)
2. রযবখক গবত (Linear-Motion)
3. বিভাবত্রক গবত(Motion-In-Two-Dimensions)
4. গবত঳ূত্র (Laws-Of-Motion)
5. যকৌবণক গবত঳ূত্র (Laws-Of-Angular-Motion)
6. কাজ, ঱বি ঑ ক্ষভতা (Work-Energy-And-Power)
7. ভ঴াকল঱ (Gravitation)
8. ঳যর েবন্দত স্পন্দন (Simple-Harmonic-Oscillation)
9. বিবতিা঩কতা (Elasticity)
10. প্রফা঴ী ঩দাথ঱ (Fluid)
11. তা঩ ঑ গযা঳ (Heat-And-Gas)
12. তা঩ভাত্রা (Temperature)
13. তা঩ গবতবফদযায প্রথভ ঳ূত্র (First Law Of Thermodynamics)
14. তা঩ বফবকযণ (Heat Radiation)
15. অফিায ঩বযফত঱ন (Change Of State)
16. তা঩গবতবফদযায বিতীয় ঳ূত্র (Second Law Of Thermodynamics)
17. তযঙ্গ ঑ ঱ব্দ (Waves & Sound)
18. ঱ব্দ (Sound)
19. ঱নব্দয গবতনফগ (Speed Of Sound)
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বদক/নবক্টয যাব঱ঃ ভান ঑ বদক উবয়ই আনে এফিং যম যকান একবট ফা উবয়বটয ঩বযফত঱ননয পনর যবক্টয যাব঱ ঩বযফবত঱ত
঴নত ঩ানয। উদা঴যণঃ ঳যণ, যফগ, ত্বযণ, ভন্দন, ফর, বযনফগ, অববকল঱জ ত্বযণ, যিৌম্বক প্রাফরয, রফদুযবতক প্রাফরয, বূ -
িুম্বনকয অনুবূ বভক প্রাফরয, ঑জন, ঩ৃষ্ঠাটান, ঳াদ্রতাতা গুণািংক, ফনরয রামাভক, যিৌম্বক রামাভক, প্রফা঴ভাত্রা ইতযাবদ।
অবদক/নস্করাযঃ ভান আনে বকন্তু বদক যনই। উদা঴যণঃ দ্রুবত, কাজ, ক্ষভতা, ঱বি, তা঩, িা঩, যিৌম্বক ঑ রফদুযবতক বফবফ,
িাজ঱, বিবতিা঩ক গুণািংক ইতযাবদ।
যবক্টয যাব঱য যমাজনঃ যনৌকায গবত, িরন্ত গাবড়নত ঩ড়ন্ত ফৃবি, ঩াবখয উড্ডয়ন।
যবক্টয যাব঱য বফনয়াজনঃ গুনটানা যনৌকা, রননযারায যঠ্রা, ঳যর যদারনকয গবত।
রবধঃ দুই ফা তনতাবধক যবক্টয যমাগ কনয একবট নতুন যবক্টয ঩া঑য়া মায়। এ নতুন যবক্টযনক রবধ যবক্টয ফনর।
অিং঱কঃ যম যবক্টয঳ভূ঴ যমাগ কনয রবধ ঩া঑য়া মায়, তানদযনক রবধয অিং঱ক ফা উ঩ািং঱ ফনর।
ভনন যাখনত ঴নফঃ

‡f±i ivwk (Vector Quantities):
‡h mKj †fŠZ ivwk‡K m¤ú~Y©iƒ‡c cÖKvk Kivi Rb¨ gvb I w`K Df‡qi cÖ‡qvRb nq Zv‡ì‡K ‡f±i ivwk e‡j|
†hgb- miY, IRb, †eM, Z¡iY, ej BZ¨vw`|
‡¯‹jvi ivwk (Scalar Quantities):
‡h mKj †fŠZ ivwk‡K ïay gvb Øviv m¤ú~Y©iƒ‡c cÖKvk Kiv hvq Zv‡ì‡K ‡¯‹jvi ivwk e‡j| †hgb- `ªæwZ, fi,
KvR, ‰`N©¨ BZ¨vw`|
‡¯‹jvi ivwk I ‡f±i ivwki g‡a¨ cv_©K¨ (Distinction between Scalar and Vector Quantities):
µwgK ‡¯‹jvi ivwk ‡f±i ivwk
1|
‡h mKj †fŠZ ivwk‡K ïay gvb Øviv m¤ú~Y©iƒ‡c
cÖKvk Kiv hvq Zv‡ì‡K ‡¯‹jvi ivwk e‡j|
‡h mKj †fŠZ ivwk‡K m¤ú~Y©iƒ‡c cÖKvk Kivi Rb¨
gvb I w`K Df‡qi cÖ‡qvRb nq Zv‡ì‡K ‡f±i ivwk
e‡j|
2|
`ªæwZ, fi, KvR, ‰`N©¨ BZ¨vw` †¯‹jvi ivwki
D`vniY|
miY, IRb, †eM, Z¡iY, ej BZ¨vw` ‡f±i ivwki
D`vniY|
3|
Kgc‡ÿ GKwU ivwki gvb k~b¨ bv n‡j ¸bdj
k~b¨ n‡Z cv‡i bv|
†Kvb ivwki gvb k~b¨ bv n‡jI ¸bdj k~b¨ n‡Z
cv‡i|
4| gvb Av‡Q wKš‘ w`K bvB| gvb I w`K Av‡Q|
5|
‡hvM, we‡qvM, ¸b, I fvM ¯^vfvweK wbq‡g Kiv
hvq|
‡hvM, we‡qvM, ¸b, I fvM ¯^vfvweK wbq‡g Kiv hvq
bv|
6|
`ywU Aw`K ivwki ¸bdj me©`v GKwU Aw`K ivwk
nq|
`ywU w`K ivwki myweav RbK ¸bdj GKwU w`K ivwk
ev GKwU Aw`K ivwk nq|
GKK †f±i (Unit Vector):
‡h †f±‡ii gvb GK Zv‡K GKK †f±i e‡j| gvb k~b¨ bq Ggb †f±i‡K Gi gvb Øviv fvM Ki‡j H w`K ivwkwUi w`‡K
GKwU GKK †f±i cvIqv hvq| g‡b Kwi A

GKwU †f±i ivwk; 0A 

A

 Gi w`‡K GKK †f±i aˆ
A
A
 

(awi)
mxgve× †f±i (Restricted Vector):
‡h †f±‡ii Avw` we›`y †Kv_vq _v‡K Zv w¯’i _v‡K Zv‡K mxgve× †f±i e‡j| O we›`y‡Z
P

ej OB †iLv eivei wµqv K‡i eySv‡j †f±i P

mxgve× †f±i hvi cv` we›`y O |
mgvb †f±i (Equal Vector):
GKB w`‡K wµqviZ `yÕwU mgRvZxq †f±‡ii gvb mgvb n‡j Zv‡ì‡K mg‡f±i
ev mgvb †f±i e‡j| wP‡Î P

I Q

mgvb †f±i|
k~b¨ ev bvj †f±i (Zero or Null Vector):
†h †f±‡ii gvb k~b¨ Zv‡K k~b¨ ev bvj †f±i e‡j| bvj †f±‡ii Avw` I †kl we›`y GKB we›`y‡Z Aew¯’Z|

01| ‡f±i (Vector) 2
m`„k †f±i ev mgvšÍÍivj †f±i (Like Vector):
mgRvZxq `yB ev Z‡ZvwaK †f±i hw` GKB w`‡K wµqv K‡i Z‡e Zv‡ì‡K m`„k †f±i ev mgvšÍÍivj
†f±i e‡j| wP‡Î A

I B

m`„k †f±i ev mgvšÍÍivj †f±i|
wem`„k †f±i (Unlike Vector):
mgRvZxq `yB ev Z‡ZvwaK †f±i hw` wecixZ w`‡K wµqv K‡i Z‡e Zv‡ì‡K wem`„k †f±i e‡j|
wP‡Î A

I B

wem`„k †f±i|
mg‡iL †f±i (Co-linear Vector):
`yB ev Z‡ZvwaK †f±i hw` GKB mij‡iLv eivei ev ci¯úi mgvšÍÍiv‡j wµqv
K‡i Z‡e Zv‡ì‡K mg‡iL †f±i e‡j| wP‡Î A

,B

,C

mg‡iL †f±i|
Ae¯’vb †f±i (Position Vector):
cÖm½ KvVv‡gvi g~j we›`yi mv‡c‡ÿ Ab¨ †Kvb we›`yi Ae¯’vb wbY©‡qi Rb¨ †h †f±i eënvi
Kiv nq Zv‡K Ae¯’vb †f±i e‡j|
e¨vL¨v t wP‡Î O n‡”Q cÖmsM KvVv‡gvi g~j we›`y Ges †h P †Kvb GKwU we›`y|

OP †f±iwU O we›`yi mv‡c‡ÿ P we›`yi Ae¯’vb wb‡`©k Ki‡Q| ZvB

OP GKwU Ae¯’vb †f±i|
Ae¯’vb †f±i‡K A‡bK mgq e¨vmva© †f±i r

Øviv cÖKvk Kiv nq| rOP



AvqZGKK †f±i (Rectangular Unit Vector):
wÎgvwÎK ¯’vbv¼ eë¯’vq ci¯úi j¤^ wZbwU Aÿ _v‡K| h_v t X,Y Ges Z Aÿ| X A‡ÿi w`‡K
wewfbœ †f±i‡K cÖKvk Kivi Rb¨ GKwU GKK †f±i iˆ eënvi Kiv nq| †Zgwb jˆ I kˆ h_vµ‡g
Y I Z A‡ÿi w`‡K GKK †f±i (Wvb cv‡k¦©i wPÎ)| iˆ , jˆ Ges kˆ †K AvqZGKK †f±i e‡j|
e¨vmva© †f±i (Radius Vector): A‡bK mgq †Kvb we›`yi Ae¯’vb‡K †h †f±‡ii mvnv‡h¨ cÖKvk
Kiv nq Zv‡K e¨vmva© †f±i

r e‡j| myZivs †Kvb we›`y P-Gi ¯’vbvsK (x,y,z) n‡j, e¨vmva©
†f±i kzjyixOPr



Øviv cÖKvk Kiv nq| Ges Gi gvb nq 222
zyxrr 

‡f±i ivwki mvgvšÍÍwiK m~Î (Law of Parallelogram) eY©bv I e¨vL¨v :
eY©bv: †Kvb KYvi Dci GKB mg‡q wµqvkxj `yÕwU †f±i ivwk‡K hw` †Kvb GK we›`y †_‡K AswKZ mvgvšÍÍwi‡Ki `yÕwU
mwbœwnZ evû Øviv wb‡`©k Kiv hvq Z‡e H we›`y ‡_‡K AswKZ mvgvšÍÍwi‡Ki KY©B †f±i `yÕwUi jwäi gvb I w`K wb‡`©k K‡i|
g‡bKwi GKwU KYvi Dci GKB mg‡q `yÕwU w`Kivwk P I Q,  †Kv‡Y wµqv Ki‡Q| OA Ges OC †iLv `yÕwU
h_vµ‡g P I Q gvb Ges Zxi wPý G‡ì w`K wb‡`©k Ki‡Q| GLv‡b AOC | GB `yÕwU w`K ivwki jwäi gvb I w`K
wbY©q Ki‡Z n‡e|
AsKb : mvgvšÍÍwiK OABC AsKb K‡i Kb© OB hy³ Kwi| Zvn‡j OB Kb©B w`Kivwk `yÕwUi gvb I w`K wb‡`©k
Ki‡e| g‡b Kwi jwäi gvb R Ges †KvY GKwU myÿ‡KvY| GLb B we›`y †_‡K OA Gi ewa©Z As‡ki Dci BN j¤^ Uvwb|

jwäi gvb wbY©qt AB I OC mgvšÍÍivj|  AOC | OBN wÎfy‡Ri ONB GK mg‡Kvb|
 OB2
= ON2
+ BN2
2
OB =(OA+AN)2
+BN2
2
OB = OA2
+2OA .AN +AN2
+BN2
2
OB = OA2
+(AN2
+BN2
) +2OA .AN
2
OB = OA2
+AB2
+2OA.
AB
AN
.AB
2
OB = OA2
+OC2
+2OA.OC Cos
 R2
= P2
+Q2
+2PQ Cos
(1).........PQCos2QPR 22

jwäi w`K wbY©q: g‡bKwi jwä R, P Gi mv‡_  †Kvb Drcbœ K‡i| A_©vr AOB  OBN mg‡KvYx wÎfy‡R
ON
BN
tan 
ANOA
BN

 tan
AB
AN
ABOA
AB
BN
AB
tan


QCosP
QSin
tan



(2).........
QCosP
QSin
tan 1


 
(1) bs mgxKiY jwäi gvb I (2) bs mgxKiY jwäi w`K wb‡`©k K‡i|
GKB mg‡q GKB we›`y‡Z wµqviZ `ywU w`Kivwki jwäi m‡e©v”P I me©wb¤§ gvb ivwk `yÕwUi †hvMdj I we‡qvM d‡ji mgvb t
P I Q, †Kv‡Y wµqv Ki‡j mvgvšÍÍwi‡Ki m~Îvbyhvqx Avgiv cvB,
(1).........PQCos2QPR 22

 R2
= P2
+Q2
+2PQ Cos
 R2
 P2
Q2
= 2PQ Cos
PQ2
QPR
Cos
222

 Avevi Avgiv Rvwb, CosGi gvb
1 †_‡K +1 Gi g‡a¨ mxgve×| A_©vr, 1Cos1 
1
PQ2
QPR
1
222


 [CosGi gvb ewm‡q]
PQ2QPRPQ2 222

PQ2QPRPQ2QP 22222
 [Dfq c‡ÿ P2
+Q2
†hvM K‡i]
222
)Q~P(R)QP( 
)Q~P(R)QP(  Kv‡RB GKB mg‡q GKB we›`y‡Z wµqviZ `yÕwU w`K ivwki jwäi m‡e©v”P I me©wb¤§ gvb
ivwk `ywUi †hvMdj I we‡qvM d‡ji mgvb| Ab¨ fv‡eI ejv hvq, GKB mg‡q GKB we›`y‡Z wµqviZ `yÕwU w`K ivwki jwäi
m‡e©v”P gvb ivwk `ywUi †hvMdj n‡Z eo n‡Z cv‡i bv I me©wb¤§ gvb ivwk `ywUi we‡qvM dj †_‡K †QvU n‡Z cv‡i bv|

`yÕwU †f±i ivwk P I Q ci¯úi †Kv‡Y  AvY&Z| G‡ì †¯‹jvi ¸Yb I †f±i ¸Yb t
†¯‹jvi ¸Yb ev WU ¸Yb t `yÕwU w`K ivwki †¯‹jvi ¸Yb GKwU Aw`K ivwk Ges Gi gvb w`Kivwk `ywUi gv‡bi
¸Ydj Ges G‡ì gaëZ©x †Kv‡Yi Cosine Gi ¸Y d‡ji mgvb n‡e|
e¨vL¨vt g‡b Kwi, P

I Q

`yÕwU w`K ivwk ci¯úi  †Kv‡Y AvbZ& A_©vr P I Q Gi gaëZ©x †KvY | G‡ì WU
¸Yb A_©vr †¯‹jvi ¸Yb  PQCosQ.P

wPÎvbyhvqx,

 ;OAP

 ;OBQ BM, OA Gi Dci j¤^ I AN, OB
Gi Dci j¤^ | myZivs OM =OB Cos 
ev, OM =Q Cos 
 )QCos(PQ.P 

= (P

Gi gvb) (OM)
= (P

Gi gvb) ( P

Gi DciQ

Gi j¤^ Awf‡ÿ‡ci gvb)
Abyiƒcfv‡e,  )(.  PCosQQPCosPQ 

= (Q

Gi gvb) (Q

Gi DciP

Gi j¤^ Awf‡ÿ‡ci gvb) A_©vr `ywU †f±i
ivwki †¯‹jvi A_©vr WU ¸Ydj ej‡Z G‡ì †h †Kvb GKwU †f±i ivwk Øviv Dnviw`‡K Aci †f±i ivwkwUi Awf‡ÿ‡ci
¸Ydj †evSvq|
(K) hw` ‡f±i ivwkØq ci¯úi mg‡Kv‡b AvbZ& _v‡K Z‡e 090CosPQQ.P 

AZGe `yÕwU †f±iivwk mg‡Kv‡Y AvbZ&
_vK‡j G‡ì †¯‹jvi ¸bdj k~b¨ n‡e wecixZµ‡g `ywU †f±i ivwki †¯‹jvi ¸bdj k~b¨ n‡j ivwk `ywU ci¯ú‡ii Dci j¤^
n‡e|
awi wZbwU AvqZKvi †f±i ivwk jˆ,iˆ I kˆ cÖ‡Z¨‡K ci¯ú‡ii Dci j¤^|
myZivs G‡ì †h †Kvb `yÕwUi †¯‹jvi ¸bdj k~b¨ n‡e| A_©vr 0ˆ.ˆˆ.ˆˆ.ˆ  ikkjji
(L) hw` `ywU †f±i ivwk GKB w`‡K wµqv K‡i Z‡e G‡ì gaëZ©x †KvY  = 0º;
†m †ÿ‡Î PQCosPQQP  0.

n‡e| GKK w`K ivwki †ÿ‡Î cvB, 1...  kkjjii

(M) hw`P

IQ

`ywU †f±i ivwk ci¯úi wecixZgyLx nq Z‡e G‡ì gaëZ©x †KvY  = 180º
nq Z‡e †m †ÿ‡Î PQCosPQQP  180.

n‡e|
†f±i ¸Yb ev µm ¸Yb t `yÕwU w`K ivwki †f±i (µm) ¸Yb GKwU †f±i ivwk hvi gvb w`Kivwk `ywUi gv‡bi ¸Ydj Ges
G‡ì gaëZ©x †Kv‡Yi Sine -Gi ¸Y d‡ji mgvb Ges GB ¸bdj ivwk `yÕwUi Z‡j j¤^fv‡e ¯’vwcZ GKwU Wvb cv‡Ki KK©
¯Œz‡K 1g †f±i ivwk ‡_‡K 2q †f±i ivwki w`‡K ÿz`ªZi †Kv‡Y Nyiv‡j GUv †h w`‡K AMÖmi nq †mB w`‡K wµqv K‡i|
e¨vL¨vt g‡b Kwi, P

I Q

`yÕwU w`K ivwk ci¯úi  †Kv‡Y AvbZ& A_©vr P I Q Gi gaëZ©x †KvY | G‡ì µm
¸Yb A_©vr ‡f±i ¸Yb PQSinQP

 GLv‡b 

GKwU GKK w`K ivwk hv )( QP

 Gi jwäi w`K wb‡`©k K‡i|
hw` RQP

 aiv nq Z‡e R Gi AwfgyL P

I Q

Gi mgZ‡ji mv‡_ Awfj¤^ eivei n‡e|
 PQSinQPR



( P Gi gvb) ( P Gi j¤^ eivei Q Gi gvb )
Avevi, )(QPSinPQ 

 QPSinPQ



PQ ( Q Gi gvb) ( Q Gi j¤^ eivei P Gi gvb )
PQQP

 A_©vr †f±i ¸Yb wewbgq m~Î †g‡b P‡j bv|

Avgiv Rvwb, Wvb cv‡Ki KK© ¯Œz‡K evgw`‡K Nyiv‡j GwU Dc‡ii w`‡K Ly‡j Av‡m Avi Wvb w`‡K Nyiv‡j wb‡Pi w`‡K AMÖmi
nq, Z`ªæc P

I Q

Gi †f±i ¸Yb evgveZ©x n‡j G‡ì jwä R

Gi AwfgyL DaŸ© w`‡K nq Ges P

I Q

Gi †f±i ¸Yb
`wÿbveZ©x n‡j G‡ì jwä R

Gi AwfgyL wb¤§gyLx n‡e hv c~‡e©i Awfgy‡Li Dëv w`‡K n‡e|
Dc‡iv³ eY©bv Abymv‡i,
1) = 0 n‡j 0QP 

n‡e KviY Sin 0º = 0
2) QP

|| n‡j 0QP 

3) QP

 n‡j PQQP 

4) jˆ,iˆ I kˆ mgKv‡Y wµqv Ki‡j jîˆkˆ,iˆkˆjˆ,kˆjîˆ 
I jˆ,iˆ I kˆ mgvšÍiv‡j wµqv Ki‡j 0kˆkˆjˆjîîˆ 
‡¯‹jvi ¸Yb wewbgq m~Î †g‡b P‡j wKš‘ †f±i ¸Yb wewbgq m~Î †g‡b P‡j bv:
g‡b Kwi `ywU †f±i P

I Q

,  †Kv‡Y AvbZ&  (1).........PQCosQ.P 

Avevi,  QPCosP.Q

(2).........PQCosP.Q 

(1) I (2) n‡Z cvB, P.QQ.P

 A_©vr †¯‹jvi ¸Yb wewbgq m~Î †g‡b P‡j| Acic‡ÿ, )QP(

 I )PQ(

 Gi gvb GKB
n‡jI G‡ì w`K wecixZ A_©vr (3).........SinPQQP 

Avevi, )(SinPQPQ 

(4).........SinPQPQ 

(3) I (4) n‡Z cvB, )PQ()QP(


PQQP 

A_©vr †f±i ¸Yb wewbgq m~Î †g‡b P‡j bv|
‡f±i †hv‡Mi wÎfyR m~Î (Triangle Law) t
`ywU †f±i‡K GKwU wÎfy‡Ri `ywU mwbœwnZ evû Øviv GKB µ‡g w`‡K I gv‡b wb‡`©k
Ki‡j wÎfyRwUi Z…Zxq evû wecixZ µ‡g w`‡K I gv‡b Dnv‡ì jwä wb‡`©k K‡i|
e¨vL¨v t awi P

I Q

GKB RvZxq `ywU †f±i| ABC wÎfy‡Ri AB Ges BC
evû h_vµ‡g †f±i P

I Q

wb‡`©k K‡i| m~Îvbymv‡i,

 BCABAC
QPR

 [

 RACb,GLv‡ ]

‡f±‡ii †hv‡Mi wewbgq m~Î (Commutative Law) t ABBA


cÖgvb t aiv hvK

 AOP Ges

 BOR `ywU †f±i O we›`y‡Z wµqv K‡i OPQR
mvgvšÍwiK c~Y© K‡i wÎfyR m~Îvbymv‡i cvB,
(1)............OQPQOP


Ges (2)............OQRQOR


(1) I (2) †_‡K cvB,

 RQORPQOP
A_©vr, ABBA

 myZvivs ‡f±i †hvM wewbgq m~Î †g‡b P‡j|
‡f±‡ii †hv‡Mi ms‡hvM m~Î (Associative Law) : )CB(AC)BA(


cÖgvb t aiv hvK

 AOP ,

 BPQ Ges

 CQR | GLb OQ, PR Ges OR a‡i
wÎf~R m~Îvbymv‡i cvB, )BA(OQPQOP


Ges )CB(PRQRPQ


GLb

 ORQROQ A_©vr DC)BA(


Avevi,

 ORPROP A_©vr D)CB(A


)CB(AC)BA(

 myZvivs ‡f±i †hvM ms‡hvM m~Î †g‡b P‡j|
e›Ub m~Î:

 C.AB.A)CB.(A Gi cÖgvY:
cÖgvY: g‡b Kwi,

CB,A I †f±i wZbwU h_vµ‡g OP,OQI QR Øviv m~wPZ Kiv n‡q‡Q| GLb wPÎ †_‡K Avgiv
cvB, )QROQ(.A)CB.(A



 OR.A)CB.(A
c¶Awf‡^j¤GiDciGi

 OROPA)CB.(A
ONA)CB.(A 

)MNOM(A)CB.(A 

Figure 1
MNAOMA)CB.(A 

c¶Awf‡^j¤GiDciGic¶Awf‡^j¤GiDciGi

 QROPAOQOPA)CB.(A

 QR.AOQ.A)CB.(A

 C.AB.A)CB.(A (cÖgvwYZ)

wÎgvwÎK ¯’vbv¼ eë¯’vvq GKwU Ae¯’vb †f±‡ii gvb wbY©q:
A_©vr 222
zyxr  Gi cÖgvY :
wÎgvwÎK ¯’vbv¼ eë¯’vq ci¯úi wZbwU †iLv OX, OY I OZ h_vµ‡g X, Y I
Z, Aÿ wb‡`©k K‡i Ges i

, j

I k

Aÿ wZbwU eivei GKK †f±i| r

GKwU Ae¯’vb †f±i| awi,

 OPr

Ges P we›`yi
¯’vbv¼ (x,y,z)| XZ Z‡ji Dci PN Ges N we›`y n‡Z X I Z A‡ÿi Dci h_vµ‡g NA I NB j¤^ AvuwK|
wPÎ n‡Z, BN =OA= x, AN=OB = z Ges NP = y
wÎfyR m~Î Abymv‡i,

 NPONOP

 NPBNOBOP

 OBNPBNOP
kˆzjˆyiˆxr 

Avevi, OPN wÎfyR n‡Z, OP2
=ON2
+NP2
 OP2
= OB2
+BN2
+NP2
 OP2
= BN2
+ NP2
+OB2
2222
zyxr 
)(222
cÖgvwYZzyxr 
zzyyxx BABABAB.A 

Gi cÖgvY:
awi, kÂjÂiÂA zyx 

I kˆBjˆBiˆBB zyx 

d‡j, )kˆBjˆBiˆ) . ( BkÂjÂiˆ(AB.A zyxzyx 

)kˆ.kˆ(BA)jˆ.kˆ(BA)iˆ.kˆ(BA
)kˆ.jˆ(BA)jˆ.jˆ(BA)iˆ.jˆ(BA
)kˆ.iˆ(BA)jˆ.iˆ(BA)iˆ.iˆ(BAB.A
zzyzxz
zyyyxy
zxyxxx




)1(BA)0(BA)0(BA)0(BA)1(BA)0(BA)0(BA)0(BA)1(BAB.A zzyzxzzyyyxyzxyxxx 

zzyyxx BA000BA000BAB.A 

oved)Pr(BABABAB.A zzyyxx 

kÂjÂiÂA zyx 


n‡j

BA

 wbY©q :
)kˆBjˆBiˆ(B)kÂjÂiˆ(ABA zyxzyx 

)kˆkˆ(BA)jˆkˆ(BA)iˆkˆ(BA
)kˆjˆ(BA)jˆjˆ(BA)iˆjˆ(BA
)kîˆ(BA)jîˆ(BA)iîˆ(BABA
zzyzxz
zyyyxy
zxyxxx




0)iˆ(BA)jˆ(BA)iˆ(BA0)kˆ(BA)jˆ(BA)kˆ(BA0BA yzxzzyxyzxyx 

)kˆ(BA)kˆ(BA)jˆ(BA)jˆ(BA)iˆ(BA)iˆ(BABA xyyxzxxzyzzy 

kˆ)BAB(Ajˆ)BAB(Aiˆ)BAB(ABA xyyxzxxzyzzy 

zyx
zyx
BBB
AAA
kˆjîˆ
BA 

(Ans.)
‡f±i wefvRb ev †f±i we‡køølb (Resolution Of Vectors) t
GKwU †f±i ivwk‡K `yB ev Z‡ZvwaK †f±i ivwk‡Z wef³ Kivi c×wZ‡K †f±‡ii wefvRb ev †f±‡ii we‡køøølb e‡j|
g‡bKwi, wP‡Î OC †iLv R

†f±iwUi gvb I w`K wb‡`©k K‡i| GLb R

†f±iwU‡K Ggb `yÕwU As‡k wef³ Ki‡Z n‡e †h, G
¸‡jv OC Gi mv‡_ h_vµ‡g I †KvY Drcbœ K‡i| GLb O we›`y †_‡K OC †iLvi mv‡_ Gi `yB cv‡k I 
†KvY K‡i OB I OA †iLv Uvbv nj| OACB mvgvšÍwiKwU c~Y© Kiv n‡j mvgvšÍÍwi‡Ki
m~Îvbymv‡i OA Ges OB evû `ywU R

†f±‡ii `ywU Dcvsk wb‡`©k Ki‡e|
g‡bKwi, wefvwRZ Dcvsk

 XOA Ges

 YOB |
(K) wP‡Î, OCABOC 
Ges  -180ºOAC GLb OAC wÎfyR we‡ePbv K‡i Avgiv cvB,
)](180[Sin
OC
Sin
AC
Sin
OA





Y)OBAC(
)(Sin
R
Sin
Y
Sin
X






 
)(Sin
RSin
X


 Ges
)(Sin
RSin
Y


 R †f±i‡K hw` mg‡Kv‡Y wefvwRZ Kiv hvq [wPÎ (L)] A_©vr
Dcvsk `ywU hw` ci¯úi j¤^ nq Z‡e Zvn‡j,+90º Sin+Sin 90º = 1
X = R SinGesY = R Sin †h‡nZz +90º 90º
Sin Sin () = Cos d‡j, X = R Cos GesY = R Sin

cÖ_g c‡Îi As‡Ki mgvavb
First Paper Math Solution
1| ‡f±i (Vector)
1| ‡f±i
1| hw` kˆ2jîˆ3A 

I kˆjˆ3iˆ2B 

nq Z‡e
 |BA|

KZ?
132
213
kˆjîˆ
BA 

kˆ11jˆ7iˆ5)29(kˆ)43(jˆ)61(iˆ 
Ans.)96.13121492511)7(5|BA| 222
(

2| hw` kˆ2jˆ3iˆ6A 

I kˆjˆ2iˆ2B 

nq Z‡e
KZ?B.A


)(Ans.82612)1)(2()2)(3()2)(6( 
3| kˆ3-jˆ2iˆ5A 

I |kˆ9-jâiˆ15B 

a Gi gvb KZ
n‡j A

I B

ci¯úi mgvšÍivj n‡e?
A

I B

ci¯úi mgvšÍivj n‡e hw` 0BA 

nq|
9-a15
325
kˆjîˆ
BA 

)30a-5(kˆ)4545(-jâ)318(iˆBA 

)30a5(kˆ)18a3(iˆBA 

2)30a5(2)18a3(BA 

cÖkœg‡Z, A

I B

ci¯úi mgvšÍivj n‡j 0BA 

0)30a5()18a3( 22

i]K‡MeK‡c¶[Dfq  0)30a5()18a3( 22
0)6(a5)6(a3 2222

0)53()6(a 222

iK‡fvMØvivK‡c¶Dfq )53(0)6(a 222

.)Ans(6a 
4| kˆ5-jˆ3iˆ2A 

I |kˆ10-jˆ2iˆmB 

m Gi
gvb KZ n‡j A

I B

ci¯úi j¤^ n‡e|
A

I B

ci¯úi j¤^ n‡j 0B.A 

n‡e|
0 zBzAyByAxBxAB.A

0506m2)10)(5()2)(3()m)(2( 
(Ans.)28-m
2
56-
m 
5| kˆjˆ2iˆ2A 

I kˆ2jˆ3iˆ6B 

n‡j BA

I
Gi gaëZ©x †KvY wbY©q Ki|
 cosABB.A

BA

I Gi gaëZ©x †KvY,
BA
B.A
Cosθ 1




42-6-12)2)(1(-
)3)(-2()6)(2(BABABABA zzyyxx.



39)1(22A 222



74949362)3(6B 222


(Ans.)02.79
21
4
Cos
)7)(3(
4
Cos
BA
B.A
Cosθ 11-1

 


6. ‡f±i kˆ2jˆ3iˆ6B 

Gi Dci ‡f±i
kˆjˆ2iˆ2A 

Gi j¤^ Awf‡ÿc wbY©q Ki|
 cosABB.A

A Gi j¤^ Awf‡ÿc,
B
B.A
cosA 


826-12)2)(1()3)(-2()6)(2(
BABABAB.A zzyyxx



74949362)3(6B 222


A Gi j¤^ Awf‡ÿc =
7
8
B
B.A


7| kˆ3jˆ2iˆB,kˆjˆ2iˆ3A 

I kˆ2jîˆC 

n‡j cÖgvY Ki †h, C).BA()CB.(A


211
321
kˆjîˆ
CB 

kˆ-jˆ5-iˆ7)2-1(kˆ)32(jˆ)34(iˆCB 

ev,

1011021)1)(-1()5)(-2()7)(3(
)CB. (AL.H.S



3-21
123
kˆjîˆ
BA 

GLb,
kˆ4jˆ10iˆ8
)2-6(kˆ)19(jˆ)26(iˆ=BA



ev,
108108-)2)(4(
)1)(10()1)(8(-C) .BA(R.H.S



R.H.SL. H. S 
A_©vr, oved.)Pr(C).BA()CB.(A


8| ,kˆ2jˆ3iÂ 

kˆjˆ2iˆB 

I kˆ4jˆ3iˆ2C 

n‡j cÖgvY Ki †h, ACABA)CB(


kˆ3jîˆ3kˆ)41(jˆ)32(iˆ)21()CB( 

231
313
kˆjîˆ
A)CB(L.H.S 

kˆ10jˆ3iˆ11)19(kˆ)36(jˆ)92(iˆ 
231
121
kˆjîˆ
AB 

Avevi,
kˆjˆ3iˆ7)2-3(kˆ)12(jˆ)34(iˆ 
231
432
kˆjîˆ
AC 

kˆ9iˆ18)36(kˆ)4-4(jˆ)126(iˆ 
kˆ10jˆ3iˆ11
kˆjˆ3iˆ7ACABR.H.S kˆ9iˆ18

 

A_©vr, oved.)Pr(ACABA)CB(


9| kˆ6-jîˆ9A 

Ges kˆ5jˆ6iˆ4B 

†f±i `ywUi
Mybdj wbY©q K‡i †`LvI †h, Giv ci¯ú‡ii Dci j¤^|

030636)5)(6()6)(1()4)(9( 
0ABCosθB.A 

0CosθC0, B0A wKš‘
90θ90CosCos θ, ev
AZGe ΒA

I ci¯ú‡ii Dc‡ii j¤^|
10| hw` kÂjÂiÂA zyx 


nq
Z‡e †`LvI †h, zzyyxx BABABAB.A 

)kˆBjˆBiˆ) . ( BkÂjÂiˆ(AB.A zyxzyx 

)kˆ.kˆ(BA)jˆ.kˆ(BA)iˆ.kˆ(BA
)kˆ.jˆ(BA)jˆ.jˆ(BA)iˆ.jˆ(BA
)kˆ.iˆ(BA)jˆ.iˆ(BA)iˆ.iˆ(BAB.A
zzyzxz
zyyyxy
zxyxxx




)1(BA)0(BA)0(BA
)0(BA)1(BA)0(BA
)0(BA)0(BA)1(BAB.A
zzyzxz
zyyyxy
zxyxxx




zzyyxx BA000BA000BAB.A 

oved)Pr(BABABAB.A zzyyxx 

11| hw` kˆ5jˆ4iˆ2A 

I kˆ3jˆ2iˆB 

‡f±i Ø‡qi jwä
†f±‡ii mgvšÍivj GKK †f±i wbY©q Ki|
k3jˆ2iˆkˆ5jˆ4iˆ2BAR 

kˆ2jˆ6iˆ3R 

R Gi mgvšÍivj GKK †f±i
R
R
aˆ 


7494369)2(63R 222


(Ans.)kˆ
7
2
jˆ
7
6
iˆ
7
3
7
kˆ2jˆ6iˆ3
R
R
aˆ 

 

12| GKB we›`y‡Z wµqvkxj `ywU mgvb gv‡bi †f±‡ii gaëZ©x †KvY KZ
n‡j G‡ì jwäi gvb †h †Kvb GKwU †f±‡ii mgvb n‡e?
R2
=P2
+Q2
+2PQCos
ev, X2
= X2
+X2
+2X.X.Cos
ev, X2
- X2
-X2
=2X2
Cos
ev, -X2
=2X2
Cos
2
2
X2
X
Cos, ev 
2
1
Cos, ev 






 
2
1
Cos, 1
ev 
(Ans.)120
GLv‡b,
awi, ‡f±i, P=Q=X
jwä, R= X
AšÍ©f~³ †KvY, 

13| Ae¯’vb †f±i kˆzjˆyiˆxr 

†K eëKjb K‡i
wKfv‡e †eM I Z¡iY cvIqv hvq?
Avgiv Rvwb,
‡eM,
dt
rd
v



)kˆzjˆyiˆ(x
dt
d

(Ans.)kˆ
dt
dz
jˆ
dt
dy
iˆ
dt
dx

Avevi, Z¡iY 





 kˆ
dt
dz
jˆ
dt
dy
iˆ
dt
dx
dt
d
dt
vd
a


(Ans.)kˆ
dt
zd
jˆ
dt
yd
iˆ
dt
xd
2
2
2
2
2
2

14| kˆ)1t2(jˆtiˆtP 2


I kˆtjˆtiˆt5Q 3


?)Q.P(
dt
d


?)QP(
dt
d


)t)(1t2()t)(t()t5)(t(Q.P 32


3423
tt2tt5Q.P 

234
tt4t2Q.P 

)tt4t2(
dt
d
)Q.P(
dt
d 234


t2t12t8)Q.P(
dt
d 23


3
2
ttt5
1t2tt
kˆjîˆ
QP



)t5t(kˆ
)tt5t10(jˆ)tt2t(iˆQP
23
5224














23
5224
t5t(kˆ
)tt5t10(jˆ)tt2t(iˆ
dt
d
)QP(
dt
d 
kˆ)t5t(
dt
d
jˆ)t5t10t(
dt
d
iˆ)tt2t(
dt
d
)QP(
dt
d
2325
24



(Ans.)kˆ)t10t3(
jˆ)5t20t5(iˆ)1t4t4()QP(
dt
d
2
43



15| kˆ3jˆ2-iˆQ,kˆ4jˆ3iˆ2P 

‡f±i Øq †h Z‡j
Ae¯’vb K‡i Zvi Dj¤^w`‡K GKwU GKK †f±i wbY©q Ki|
Avgiv Rvwb,
`ywU †f±‡ii µm ¸bdj †f±i `ywU Øviv MwVZ mgZ‡ji Dci j¤^ nq| †mB
j¤^ †f±‡ii mgvšÍivj GKK †f±iB n‡e mgZ‡ji Dj¤^ w`‡K GKK
†f±i| awi, †mB †f±i aˆ ,
QP
QP
a 



ˆ
)34(kˆ)46(jˆ)89(iˆ
321
432
kˆjîˆ
QP 



kˆ7jˆ10iˆQP 

P

I Q

‡h Z‡j Aew¯’Z Zvi Dj¤^ w`‡K †f±i
222
)7()10()1(
)kˆ7jˆ10iˆ(
QP
)QP(
aˆ





 

.)(
150
)kˆ7jˆ10iˆ(
aˆ Ans


16| †Kvb GKwU KYvi Ae¯’vb †f±i
jˆ][5.3msiˆ4.2m])t[(3.5msr 11 


n‡j
†eM V wbY©q Ki|
Avgiv Rvwb,
dt
rd
V


 jˆt][5.3msiˆ4.2m])t[(3.5ms
dt
d
V 11 

jˆ5.3iˆ3.5V  (Ans.)
17| cÖgvY Ki t     22
22
BAB.ABA 

   
22
.BABALHS


   
22
cossinˆ  ABAB 
 2222222
cosBAsinBAˆ
 222222
cosBAsinBA.1
  2222
cossinBA
2222
1. BABA  (Proved).... SHRSHL 
18| kˆjˆ2iˆP 

Ges kˆ3jˆ6iˆ3Q 

n‡j †`LvI †h, P

I
Q

ci¯úi mgvšÍivj|
P

I Q

ci¯úi mgvšÍivj n‡e hw` 0QP 

nq|
)66(kˆ)33(jˆ)66(iˆ
363
121
kˆjîˆ
QP 



0000)0(kˆ)0(jˆ)0(iˆQP 


0 QP

0 QP

  P

I Q

ci¯úi mgvšÍivj|
(cÖgvwYZ)
19| kˆ4jˆ3iˆ2P 

Ges kˆ3jîˆ2Q 

‡f±i Øq †h
Z‡j Aew¯’Z Zvi Dj¤^w`‡K GKwU GKK †f±i wbY©q Ki|
Avgiv Rvwb,
`ywU †f±‡ii µm ¸bdj †f±i `ywU Øviv MwVZ mgZ‡ji Dci j¤^
nq| †mB j¤^ †f±‡ii mgvšÍivj GKK †f±iB n‡e mgZ‡ji Dj¤^
w`‡K GKK †f±i| awi, †mB †f±i nˆ ,
QP
QP
n 



ˆ
)62(kˆ)86(jˆ)49(iˆ
312
432
kˆjîˆ
QP 



kˆ8jˆ2iˆ13QP 

Avevi, 222
8213QP 

237QP 

237
ˆ8ˆ2ˆ13
ˆ
kji
QP
QP
n




 

.)(
237
8ˆ
237
2ˆ
237
13
ˆ Ansjin 






20| kˆjˆ4iˆ4P 

Ges kˆjˆ2iˆ2Q 

†f±iØq
GKwU mgvšÍwi‡Ki `ywU mwbœwnZ evû wb‡`©k Ki‡j Gi †ÿÎdj wbY©q
Ki|
Avgiv Rvwb, `ywU †f±i GKwU mgvšÍwi‡Ki `ywU mwbœwnZ evû wb‡`©k
Ki‡j H mgvšÍwi‡Ki †ÿÎdj n‡e †f±i `ywUi µm ¸Yd‡ji gv‡bi
mgvb| QP

mgvšÍwi‡Ki †ÿÎdj|
GLb,
)88(kˆ)24(jˆ)24(iˆ
122
144
kˆjîˆ
QP 



jˆ6iˆ6QP 

49.87266 22
 QP

GKK (Ans.)
21| kˆ3-jˆmiˆ2P 

I |kˆ15-jˆ5iˆ10Q 

m Gi gvb KZ n‡j
P

IQ

ci¯úi mgvšÍivj n‡e?
P

I Q

ci¯úi mgvšÍivj n‡e hw` 0QP 

nq|
15-510
3m2
kˆjîˆ
QP



m)10-10(kˆ)3030(-jˆ)15m15(iˆQP  

m)1010(kˆ)15m15(iˆQP 

2m)1010(2)15m15(QP 

cÖkœg‡Z, P

IQ

ci¯úi mgvšÍivj n‡j, 0QP 

02m)1010(2)15m15( 
i]K‡MeK‡c¶[Dfq  0m)1010()15m15( 22
0)1(m10)1(m215 222

0)1015()1(m 222

iK‡fvMØvivK‡c¶Dfq )1015(0)1(m 222

.)Ans(1m 
22| kjiA ˆˆ2ˆ2 

Ges kjiB ˆ2ˆ3ˆ6 

`yÕwU ‡f±i ivwk|
G‡ì j¤^ Awfgy‡L GKwU GKK †f±i wbY©q Ki|
Avgiv Rvwb,
`ywU †f±‡ii µm ¸bdj †f±i `ywU Øviv MwVZ mgZ‡ji Dci j¤^ nq| †mB
j¤^ †f±‡ii mgvšÍivj GKK †f±iB n‡e mgZ‡ji Dj¤^ w`‡K GKK
†f±i| awi, †mB †f±i nˆ ,
QP
QP
n 



ˆ
)126(ˆ)64(ˆ)34(ˆ
236
122
ˆˆˆ


 kji
kji
QP

kjiQP ˆ18ˆ10ˆ 

Avevi, )18()10(1 222
QP

425 QP

425
ˆ18ˆ10ˆ
ˆ
kji
QP
QP
n




 

(Ans.)

 ফাস্তফ যক্ষনত্র ঩যভবিবত ঑ ঩যভগবত অ঳ম্ভফ।
঩ৃবথফীয গবত িরন ঘূণ঱ন গবত।
 ঘবড়য কাটায গবত ঩ম঱াফৃত্ত গবত।
 ভ঴াবফনেয ঳ফ ফস্ত্ত্তই গবত঱ীর, তাই বনশ্চর প্র঳ঙ্গ কাঠ্ানভা ঩া঑য়া মায় না।
 যনৌকায মাত্রী যনৌকায ঳ান঩নক্ষ আন঩বক্ষক বিবতনত থানক বকন্তু তীনযয ঳ান঩নক্ষ আন঩বক্ষক গবতনত থানক।
 ভুিবানফ উ঩য যথনক বননি ঩ড়ন্ত ফস্ত্ত্ত ঳ভত্বযনণয একবট ফাস্তফ উদা঴যণ এফিং খাড়া উ঩নযয বদনক বনবক্ষপ্ত ফস্ত্ত্তয ভন্দন ঳ভ-
ভন্দননয একবট ফাস্তফ উদা঴যণ।
঩ড়ন্ত ফস্ত্ত্তয ৩বট ঳ূত্র আবফষ্কায কনযনেন গযাবরবর঑ বকন্তু প্রভাণ কনযনেন বনউটন।
 অ঳ভ দ্রুবতনত ফা অ঳ভ যকৌবনক যফনগ ঘুযনত থাকনর কণাবটয ৩বট ত্বযণ থানক। মথা-যকদ্রতাভুখী ত্বযণ, স্প঱঱ীত্বযণ ঑ যকৌবনক
ত্বযণ।
ফরবফদযা ভূরত দুই প্রকায। মথা:
১। বিবতবফদযা: (ক) ঩যভবিবত (খ) আন঩বক্ষক বিবত।
২। গবতবফদযা: (ক) ঳ৃবতবফদযা (খ) ির বফদযা।
গবতয প্রকায যবদঃ
(i) রযবখক গবত ফা একভাবত্রক গবতঃ যকান ফস্ত্ত্তয গবত মবদ একবট ঳যর যযখায উ঩য ঳ীভাফদ্ধ থানক। তা঴নর তায
গবতনক রযবখক গবত ফা একভাবত্রক গবত ফনর। যমভন-য঳াজা ঳ড়নক গাবড়য গবত।
(ii) ঳ভতরীয় ফা বিভাবত্রক গবতঃ গবত ঳ভতনরয উ঩য ঳ীভাফদ্ধ। যমভন- ব঩঩ঁড়ায গবত, ভানফ঱নরয গবত।
(iii) িাবনক গবত ফা বত্রভাবত্রক গবতঃ যকান ফস্ত্ত্তয গবত মবদ যম যকান বদনক গবত঱ীর ঴নত ঩ানয তনফ তায গবতনক িাবনক গবত ফা
বত্রভাবত্রক গবত ফনর।
বফববন্ন প্রকায গবতয রফব঱িয ঑ উদা঴যণঃ
১. িরন গবত এই গবতনত ফস্ত্ত্তয প্রবতবট কণা একই বদনক ঳ভান দূযত্ব অবতক্রভ কযয।
উদা঴যণঃ যযর঩নথয উ঩য িরন্ত যযরগাবড়য ফবগয গবত।
২. ঘূণ঱ন গবত এই গবতনত ফস্ত্ত্ত একবট বনবদি বফন্দু ফা অক্ষনক যকদ্রতা কনয িক্রাকানয ঩বযরামভন কনয।
উদা঴যণঃ রফদুযবতক ঩াখায গবত, ঘবড়য কাঁটায গবত,
৩. িরন ঘূণ঱ন ফা বভশ্র ফা জবটর গবত এই গবতনত ফস্ত্ত্তয িরন ঑ ঘূণ঱ন দুবট গবতই থানক।
উদা঴যণঃ িরন্ত ঳াইনকর ফা গরুয গাবড়য িাকায গবত, ঳ূনম঱য িাযবদনক ঩ৃবথফীয গবত এফিং রাবটনভয গবত।
৪. ঩ম঱াফৃত্ত গবত এই গবতনত ফস্ত্ত্ত একবট বনবদ঱ি ঳ভয় ঩য ঩য একই ঩থ অবতক্রভ কনয একই বদনক িরনত থানক।
উদা঴যণঃ ঘবড়য কাঁটা, ইবিননয ব঩স্টন, রফদুযবতক ঩াখা, যদারক ব঩ন্ড ইতযাবদয গবত।
৫. যদারন গবত এই গবতনত ফস্ত্ত্তবট বনবদ঱ি ঳ভয় অন্তয অন্তয এবদক ঑বদক যদার যদয়।
উদা঴যণঃ যদারক ঘবড়য গবত, ঳যর যদারনকয

Mo‡eM (Average Velocity) :
msÁv: †h †Kvb mgq eëav‡b †Kvb e¯‘i M‡o cÖwZ GKK mg‡q †h miY nq Zv‡K e¯‘wUi Mo †eM e‡j|
e¨L¨v: t mgq eëav‡b †Kvb e¯‘i miY r

 n‡j Mo †eM
t
rΔ
v




n‡e|
‡eM (Velocity):
msÁv: mgq eëavb k~‡bï KvQvKvwQ n‡j mg‡qi mv‡_ e¯‘i mi‡Yi nvi‡K †eM e‡j|
e¨L¨v: t mgq eëav‡b †Kvb e¯‘i miY r

 n‡j †eM
t
rΔ
limv
0t 




wKš‘
t
rΔ


n‡”Q Mo †eM v

| myZivs vlimv
0t



A_©vr mgq eëavb k~‡bï 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‡ì †eM Amg‡eM|
ZvrÿwbK †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 ZvrÿwbK †eM e‡j| ZvrÿwbK †e‡Mi w`K e¯‘wUi H gyû‡Z©i Ae¯’v‡b AswKZ MwZc‡_i
¯úk©K eivei|
Z¡iY (Acceleration) :
mg‡qi mv‡_ †eM e„w×i nvi‡K Z¡iY e‡j| t mgq eëav‡b e¯‘i †e‡Mi cwieZ©b v

 n‡j Z¡iY
t
v





a n‡e| Ab¨fv‡e
ejv hvq mgq eëavb k~‡bï KvQvKvwQ n‡j mg‡qi mv‡_ e¯‘i †eM e„w×i nvi‡K Z¡iY e‡j| t mgq eëav‡b e¯‘i †e‡Mi
cwieZ©b v

 n‡j Z¡iY
t
v
lim
0t 





a n‡e|
mgZ¡iY ev mylg Z¡iY (Uniform Acceleration) :
GKB w`‡K GKB mgq eëav‡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 cÖK…ó 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

  
t
0
xx
v
v
x dtdv
x
xo
“eKaªaa 
   
t
o
x
v
v
x tv
x
x0
a
 0tvv xxox  a
tvv xxox a mgxKiYwU cÖwZcv`b Kiv nj|
(L) t)vv(
2
1
xx xxoo  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 = x0 Ges Avw`‡eM vx = vxo | Av‡iv awi, t mgq ci e¯‘wUi Ae¯’vb x Ges Ges
†kl‡eM vx | Mo‡e‡Mi msÁv n‡Z Avgiv Rvwb, ÿz`ªvwZÿz`ª mgq eëav‡b †eM I mgq eëav‡bi ¸bd‡ji mgwó wb‡q
Zv‡K †gvU mgq w`‡q fvM Ki‡j H mg‡qi Mo‡eM e‡j| myZivs KYvwUi t mgq ci Mo‡eM xv n‡j,
dtv
t
v
t
0
xx 
1





dt
dx
xvdt
dt
dx
t
1
v
t
0
x 
(1)............tvxx xo 
mymg Z¡i‡Y Pjgvb e¯‘wUi †ÿ‡Î †eM xv mg‡qi mv‡_ mymgfv‡e cwiewZ©Z nq e‡j †h †Kvb mgq eëav‡b Zvi Mo gvb H
mgq eëav‡bi ïiæ I †k‡li †e‡Mi gvbØ‡qi mgwói A‡a©K| A_©vr xxxox v),vv(
2
1
v  Gi GB gvb (1) bs
mgxKi‡Y ewm‡q cvB, t)vv(
2
1
xx xxoo  t)vv(
2
1
xx xxoo  mgxKiYwU cÖwZcv`b Kiv nj|
(M) 2
xxoo
2
1
tatvxx  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|
myZivs,
dt
dv
a x
x 
dtadv xx  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,
  
t
0
xx
x
xo
æeKaªadtadv
v
v
x 
   t
o
v
v tav x
xo xx 
 0xxox  tavv
(1)...............xxox tavv  †h †Kvb gyû‡Z© e¯‘i miY e„w×i nvi‡K †eM e‡j| D³
(1) mgxKi‡Y
dt
dx
v x ewm‡q cvB, tav
dt
dx
xxo 

xxo dttadtvdx 

t
o
t dtxa
t
o
dtxov
x
xo
dx
    t
atvx
t
o
x
t
oxo
x
xo 






2
2
   0
2
1
0 2
 tatvxx xxoo
2
2
1
tatvxx xxoo  mgxKiYwU cÖwZcv`b Kiv nj|
ev,
2
2
1
tatvxx xxoo  2
2
1
tatvS xxo  mgxKiYwUI cÖwZcv`b Kiv nj| w¯’i
Ae¯’vb †_‡K mgZ¡i‡Y Pjgvb e¯‘i †ÿ‡Î, 0xov Ges æeKaªxa ,d‡j
2
2
1
0 tS  æeKaª 2
tS æeKaª
2
tS Kv‡RB, w¯’i Ae¯’vb †_‡K mgZ¡i‡Y Pjgvb e¯‘i AwZµvšÍ `yiZ¡ mg‡qi e‡M©i mgvbycvwZK|
(N) )xx(2vv ox
2
xo
2
x  a 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|
myZivs
dt
dvx
x aiYZ¡
dt
dx
xd
dvx
x  a




 xx
x
x v
dt
dx
v
xd
dv
a
dxdvv xxx a
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,
 
x
x
x
v
v
xx
o
x
ox
dxdvv a
 x
xx
o
v
v
2
x
x
2
v
x
xo
a






)xx(
2
vv
ox
2
xo
2
x


 a
)xx(2vv ox
2
xo
2
x  a mgxKiYwU cÖwZcv`b Kiv nj|

(O) )1t2(a
2
1
vs ot  ev, a
2
)1t2(
vs ot

 cÖwZcv`b :
g‡bKwi, vo Avw`‡e‡M Ges a mgZ¡i‡Y AB mij‡iLv eivei Pjgvb KYvwUi Avw` Ae¯’vb A we›`y‡Z| (t-1) †m‡K‡Û Ges t
†m‡K‡Û hw` Gi Ae¯’vb h_vµ‡g C I B we›`y‡Z nq Z‡e t Zg †m‡K‡Û e¯‘ KYvwU BC `~iZ¡ AwZµg Ki‡e| MwZi
m~Îvbyhvqx t †m‡K‡Û e¯‘ KZ…K AwZµvšÍ `~iZ¡ AB
(1)...............at
2
1
tvAB 2
0 
(t-1) †m‡K‡Û e¯‘ KZ…K AwZµvšÍ `~iZ¡ AC
(2)...............)1t(a
2
1
)1t(vAC 2
0 
Kv‡RB t Zg †m‡K‡Û e¯‘ KZ…K AwZµvšÍ Í `yiZ¡ st n‡j
st = CB
ACABst 













 )1t(a
2
1
)1t(vat
2
1
tvs 2
0
2
0t
)1t(a
2
1
)1t(vat
2
1
tvs 2
0
2
0t 
)1t2t(a
2
1
vtvat
2
1
tvs 2
00
2
0t 
a
2
1
atat
2
1
vtvat
2
1
tvs 2
00
2
0t 
a
2
1
atvs 0t 
a
2
1
atvs 0t 
)1t2(a
2
1
vs ot  ev, a
2
)1t2(
vs ot

 mgxKiYwU cÖwZcv`b Kiv nj|
Lvov Dc‡ii w`‡K wbwÿß e¯‘i †ÿ‡Î me©vwaK D”PZv wbY©q :
g‡bKwi, †Kvb GKwU e¯‘‡K Lvov Dc‡ii w`‡K v0 †e‡M †Qvov nj| G †ÿ‡Î e¯‘i Dci wµqviZ Z¡iY = g, KviY g
c„w_exi †K›`ªvwfgyLx| Kv‡RB h D”PZvq v MwZ‡eM AR©b Ki‡j MwZi mgxKiY †_‡K Avgiv cvB, gH2vv 2
o
2
 Kv‡RB
e¯‘wU D‡aŸ© DV‡Z _vK‡j Gi †kl †e‡Mi gvb µgkt Kg‡Z _vK‡e| Kv‡RB †h we›`y‡Z wbwÿß e¯‘i †eM k~b¨ H we›`yB
e¯‘i MwZ c‡_i m‡e©v”P we›`y wb‡`©k K‡i| myZivs †h we›`y‡Z e¯‘i †kl ‡eM k~b¨, †mB we›`yi D”PZv H n‡j ,
gH2v0 2
o
2

g2
v
H
2
o
 Kv‡RB Lvov Dc‡ii w`‡K wbwÿß e¯‘i †ÿ‡Î
g2
v2
o
-B nj e¯‘ KZ©…K AwZµvšÍ Í m‡e©v”P
D”PZv|
Lvov Dc‡ii w`‡K wbwÿß e¯‘i †ÿ‡Î me©vwaK D”PZvq †cŠQ‡Z mgq wbY©q :
g‡bKwi, †Kvb GKwU e¯‘‡K Lvov Dc‡ii w`‡K vo †e‡M †Qvov nj| G †ÿ‡Î e¯‘i Dci wµqviZ Z¡iY = -g, KviY g
c„w_exi †K›`ªvwfgyLx| t mgq c‡i †Kvb wbw`©ó D”PZvq e¯‘wUi †eM v n‡j, MwZi mgxKiY n‡Z cvB, v = vo  gt Kv‡RB
†`Lv hv‡”Q †h, Dc‡ii w`‡K wbwÿß e¯‘i †eM ax‡i ax‡i Kg‡Z _v‡K| †Kvb wbw`©ó D”PZvq hLb gt Gi gvb vo Gi mgvb
ZLb †kl †eM v = 0 n‡e| A_©vr e¯‘wU Avi Dc‡i DV‡Z cvi‡e bv| GB D”PZvB m‡Ÿ©v”P D”PZv| awi, m‡e©v”P D”PZvq

MgbKvj t, myZiv&s 0 = vo gt
g
v
t o
 Kv‡RB Lvov Dc‡ii w`‡K wbwÿß e¯‘i †ÿ‡Î
g
vo
-B nj e¯‘ KZ©…K AwZµvšÍ
m‡e©v”P D”PZvq †cŠQ‡bvi mgq|
Lvov Dc‡ii w`‡K wbwÿß e¯‘i †ÿ‡Î DÌv‡bi, cZ‡bi, DÌvb cZ‡bi mgq wbY©q :
g‡bKwi, †Kvb GKwU e¯‘‡K Lvov Dc‡ii w`‡K vo †e‡M †Qvov nj| G †ÿ‡Î e¯‘i Dci wµqviZ Z¡iY = g, KviY g
c„w_exi †K›`ªvwfgyLx| t mgq c‡i h D”PZvq e¯‘wUi †eM v n‡j MwZi mgxKiY n‡Z cvB,
v = vo – gt ... ... ... ... (1)
Ges )2(.........gH2vv 2
o
2

myZivs †`Lv hv‡”Q †h, Dc‡ii w`‡K wbwÿß e¯‘i †eM ax‡i ax‡i Kg‡Z _v‡K| GB †eM GK mgq 0 (k~b¨) n‡e| A_©vr
e¯‘wU Avi Dc‡i DV‡e bv| d‡j e¯‘wU †h D”PZvq D‡V ZvB-B e¯‘ KZ©…K AwZµvšÍ me©vwaK D”PZv H| awi GB D”PZvq
†h‡Z mgq jv‡M t1 myZivs (1) bs mgxKiY n‡Z cvB,
0 = vo – gt1
)3(...............
g
v
t o
1 
me©vwaK D”PZvq DÌvb Kvj
g
v
t o
1  , Avevi (2) mgxKiY n‡Z cvB,
gH2v0 2
o 
)4(............
g2
v
H
2
o
 me©vwaK D”PZvq e¯‘i †kl‡eM k~b¨| Kv‡RB m‡e©v”P D”PZv n‡Z f~wg‡Z
covi mgq Gi Avw`‡eM n‡e k~b¨ Ges ïay AwfKl©R Z¡i‡Y e¯‘wU wb‡P co‡Z _vK‡e| m‡e©v”P we›`y †_‡K f~wg‡Z wd‡i
Avm‡Z t2 mgq jvM‡j, MwZi mgxKiY †_‡K cvB,
2
22 gt
2
1
t0H 
g2
v
Hgt
2
1 2
02
2 
2
2
o2
2
g
v
t 
)5(............
g
v
t o
2 
DÌvb cZ‡bi †gvU mgq T n‡j T = t1+t2
g
v
g
v
T oo

g
v2
T o
 BnvB DÌvb cZ‡bi †gvU mgq|
Lvov Dc‡ii w`‡K wbwÿß e¯‘ fywg‡Z wd‡i Avmvi mgq wbY©q :
g‡bKwi, †Kvb GKwU e¯‘‡K Lvov Dc‡ii w`‡K vo Avw`‡e‡M †Qvov nj| G †ÿ‡Î e¯‘i Dci wµqviZ Z¡iY = g,
KviY g c„w_exi †K›`ªvwfgyLx| d‡j e¯‘wUi †eM ax‡i ax‡i Kg‡Z _vK‡e Ges GKwU wbw`©ó D”PZvq †eM k~b¨ n‡e A_©vr
e¯‘wU Avi Dc‡i DV‡ebv| Kv‡RB GB D”PZvq e¯‘ K©Z…K AwZµvšÍ Í m‡e©v”P D”PZv| awi m‡e©v”P D”PZv H, d‡j MwZi
mgxKiY †_‡K cvB, gH2vv 2
o
2

gH2v0 2
o 

(1).........
g2
v
H
2
o

Avevi m‡e©v”P D”PZv †_‡K fywg‡Z cÖZ¨veZ©bKv‡j e¯‘i Avw`‡eM k~b¨ n‡e Ges ïay gvÎ AwfKl©R Z¡i‡Y e¯‘wU wb‡P co‡Z
_vK‡e| m‡e©v”P we›`y †_‡K fywg‡Z wd‡i Avm‡Z e¯‘i t mgq jvM‡j MwZi mgxKiY †_‡K cvB,
2
gt
2
1
t0H 
2
2
o
gt
2
1
2g
v

2
o2
g
v
t 






(2)............
g
v
t o
 BnvB m‡e©v”P D”PZv n‡Z cZbKvj|
awi, fywg‡Z wd‡i Avm‡Z †h mgq jv‡M †mB mg‡qi †k‡l e¯‘i †eM = v
myZivs v = 0 + gt
g
v
gv o

ovv  myZivs f~wg †_‡K e¯‘‡K †h †e‡M Dc‡ii
w`‡K wb‡ÿc Kiv nq, e¯‘wU wd‡i G‡m †mB †e‡M f~wg‡Z AvNvZ K‡i|
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,
tv
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, 2
th
¯^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 Íc‡o|
djvdjt evqyk~b¨ ¯’v‡b mKj e¯‘ mgvb mg‡q mgvb c_ AwZµg K‡i|

mgZ¡iY MwZi †ÿ‡Î †eM ebvg mgq (v  t)†jLwPÎ AsKb Ges †jLwPÎ n‡Z 2
o at
2
1
tvs  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|
= AO×OQ + 2
1
×AR×PR
s = AO×OQ + 2
1
×OQ×PR [∵ AR = OQ ]
wKš‘ AB ‡iLvi Xvj n‡”Q KYvwUi Z¡iY a,
AR
PR
a 
PR = a×AR
= a×OQ
s = AO×OQ + 2
1
×OQ×a×OQ
 s = AO×OQ + 2
1
×a×OQ2
 2
o at
2
1
tvs  mgxKiYwU cÖwZcv`b Kiv nj| 

2| ‰iwLK MwZ (Linear Motion)
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?
g‡b Kwi,
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 =
2
u
Ges †kl †eM n‡e 0 (k~b¨)|
Avgiv Rvwb, cÖ_g As‡ki Rb¨
2a(0.04)u
2
u 2
2






4
u
u0.08a
2
2

)1(...............
0.32
3u
0.084
3u
a
22



wØZxq As‡ki Rb¨,
ax2
2
u
0
2







x
32.0
u3
2
2
u
0,
22






ev
4
u
0.32
x6u
,
22
ev
(Ans.)m0.0133
46
0.32
x 


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?
(K) 2
gt
2
1
uth 
2
t9.8
2
1
005 
2
t9.405 
9.4
50
t2

9.4
50
t 
t = 3.19 s (Ans.)
Avevi,
(L) v = u + gt
⇒ v = 0 + 9.8 × 3.19
v = 31.26 ms-1
(Ans.)
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
ev, 0 = 202
– 2(3)s
ev, 6s = 400
6
400
s ev,
(Ans.)m66.7s 
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,
2ghuv 22

5.19.82u(0.7) 22

5.19.82(0.7)u 22

96.9949.0u2

45.100u2

(Ans.)ms10.0245.100u -1

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,
2as2u2v 
603210v 22

360100v2

460v2

)Ans(1ms45.21447.21460v 
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|
Avgiv Rvwb,
3 †mt c‡i †eM
v1 = u gt1
ev, v1 = 989.8×3
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)
GLv‡b,
Avw`‡eM, u = 20 ms-1
g›`b, a = 3 ms-2
‡kl‡eM, v = 0
_vgvi Av‡M e¯‘wU KZ…K
AwZµvšÍ `~iZ,¡ s = ?
GLv‡b,
D”PZv, h =50m
Avw`‡eM, u = 0
g = 9.8 ms-2
(K) mgq, t = KZ?
(L) ‡kl †eM, v = KZ ?
GLv‡b,
D”PZv, h =5.1m
g = 9.8 ms-2
‡kl †eM, v =0.70ms-1
Avw` `ªæwZ, u =?
GLv‡b,
Z¡iY, a = 3ms-2
miY, s = 60m
‡kl‡eM, v = ?
GLv‡b,
mgq, t1 = 3S
mgq, t2 = 17S
†kl‡eM, v1 =?
†kl‡eM, v2 =?

7| t3t
3
1
s 3
 m~Îvbymv‡i GKwU e¯‘ mij †iLvq Pj‡Q|
2 †m‡KÛ ci Gi †eM KZ n‡e?
Avgiv Rvwb,
dt
ds
v 






 t3t
3
1
dt
d
v 3
3t3
3
1
v 2

3tv 2

32v 2
 [t Gi gvb ewm‡q]
7v  GKK („Ans.)
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
s75.02150 2

m
75.02
1515
s



(Ans.)m150s 
Avevi, v = u  at
t75.0150 
s
75.0
15
t 
(Ans.)s20t 
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|
Avgiv Rvwb,
2
111 at
2
1
uts 
2
)1(a
2
1
01 
2
a
1 
2
ms2a 

GLb cÖ_g †_‡K s2 = (1m+1) =2m `~iZ¡ AwZµg Ki‡Z mgq
jv‡M = t2
2
222 at
2
1
uts 
2
2t2
2
1
02 
2t2
2 
s414.12t2 
‡k‡li 1m `~iZ¡ AwZµg Ki‡Z mgq
jv‡M, .)Ans(s414.0s)1414.1(ttt 12 
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 wcQ‡b †d‡j P‡j hv‡e,
t mgq †Ub KZ…K AwZµvšÍ `~iZ¡,
2
at
2
1
0x 
2
t10
2
1
x 
(1).........t5x 2

t mg‡q Mvwo KZ…K AwZµvšÍ `~iZ¡, Vtx 
(2).........t100x 
kZ©g‡Z †Uªb hLb MvwowU‡K AwZµg Ki‡e ZLb xx  n‡e|
t100t5 2

5
100
t  (Ans.)s20t 
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,
as2uv 22

625a201252

2
2
ms
6252
125
a 


.)Ans(ms5.12a 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,
2
gt
2
1
uth 
2
t8.9
2
1
064 
2
t9.464 
.)Ans(.s61.3t
9.4
64
t 
GLv‡b,
mgq, t = 2 Sec
‡eM, v =?
GLv‡b,
Avw`‡eM, u = 54 kmh-1
1
ms
3600
100054 

=15ms-1
g›`b, a = 0.75ms-2
†kl‡eM, v = 0
mgq, t = ?
miY, s=?
GLv‡b,
Avw`‡eM, u = 0
mgq, t1 = 1s
miY, s1 =1m
Z¡iY, a=?
GLv‡b,
Mvwoi mg‡eM, V = 100ms-1
‡Uª‡bi Z¡iY, a = 10ms-2
mgq, t = ?
GLv‡b,
Avw`‡eM, u = 0
AwZµvšÍ `~iZ¡, s = 625m
‡kl †eM, v = 125 ms-1
Z¡iY, a =?
GLv‡b,
Avw`‡eM, u = 0
AwZµvšÍ `~iZ¡, h = 64m
fi, m= 5kg
mgq, t =?

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|
Avgiv Rvwb,
2
111 at
2
1
uts 
2
)1(
2
1
02 a
2
2
a

2
ms4a 

GLb cÖ_g †_‡K s2 = (2m+2m) = 4m `~iZ¡ AwZµg Ki‡Z
mgq jv‡M = t2
2
222 at
2
1
uts 
2
24
2
1
04 t
2t2
2 
s414.12t2 
‡k‡li 2m `~iZ¡ AwZµg Ki‡Z mgq
jv‡M, .)Ans(s414.0s)1414.1(ttt 12 
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?
Avgiv Rvwb,
2
111 at
2
1
uts 
2
2
2
1
230  au
)1(..........15 au
cÖ_g †_‡K t2= (2+4)= 6 †m‡K‡Û e¯‘wU hvq s2=(30+150)m=180m
2
222
2
1
atuts 
2
6
2
1
6180  au
)2(..........303  au
)1(..........15 au
we‡qvM K‡i, 2a= 15 2
5.7 
 msa
GLb (1) bs mgxKi‡Y a Gi gvb ewm‡q,
155.7 u 1
5.7 
 msu
6 ‡m‡K‡Ûi c‡ii †m‡KÛ A_©vr 7g †m‡K‡Û AwZµvšÍ `~iZ¡,
)12(
2
 t
a
ust
)172(
2
5.7
5.77  s
13
2
5.7
5.77  s
75.485.77  s
)(25.567 Ansms 
GLv‡b,
Avw`‡eM, u = 0
mgq, t1 = 1s
miY, s1 =1m
Z¡iY, a =?
GLv‡b,
`~iZ¡, s1 = 30m
mgq, t1 = 2s
miY, s2 = (30+150)
=180m
miY, s7 =?

cÖkœt wb¤œwjwLZ mgxKiY¸wj mivmwi †f±i iƒ‡c cÖwZcv`b Ki|
)rr(a2vvta
2
1tvrr)tvv(
2
1rrtavv
0
2
000000
22 
 N)M)L)K) ((((
|cÖwZcv`bK)( tavv
0


awi, GKwU e¯‘ a mgZ¡i‡Y †Kvb Z‡j MwZkxj| GB MwZi cÖviw¤¢K kZ©vw` nj hLb oit ZLb Avw` Ae¯’vb †f±i
0rr

 Ges Avw`‡eM 0vv

 Avevi, ttf  mg‡q e¯‘wUi Ae¯’vb †f±i r

Ges †eM v

|
‡h †Kvb gyn~‡Z© mg‡qi mv‡c‡ÿ e¯‘i †eM e„w×i nvi‡K Z¡iY e‡j|
dt
dv
a


iYZ¡
dtavd

ev,
hLb t = 0 ZLb 0vv

 Ges hLb t = t ZLb vv

 GB mxgvi g‡a¨ Dc‡iv³ mgxKiY‡K mgvKjb K‡i cvB,
 
t
0
dtd
0
av
v
v



    tv
v tav o 0
 
 
)(0 otavv 

tavv

 0
tavv

 0
)(
2
1rr
00
|cÖwZcv`b(L) tvv


awi, GKwU e¯‘ a mgZ¡i‡Y †Kvb Z‡j MwZkxj| GB MwZi cÖviw¤¢K kZ©vw` nj hLb 0it ZLb Avw` Ae¯’vb †f±i
0rr



Ges †eM v

|
myZivs KYvwUi t mgq e¨eav‡b Mo†eM V

n‡j

t
dtv
t
V
0
1 
 
t
dt
rd
vdt
dt
rd
t
V
0
][
1





r
r
rd
t
V



0
1
r
rr
t
V



0
][
1

)(
1
0rr
t
V


tVrr

 0
tvvrr )( 02
1
0

  )( 02
1
vvV

 
tvvrr )( 02
1
0



3| wØgvwÎK MwZ (Motion In Two Dimention) 2
|cÖwZcv`b(M)
2
00
t
2
1rr atv


0rr



Ges †eM v

|
dt
dv
a


iYZ¡
dtavd

ev,
hLb ti = 0 ZLb 0vv

 Ges hLb tf = t ZLb vv

 
t
0
dtd
0
av
v
v



   tv
v tav o 0
 
 
)(0 otavv 

tavv

 0
tavv

 0
Avevi, †h‡nZz †Kvb gyn~‡Z© mg‡qi mv‡c‡ÿ e¯‘i Ae¯’vb †f±i e„w×i nvi‡K †eM e‡j, ZvB msÁvbymv‡i,
dt
rd
v


 ewm‡q cvB, tav
dt
rd 

 0
tdtadtvrd

 0
hLb ti = 0 ZLb 0rr

 Ges hLb tf = t ZLb rr

 
t
0
t
0
0
r
r
tdtadtvrd
0



 tttr
r atvr 0200 ][][
2
0
 
 
)0(
2
)0( 2
00  tatvrr

2
2
1
00 tatvrr


2
2
1
00 tatvrr


2
0
2
0
|cÖwZcv`b(N) )rr(avv2 

0rr



Ges †eM v

|
dt
dv
a


iYZ¡
dtavd

ev,
hLb t = 0 ZLb 0vv

 Ges hLb t = t ZLb vv

 
t
0
dtd
0
av
v
v



    tv
v tav o 0
 
 

)(0 otavv 

tavv

 0
tavv

 0
Dfq cÿ‡K GB mgxKiY w`‡q ¸Y K‡i cvB,
)).((. 00 tavtavvv


2
0000 ..... taatavtavvvvv


2
000 ..2.. taatavvvvv


)
2
1
.(2.. 2
000 tatvavvvv


  



 )(
2
1
.2.. 0
2
0000 rrtatvrravvvv



 0
2
0
2
.2 rravv


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|
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
x = voCoso t + 2
1
axt2
ev, x = voCoso t + 0 [ax= 0]
ev, x = voCoso t
)1...(....................
ooCosv
x
t


Dj¤^ w`‡K Z¡iY ay=g;
Dj¤^ w`‡K miY y; Abyiƒcfv‡e
y=voSinot 2
1
gt2

2
o 2
1









oooo
o Cosv
x
g
Cosv
x
Sinvy

ev, [t Gi gvb ewm‡q]
2
22
2
tan x
Cosv
g
xy
oo
o 







ev,






 c
oθCosov
g
bθcxbxy o 22
2
2
tan, GesæeKaªawi
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
tmg‡q AwfKl©RZ¡iYnxb Abyf~wgK miY x = vot
)(tvx ... ...... ...... ...o
1
222

tmg‡q Dj¤^ miY y = 0.t + 2
1
gt2
y = 2
1
gt2
... .... .... .... .... .... .... (2)
(1) ‡K (2) Øviv fvM K‡i cvB
2
2
1
222
gt
tv
y
x o

g
v2
y
x 2
o
2

y
g
v
x o







2
2 2






 æeKaªawi a
v
ayx 4
g
2
,4
2
o2
Dc‡iv³ mgxKiYwU GKwU Awae„‡Ëi mgxKiY| ZvB wbwÿß e¯‘i MwZc_ GKwU Awae„Ë (c¨viv‡evjv)|
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 ov †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
0 = voSino gt
)3....(..............................
g
Sinv
t oo 

(L) m‡e©v”P D”PZvt
g‡bKwi, m‡e©v”P D”PZv = H
 H = voSinot  2
1
gt2
 qewm‡gvbGiZn‡bs t(3)
2
2
1







g
Sinv
g
g
Sinv
SinvH oooo
oo

 
   
2
22
g
Sinv
g
Sinv
H oooo 
 
(4).....................
2g
Sinv
H o
22
o 
 
(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|
 t mg‡q Dj¤^ w`‡K miY y = voSinot  2
1
gt2
GB mgxKi‡Y mgq t = T Ges miY y = 0 ewm‡q cvB,
0 = voSinoT  2
1
gT2

ev, 2
1
gT2
 = voSinoT
(5)..................
2
T
g
Sinθv oo
 
(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
g
Sinv
CosvR oo
oo


2
 [(5) bs n‡Z T Gi gvb ewm‡q]
g
CosSinv
R ooo 22

.......(6)....................
22
g
Sinv
R oo 

(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

 me©vwaK cvjøv
4522
max
g
Sinv
R
o
o 

092
max
g
Sinv
R
o
o

12
max
g
v
R o 

(7)...............
2
max
g
v
R o

cªkœt îwLK †eM I †KŠwbK †e‡Mi msÁv `vI Ges G‡ì g‡a¨ m¤úK© ¯’vcb Ki| |KicÖgvbev, , rvrv

  ev
|KicÖgvYev, rv

 
‰iwLK †eM (Linear Velocity)t wbw`©ó w`‡K îwLK c‡_ †Kvb e¯‘ GKK mg‡q †h `yiZ¡ AwZµg K‡i Zv‡K H e¯‘i ‰iwLK
†eM e‡j| îwLK †eM‡K v Øviv cÖKvk Kiv nq| wbw`©ó w`‡K e¯‘ t mg‡q d `~iZ¡ AwZµg Ki‡j †eM
t
d
v  n‡e| †eM
GKwU †f±i ivwk| îwLK †e‡Mi GKK ms-1
‡KŠwYK †eM (Angular Velocity) t mgq eëavb k~‡bï 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
mg‡q  ‡KvY Drcbœ Ki‡j †KŠwbK †eM
t

  n‡e| †KŠwbK †e‡Mi GKK rad s-1
Gi gvÎv n‡”Q 1-T
TL
L





mgqe¨vmva©
Pvc
mgq
KvY†
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,
T
2π
ω 
ev, )1...(............
ω
2π
T 
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
cwiwai mgvb c_ r `~iZ¡ AwZµg KiZ|  îwLK †eM
2
T
πr
v 
)2...(............
2
v
πr
T 
(1) bs I (2) mgxKiYØq n‡Z cvB
v
r22 




v
r


1
 v = r A_©vr ‰iwLK †eM = †KŠwbK †eM × e„‡Ëi e¨vmva©|
v = r mgxKi‡Yi ‡f±i iƒc:
g‡b Kwi, (3).........ru

  
u

‡f±‡ii gvb ][r90sin rru

  
µm ¸Y‡bi wbqg Abymv‡i, u,

evr †f±‡ii AwfgyL Ges v

†f±‡ii AwfgyL Awfbœ| Avevi v = r| †`Lv hv‡”Q †h,
gvb I w`K we‡ePbvq vu

I ‡f±i Awfbœ|
(4).........vu


(3) I (4) n‡Z rv

  (cÖgvwYZ)
cÖkœ: †KŠwbK †eM I îwLK †e‡Mi g‡a¨ cv_©K¨ eY©bv Ki|
†KŠwbK †eM I îwLK †e‡Mi g‡a¨ cv_©K¨ (Distinction between Angular Velocity and Linear Velocity)
µwgK †KŠwbK †eM îwLK †eM
1|
†KŠwbK c‡_ e¯‘i †KŠwbK mi‡bi nvi‡K †KŠwbK †eM
e‡j|
wbw`©ó w`‡K ‰iwLK c‡_ †Kvb GKwU e¯‘i ¯’vb
cwieZ©‡bi nvi †K ‰iwLK †eM e‡j|
2| G‡K Øviv cÖKvk Kiv nq| G‡K Øviv v cÖKvk Kiv nq|
3| Gi mgxKiY
t

 Gi mgxKiY
t
s
v 
4| Gi gvÎv mgxKiY [ T-1
] Gi mgxKiY [ LT-1
]
5| Gi GKK †iwWqvb/ †m‡KÛ Gi GKK wgUvi/ †m‡KÛ
‡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
r
v
m
2
 |
†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
r
v
m
2
 |
m fi wewkó GKwU e¯‘ r e¨mv‡a©i e„ËvKvi c‡_ v mg`ªæwZ‡Z Nyi‡Q| (1) ‡`LvI †h, j¤^ Z¡iY r
r
v
a 2
2
 ev (2)
j¤^ Z¡i‡Yi ivwkgvjv wbb©q Ki| (3) cÖgvY Ki †h, †K›`ªgyLx ej rm
r
v
mF 2
2
 ev, (4) e„ËvKvi c‡_ mg`ªæwZ‡Z
N~b©vqgvb †Kvb e¯‘i Dci wµqvkxj †K›`ªgyLx e‡ji gvb I w`K wbY©©q |
aiv hvK, m f‡ii †Kvb e¯‘ r e¨vmv‡a©i e„ËvKvi c‡_ v mg`ªæwZ‡Z Ges  †KŠwbK †e‡M AveZ©biZ Av‡Q| awi AwZ ÿz`ª
mgq t eëav‡b e¯‘wU A n‡Z B we›`y‡Z A‡m| A we›`y‡Z e¯‘wUi †eM H we›`y‡Z ¯úk©K AC eivei| B we›`y‡Z e¯‘wUi †eM
H we›`y‡Z ¯úk©K BD eivei| BD †K †cQ‡b ewa©Z Ki‡j AC I BD Gi wgjb we›`y nq E|

GLb, OAEB PZzf©~‡R,
 AEB+  AOB = `yB mg‡KvY|
Avevi,  AEB+  BEC = `yB mg‡KvY|
  AOB = BEC =  awi,
A we›`y‡Z e¯‘i †e‡Mi Dj¤^ Dcvsk, vy = 0
Ges AbyfywgK Dcvsk, vx = v
B we›`y‡Z e¯‘i †e‡Mi AC eivei †e‡Mi Dj¤^ Dcvsk, vsinvy 
Ges AbyfywgK Dcvsk, vcosvx 
 t AwZ ÿz`ª mgq myZivs AwZ ÿz`ª|
 sin Ges 1c os
B we›`y‡Z e¯‘i †e‡Mi †e‡Mi Dj¤^ Dcvsk, vvy 
Ges AbyfywgK Dcvsk, vvx  G‡Z †`Lv hv‡”Q, AbyfywgK eivei †e‡Mi Dcvs‡ki †Kvb cwieZ©b nq bv|
‡e‡Mi Dj¤^ Dcvs‡ki cwieZ©‡bi Kvi‡Y Z¡iY, a n‡j,
t
v
a
0


t
v

v 



 

t

r
v
v 




r
v

r
r
r
r
v
a 2
222



†K›`ªgyLx ej, rm
r
v
mmaF 2
2
 (cÖgvwYZ)
cÖkœ: Awfj¤^ Z¡iY ev e¨vmva©gyLx Z¡iY ev †K›`ªgyLx Z¡iY:
Awfj¤^ Z¡iY ev e¨vmva©gyLx Z¡iY ev †K›`ªgyLx Z¡iY t
‡Kvb e¯‘ hLb e„ËvKvic‡_ Nyi‡Z _v‡K ZLb e„‡Ëi e¨vmva© eivei e„‡Ëi †K‡›`ªi w`‡K wµqvkxj Awf‡K›`ª e‡ji Rb¨ †h
Z¡i‡Yi m„wó nq Zv‡K e¨vmva©gyLx Z¡iY ev Awfj¤^ Z¡iY ev †K›`ªgyLx Z¡iY e‡j| Gi GKK wgUvi/†m‡KÛ2
|
cÖkœ: ‡KŠwbK Z¡iY Kv‡K e‡j?
‡KŠwbK Z¡iYt hLb †Kvb e¯‘KYv Amg †KŠwbK †e‡M Ny‡i, ZLb e¯‘wUi †KŠwbK †eM cwieZ©‡bi nvi‡K †KŠwbK Z¡iY
e‡j A_ev, mg‡qi mv‡_ Amg †KŠwbK †eM cwieZ©‡bi nvi‡K †KŠwbK Z¡iY e‡j| G‡K  Øviv cÖKvk Kiv nq| Gi GKK
†iwWqvb/†m‡KÛ2
|
g‡bKwi, eËvKvi c‡_ Nyb©vqgvb e¯‘KYvi Avw`‡KŠwbK †eM i Ges t mgq ci Gi †kl †KŠwbK †eM f Kv‡RB
†KŠwbK Z¡iY
t
if 




cÖkœ: mg`ªæwZ‡Z Pjgvb e¯‘i Z¡ib _v‡K bv, wKš‘ e„ËvKvi c‡_ mg`ªwZ‡Z Pjgvb e¯‘i Z¡iY _v‡K †Kb? e¨vL¨v Ki|
mg`ªæwZ‡Z Pjgvb e¯‘i Z¡ib _v‡K bv, wKš‘ e„ËvKvi c‡_ mg`ªwZ‡Z Pjgvb e¯‘i Z¡iY _v‡K t
‡e‡Mi gvb n‡”Q `ªæwZ Ges †e‡Mi cwieZ©‡bi nvi n‡”Q Z¡iY| †Kvb e¯‘ hLb mij c‡_ mg `ªæwZ‡Z P‡j ZLb †e‡Mi
gv‡bi †Kvb cwieZ©b nq bv Avi mij c‡_ Pjvi Rb¨ w`‡Ki I †Kvb cwieZ©b
nq bv| d‡j e¯‘i †Kvb Z¡iY _v‡K bv|
wKš‘ e„ËvKvi c‡_ Nyievi mgq e¯‘i wbqZ w`‡Ki cwieZ©b nq, KviY
†e‡Mi AwfgyL me©`vB e„‡Ëi ¯úk©K eivii nq| Gfv‡e AbeiZ w`K cwiewZ©Z
n‡Z _v‡K e‡j e¯‘ mg`ªæwZ‡Z Pj‡jI †eM mgvb _v‡Kbv| †e‡Mi GB cwieZ©‡bi
d‡j Z¡i‡Yi m„wó nq| GB Z¡i‡Yi AwfgyL e„ËvKvi c‡_i †K›`ª eivei n‡q _v‡K|
G Rb¨ e„ËvKvi c‡_ mg`ªæwZ‡Z Pjgvb e¯‘i Z¡iY _v‡K|

3| wØgvwÎK MwZ (Motion In Two Dimention)
1| GKwU cÖv‡mi AbyfywgK cvjøv 96m Ges Avw`‡eM 66 ms-1
|
wb‡ÿc †KvY KZ?
Avgiv Rvwb,
g
2Sinv
R o
2
o 

2
0
o
v
gR
Sin2θ,

ev
2o
66
9.896
Sin2

ev,
(0.2159)Sin2θ, 1
o

ev
)Ans.(24.6θ
47.12θ2
o
o

ev,
2| 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,
g2
)Sin θ(v
H
2
oo

 
8.92
2
60Sin40
H



 
8.92
2
60Sin40
H



 
8.92
2
86602.040
H



 
8.92
2
6408.34
H


8.92
9850.1199
H


.)m (Ans22.61H 
g
oθ2Sinv
R
2
0

Avevi,
8.9
)602Sin (40
R
2


8.9
)602Sin (40
R
2


8.9
)120Sin1600
R


8.9
86602.01600
R


8.9
632.1385
R 
m (Ans.)39.141R 
3| nvB‡Wªv‡Rb cigvbyi g‡W‡ji GKwU B‡jKUªb GKwU †cÖvU‡bi Pviw`‡K
5.2 ×10 -11
m e¨vmv‡a©i GKwU e„ËvKvi c‡_ 2.18 ×106
ms-1
†e‡M
cÖ`wÿb K‡i| B‡jKUª‡bi fi 9.1 ×10-31
kg n‡j †K›`ªgyLx ej KZ?
Avgiv Rvwb,
r
mv
F
2

11-
2631-
102.5
)1018.2(101.9
F



N (Ans.)10316.8F 8

4| 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|
Avgiv Rvwb,
rmF 2

r
t
n2π
mF
2







75.0
60
901416.32
25.0F
2





 

(Ans.)N65.16F 
5| 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,
g
Sinv2
T oo 

8.9
90Sin2.92
T


8.9
12.92
T


8.9
4.18
T 
.)Ans(s877.1T 
6| 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|
Avgiv Rvwb,
2
2
00
0 x
)cosv(2
g
x)(tany


2
2
0
)50(
)30cosv(2
g
x)30(tany


GLv‡b,
AbyfywgK cvjøv, R = 96 m
Avw`‡eM, vo = 66ms-1
AwfKl©R Z¡iY, g = 9.8ms-2
wb‡ÿc †KvY,  = ?
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 = ?
GLv‡b,
fi, m = 0.250 kg
e¨vmva©, r = 0.75 m
mgq t = 1 min.
= 60s.
cvKmsL¨v, n = 90 cvK|
Uvb, F = ?
GLv‡b,
Avw`‡eM, vo = 9.2 ms-1
wb‡¶c †KvY, 0º
AwfKl©R Z¡iY,
g = 9.8ms-2
DÌvb cZ‡bi †gvU
mgq T =?
GLv‡b,
Avw`‡eM, vo = 50 ms-1
wb‡¶c †KvY, º
AwfKl©R Z¡iY,
g = 9.8ms-2
AbyfywgK `~iZ¡, x=50m
Dj¤^ `~iZ¡, y=?

2
2
0
)50(
)30cosv(2
g
50)30(tany 


2
2
)50(
)866025403.050(2
8.9
50577350269.0y 


533333345.686751346.28y 
(Ans.)m33.22y 
7| GKwU cÖv‡mi AbyfywgK cvjøv 79.53 m Ges wePiYKvj 5.3 s
n‡j wb‡ÿc †KvY I wb‡ÿc †eM KZ?
Avgiv Rvwb,
g
2Sinv
R o
2
o 

8.9
2Sinv
53.79 o
2
o 

(1).....394.7792Sinv o
2
o 
Aevi,
g
Sinv2
T oo 

8.9
Sinv2
3.5 oo 

(2).........94.51Sinv2 oo 
(1) bs mgxKiY‡K (2) bs mgxKiY Øviv fvM K‡i cvB,
94.51
394.779
Sinv2
2Sinv
oo
o
2
o



94.51
394.779
Sinv2
CosSin2v
oo
oo
2
o




(3)........15Cosv o0 
(2) bs mgxKiY‡K (3) bs mgxKiY Øviv fvM K‡i cvB,
15
94.51
Cosv
Sinv2
o0
oo



2
463.3
tan o 
732.1tan o 
732.1tan 1
o


.)Ans(60o 
(3) bs mgxKi‡Y 0 Gi gvb ewm‡q cvB,
1560Cosv0 
15
2
1
v0 
(Ans.)ms30v 1
0


8| 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,
2
2
00
0 x
)cosv(2
g
x)(tany


2
22
0
)20(
30cos2
8.9
2030tan5


v
75.02
4008.9
02577350269.05 2
0 


v
5.1
3920
54.115 2
0 

v
54.6
5.1v
3920
2
0



5.154.6
3920
v2
0

 .)Ans(ms20v 1
0


9| 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,
g
Sinθv2
T oo
 
8.9
90Sin482
T


8.9
1482
T


.)(.795.9T Anss 
Avevi,
2g
Sinv
H o
22
o 

9.82
)90Sin(48
H
22

 
(Ans.)117.5mH  
10| GKwU MÖv‡gv‡dvb †iKW© cÖwZ wgwb‡U 45 evi Ny‡i| Gi †K›`ª †_‡K
9cm `~‡i †Kvb we›`yi `ªæwZ KZ?
Avgiv Rvwb,
rv 
r
t
n2
v


60
09.04514.32
v


.)Ans(ms42.0v 1

GLv‡b,
AbyfywgK cvjøv, R = 79.53 m
wePiYKvj, T=5.3s
wb‡¶c ‡eM, vo = ?
wb‡¶c †KvY,  = ?
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 = ?
GLv‡b,
†eM, v0 = 48 ms-1
wb‡¶c †KvY, =º
DÌvb cZ‡bi †gvU mgq,T =?
D”PZv, H =?
GLv‡b,
mgq, t = 1m =60s
cvKmsL¨v, n=45
e¨vmva©,r =9cm=0.09m
`ªæwZ, v =?

 এযবযস্টটর ভনন কযনতন ফনরয ঳ম্পক঱ গবতয ঳ানথ বকন্তু বনউটননয ঳ূত্র অনুমায়ী ফনরয ঳ম্পক঱ গবতয ঩বযফত঱ননয ঳ানথ।
1 বনউটন = 7.2324 ঩াউন্ডার ঑ 1 ঩াউন্ডার= 13825.728 ডাইন।
 উৎ঩বত্ত অনু঳ানয ফর দুই প্রকায। মথা-১। যভৌবরক ফর ২) মাবিক ফর।
 বফনে যভৌবরক ফর ৪বট। মথাঃ (১) ভ঴াকল঱ ফর (২) দুফ঱র বনউবিয় ফর (৩) তাবড়ৎ যিৌম্বক ফর (৪) ঳ফর বনউবিয় ফর।
 যভৌবরক ফরগুনরায আন঩বক্ষক তীব্রতায অনু঩াত।মথাক্রনভ 1:10
30
:10
40
:10
42
[নমখানন ভ঴াকল঱ ফনরয তীব্রতা 1]
 ঳ারাভ ঑য়ানয়ন ফাগ঱ প্রভান কনযনেন দুফ঱র ঑ তাবড়ৎ যিৌম্বক ফর একই ফনরয দুবট ববনড়ফরূ঩।
ভ঴াকল঱ ফরঃ ঳ফনিনয় দুফ঱র। Gravitation নাভক এক প্রকায কণায ঩াযষ্পবযক বফবনভনয়য ভাধযনভ ভ঴াকল঱ ফর কাম঱কয।
 দুফ঱র বনউবিয়ায ফরঃ Intermediate vector bosons নাভক কণায ঩াযস্পবযক বফবনভনয়য ভাধযনভ এ ফনরয ঳ৃবি।
 Photon নাভক এক প্রকায কণায ঩াযস্পবযক বফবনভনয়য ভাধযনভ তাবড়ৎ যিৌম্বক ফর কাম঱কয ঴য়।
 ঳ফর বনউবিয়ায ফরঃ বনউবিয়ায ফর আকল঱ণ ধভ঱ী। স্বল্প ঩াল্লা বফব঱ি এফিং আধান বনযন঩ক্ষ । যভ঳ন কণায ঩াযস্পবযক বফবনভনয়য
ভাধযনভ ঳ফর বনউবিয়ায ফর ঳ৃবি ঴য়।
 ২য় ঳ূত্র যথনক ১ভ ঳ূত্র ঩া঑য়া মায় অথ঱াৎ ১ভ ঳ূত্র ২য় ঳ূনত্রয একবট রূ঩।
 ১ভ ঳ূত্র যথনক ঩া঑য়া মায়ঃ ফর ঑ জড়তা
 ২য় ঳ূত্র যথনক ঩া঑য়া মায়ঃ ফনরয অববভুখ, ফনরয ঩বযভা঩, ফনরয গুণগত রফব঱িয, ত্বযনণয ঳ানথ ফনর ঳ম্পক঱, ফনরয নীযন঩ক্ষ নীবত।
 বিবত জড়তায দৃিান্তঃ
১। কান঩য উ঩য য঩াি কাড঱ ঑ তায উ঩য ভুদ্রা যযনখ যটাকা বদনর ভুদ্রা কান঩ ঩নড় মানফ।
২। ঴ঠ্াৎ গাবড় িরনত শুরু কযনর আনযা঴ী ব঩েনন য঴নর ঩নড়।
৩। কযাযানভয যফানড঱ একবট গুবটয উ঩য আয একবট গুবট যযনখ স্ট্রাইক িাযা আঘাত কযনর নীনিয গুবট িনর মায়।
৪। ধুবরমুি য঩ালাক দবড় বদনয় আঘাত কযনর ভয়রা দূয ঴নয় মায়।
৫। কানিয জানারায় ফুনরট যোড়নর একবট বেদ্র বকন্তু বির যোড়নর একাবধক বেদ্র ঴য়।
৬। যঘাড়া ঴ঠ্াৎ যদৌড়ানত শুরু কযনর আনযা঴ী ব঩েনন য঴নর ঩নড়।
 গবত জড়তায দৃিান্তঃ
১। িরন্ত ফা঳ গঠ্াৎ যথনভ যগনর মাত্রী ঳াভনন য঴নর ঩নড়।
২। যদৌড় বদনয় রাপ বদনর।
৩। ধাফভান যঘাড়া যথনক উ঩নযয বদনক রাপ যদয়া।
৪। িরন্ত গাবড়য কাভযায় যকান আনযা঴ী য঳াজা উ঩নযয বদনক বকেু যোড়নর।
৫। ঳যর যদারনক একফায দুবরনয় বদনর অননক্ষণ দুরনত থানক।
 ঘাত ফনরয দৃিান্তঃ
১। ফযাট িাযা ফর আঘাত কযা।
২। ঴াতুবড় বদনয় ব঩ন য঩াতা।
৩। স্ট্রাইকায িাযা গুবটনক আঘাত কযা।
৪। যেনন যেনন ঳িংঘল঱।
৫। ঩া বদনয় পু টফর বকক কযা।
৬। বফনফাযণ।
৭। কাভান ঴নত যগারা যোড়া।

RoZv (Inertia):
c`v_© †h Ae¯’vq Av‡Q †mB Ae¯’vq _vK‡Z PvIqvi †h cÖeYZv ev †mB Ae¯’v ivL‡Z PvIqvi †h ag© Zv‡K RoZv e‡j|
RoZv `yB cÖKvi: h_v (1) w¯’wZ RoZv (2) MwZ RoZv
(1) w¯’wZ RoZv (Inertia of rest) t w¯’i e¯‘ w¯’i _vK‡Z Pvq e¯‘i GB ¸b‡K e¯‘i w¯’wZ RoZv e‡j|
D`vniYt w¯’i Mvwo nVvr †Q‡o w`‡j hvwÎiv wcQb w`‡K †n‡j c‡o| KviY Mvwo Pvjy nIqvi ms‡M ms‡M hvwÎ‡ì
kix‡ii wb‡Pi Ask Mvwoi mv‡_ mshy³ _vKvq mvg‡bi w`‡K GwM‡q hvq| wKš‘ kix‡ii Dc‡ii Ask w¯’wZ RoZvi Rb¨
wcwQ‡q c‡o|
(2) MwZ RoZv (Inertia of motion) t MwZkxj e¯‘ MwZkxj _vK‡Z Pvq e¯‘i GB ¸b‡K e¯‘i MwZ RoZv e‡j|
D`vniYt PjšÍ Mvwo nVvr †eªK Ki‡j hvwÎiv mvg‡bi w`‡K Sz‡K c‡o| KviY Mvwo ‡eªK Kivi Kvi‡Y Mvwo †_‡g
hvIqvi mv‡_ mv‡_ hvwÎ‡ì kix‡ii wb‡Pi Ask Mvwoi mv‡_ mshy³ _vKvq ‡_‡g hvq wKš‘ kix‡ii Dc‡ii Ask MwZ
RoZvi Rb¨ mvg‡bi w`‡K GwM‡q hvq|
ej (Force):
hv w¯’i e¯‘i Dci wµqv K‡i e¯‘‡K MwZkxj K‡i ev Ki‡Z Pvq A_ev MwZkxj e¯‘i Dci wµqv K‡i MwZi cwieZ©b K‡i ev
Ki‡Z Pvq Zv‡K ej e‡j| ej‡K F Øviv cÖKvk Kiv nq| m f‡ii e¯‘i Z¡iY a n‡j ej F=ma n‡e| ej GKwU †f±i
ivwk|
‡gŠwjK ej (Fundamental Forces) t
‡h mKj ej ¯^vaxb A_©vr ‡h mKj ej Ab¨ †Kvb ej †_‡K Drcbœ nq bv Zv‡K eis Ab¨vb¨ ej GB mKj e‡ji †Kvb bv
†Kvb iƒ‡ci cÖKvk Zv‡ì‡K †gŠwjK ej e‡j| †gŠwjK ej Pvi cÖKvi, h_v: (1) gnvKl© ej (2) ZvwoZ †PŠ¤^K ej (3)
mej wbDwK¬q ej I (4) `~e©j wbDwK¬q ej
(1) gnvKl© ej t gnvwe‡k¦i †h †Kvb `ywU e¯‘i ga¨Kvi cvi¯úwiK AvKl©b ej‡K gnvKl© ej e‡j|
(2) ZvwoZ †PŠ¤^K ej t `ywU PvwR©Z KYv Zv‡ì Pv‡R©i Kvi‡Y G‡K Ac‡ii Dci †h AvKl©Y ev weKl©b ej cÖ‡qvM K‡i
Zv‡K ZvwoZ †PŠ¤^K ej e‡j|
(3) mej wbDwK¬q ej t cigvbyi wbDwK¬qv‡m wbDwK¬q Dcv`vb Z_v wbDwK¬qb¸‡jv‡K GK‡Î Ave× iv‡L †h kw³kvjx ej
Zv‡K mej wbDwK¬q ej e‡j|
(4) `~e©j wbDwK¬q ej t †h ¯^í cvjøvi I ¯^í gv‡bi ej wbDwK¬qv‡mi g‡a¨ †gŠwjK KYv¸wji g‡a¨ wµqv K‡i A‡bK
wbDwK¬qv‡m Aw¯’wZkxjZvi D™¢e NUvq Zv‡K `~e©j wbDwK¬q ej e‡j|

04| MwZm~Î (Laws Of Motion) 2
‡gŠwjK ej mg~n Av‡cwÿK wZeªZv cvjøv
1) gnvKl© ej 1 Amxg
2) ZvwoZ †PŠ¤^K ej 1039 Amxg
3) mej wbDwK¬q ej 1041
10-15
m
4) `~e©j wbDwK¬q ej 1030
10-16
m
fi‡eM (Momentum) t fi I †e‡Mi ¸bdj‡K fi‡eM e‡j| fi‡eM‡K P Øviv cÖKvk Kiv nq| m f‡ii †Kvb e¯‘i †eM
v n‡j fi‡eM P = mv n‡e| fi‡e‡Mi GKK kg-ms-1
| Gi gvÎv [MLT-1
]
wbDU‡bi MwZi 1g m~Î (Newton's 1st
Law of Motion) t
eY©bvt evwn¨K ej cÖ‡qvM bv Ki‡j w¯’i e¯‘ wPiKvj w¯’i Ges MwZkxj e¯‘ mg‡e‡M mij c‡_ Pj‡Z _vK‡e|
e¨L¨vt evB‡i †_‡K ej cÖhy³ bv n‡j (1) w¯’i e¯‘ wPiKvj w¯’i _vK‡e Ges
(2) MwZkxj e¯‘ mg‡e‡M mij c‡_ Pj‡Z _vK‡e|
myZivs †`Lv hvq †h, MwZi cÖ_g m~‡Îi `ywU Ask| cÖ_g Ask †_‡K w¯’i e¯‘i w¯’wZkxj _vKvi cÖeYZv m¤ú‡K© ¯úó
aviYv cvIqv hvq| GB cÖeYZv‡K w¯’wZ RoZv e‡j| Avevi wØZxq As‡k MwZkxj e¯‘i MwZkxj _vKvi cÖeYZv jÿbxq |
GB cÖeYZv‡K MwZ RoZv e‡j| G Kvi‡Y wbDU‡bi MwZi 1g m~Î‡K RoZvi m~Î I ejv nq| G m~Î †_‡K e‡ji msÁv I
cvIqv hvq|
wbDU‡bi MwZi 2q m~Î (NewtonÕs 2nd
Law of Motion)t
eY©bvt e¯‘i fi †eM cwieZ©‡bi nvi e¯‘i Dci cÖhy³ e‡ji mgvbycvwZK Ges ej †h w`‡K wµqv K‡i e¯‘i fi †e‡Mi
cwieZ©b I †mB w`‡K N‡U|
e¨L¨vt G m~‡Îi e¨vL¨v nj amF

 cÖwZcv`b|
amF

 cÖwZcv`b t
g‡b Kwi, †Kvb MwZkxj e¯‘KYvi fi‡eM P

| F

e‡ji wµqvq ÿz`ªvwZÿz`ª mgq e¨eavb dt AeKv‡k KYvi fi‡e‡Mi
cwieZ©b Pd

n‡j fi‡eM cwieZ©‡bi nvi
dt
Pd

n‡e| MwZi 2q m~Îvbymv‡i, fi‡eM cwieZ©‡bi nvi cÖhy³ e‡ji
mgvbycvwZK| A_©vr, F
dt
Pd 


e¯‘KYvi fi m Ges †eM v n‡j fi‡eM vmP

 ewm‡q cvB,
F
dt
)vm(d 

F
dt
vd
m







dt
vd
aFma




iYZ¡
(1)................Fkma

 GLv‡b k GKwU mgvbycvwZK aªæeK; hvi gvb GKK e‡ji
msÁv Øviv wba©viY Kiv hvq| ‡h ej GKK f‡ii Dci wµqv K‡i GKK Z¡iY m„wó K‡i Zv‡K GKK ej e‡j|
A_©vr 1F 

, m=1 Ges 1a 

nq Z‡e mgxKiY (1) n‡Z cvB|
1×1=k×1  k=1 mgxKi‡Y ewm‡q cvB, Fam



Intermediate physics 1st paper

Intermediate physics 1st paper

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