Conservation of linear momentum — derivation from Newton's third law: Question bank

1.4.3 · D5 · Physics › Momentum & Collisions › Conservation of linear momentum — derivation from Newton's t

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Ek particle ki momentum $p = m v$ ek vector hai — iska ek direction hota hai, isliye kisi chosen axis ke saath yeh positive ya negative ho sakta hai. (Yahan $m$ = mass, $v$ = velocity.)
Pure system ki total momentum sab particles ka vector sum hai, $P = \sum_{i} m_{i} v_{i}$ . Dhyan do — capital $P$ (poora system) versus small $p$ (ek particle).
Total mass $M = \sum_{i} m_{i}$ sabhi individual masses ka sum hai. Iske saath, total momentum ko $P = M v_{c m}$ likh sakte hain, jahan $v_{c m}$ system ke centre of mass ki velocity hai.
Frame velocity $u$ ek naye observer ki constant velocity hai purane observer ke relative. Ek aisi frame se dekhne par jo $u$ par move kar rahi hai, har velocity $u$ se choti dikhti hai, isliye observed total momentum $P - M u$ ban jaati hai.
Total momentum tab hi constant rehti hai jab system par net external force zero ho. Zero net external force wale system ko isolated system kehte hain.
Subscripts batate hain kaun sa object hai: $m_{1}, v_{1}$ matlab "body 1 ki mass aur velocity"; $p_{2}$ matlab "body 2 ki momentum". Gun-and-bullet problem mein hum $m_{b}, v_{b}$ likhte hain bullet ke liye aur $m_{g}, v_{g}$ gun ke liye. $F$ ke baad jo letters aate hain woh from→on padhte hain: $F_{12}$ = force on body 1 from body 2. Jab hum $F_{ij}$ likhte hain, index $i$ woh body hai jise push kiya ja raha hai aur $j$ woh body hai jo push kar raha hai.

Figure 1 dikhata hai ki "momenta ko signs ke saath add karo" actually kaisa lagta hai — ise poore page ke liye dhyan mein rakho.

Figure — Conservation of linear momentum — derivation from Newton's third law

Figure 1 — Gun-and-bullet rest se: signed momenta $+ 4$ aur $- 4$ add hokar initial zero dete hain.

True or false — justify

True or false: Momentum har collision mein conserved hoti hai.

False — yeh tabhi conserved hoti hai jab chosen system par net external force zero ho; interval ke dauran wall, friction, ya gravity ise change kar sakti hai.

True or false: Perfectly inelastic collision mein, momentum isliye lost hoti hai kyunki objects deform hona band kar dete hain.

False — momentum poori tarah conserved rehti hai; sirf kinetic energy lost hoti hai (heat, sound, deformation mein). Saath move karne wale objects phir bhi saari original momentum carry karte hain.

True or false: Agar total kinetic energy unchanged hai, toh collision elastic hai.

True — elastic collision define hi aise hoti hai jahan kinetic energy conserved ho; momentum dono elastic aur inelastic cases mein conserved hoti hai, isliye KE distinguishing test hai.

True or false: Ek akela isolated particle apni momentum spontaneously change kar sakta hai.

False — koi external force nahi, isliye

d p / d t = F = 0

, matlab uski momentum frozen hai. Internal forces ko push karne ke liye ek doosri body chahiye.

True or false: Agar kisi event se pehle total momentum zero hai, toh baad mein bhi zero honi chahiye.

True — zero ek constant hai, aur ek conserved quantity jo zero se start kare woh zero hi rehti hai (jaise rest se explosion: fragments ki momenta vector-sum karke zero aani chahiye).

True or false: Newton's third law akele do interacting bodies ke liye momentum conservation prove karta hai.

True internal pair ke liye —

F_{12} = - F_{21}

unke momentum changes ko cancel karta hai; lekin total constant rehne ke liye "no external force" bhi chahiye.

True or false: Ek system ki momentum

x

ke saath conserved ho sakti hai lekin

y

ke saath nahi.

True — momentum ek vector hai, isliye har component independently conserve hota hai. Agar external force sirf

y

direction mein hai (jaise gravity), toh

x

-momentum constant rehti hai jabki

y

-momentum change hoti hai.

True or false: Bhaari objects hamesha halke objects se zyada momentum carry karte hain.

False — momentum

m v

hai, ek product. Ek fast light bullet ek slow heavy truck se zyada momentum rakh sakti hai; aur ek stationary heavy object ki momentum zero hoti hai.

True or false: Momentum har reference frame mein conserved hoti hai, ya kisi mein bhi nahi.

True — agar

P

ek inertial frame mein constant hai, toh dusre frame mein jo constant frame-velocity

u

par move kar raha hai, observed total

P - M u

hai (jahan

M

total mass hai), jo bhi constant hai. Conservation frame-independent hai, chahe

P

ki value frame se frame mein alag ho.

Spot the error

Error hunt: "Gun aur bullet rest se start karte hain, isliye firing ke baad unki speeds equal honi chahiye."

Galat — unki momenta equal aur opposite hain, speeds nahi. Kyunki

m_{b} v_{b} = m_{g} v_{g}

(rest se momentum sum zero hoga), halki bullet ko badi speed milti hai aur bhaari gun ko choti recoil speed.

Error hunt: "Do carts saath chipak jaate hain, isliye maine dono final velocities alag-alag

v_{1}^{'}

aur

v_{2}^{'}

rakhi hain."

Galat — chipakna matlab ek common final velocity

v^{'}

. Do unknowns likhna woh freedom invent karta hai jo physics ne hata di hai.

Error hunt: "KE before = KE after, isliye momentum before = momentum after — same statement hai."

Galat — yeh alag-alag conservation laws hain. KE (

\frac{1}{2} m v^{2}

, ek scalar) destroy ho sakti hai jabki momentum (

m v

, ek vector) bachti hai; inhe equal karna ek scalar ko vector se confuse karna hai.

Error hunt: "Mass

m_{b} = 0.01

kg ki bullet

v_{b} = 400

m/s par jaati hai aur mass

m_{g} = 2

kg ki gun

v_{g} = 2

m/s par recoil karti hai, toh baad mein total speed

402

m/s hai — lekin pehle

0

thi. Momentum toot gayi!"

Galat — tum momenta ko axis ke saath signs ke saath add karte ho, speeds nahi. Rightward ko

+

maano, bullet ki momentum

+ m_{b} v_{b} = + 4

kg·m/s hai aur gun ki

- m_{g} v_{g} = - 4

kg·m/s; unka vector sum

4 - 4 = 0

hai, exactly starting value. Speeds unsigned magnitudes hain aur aise sum nahi ho sakti — momentum ka direction precisely woh information hai jo total ko cancel karta hai (dekho Figure 1).

Error hunt: "Ball wall se bounce karke wapas aati hai, isliye ball ke liye momentum conserved hai."

Galat — wall ball par external force lagaati hai, uski momentum reverse karke. Momentum sirf ball + wall + Earth system ke liye conserved hai, akeli ball ke liye nahi.

Error hunt: "Collision ke dauran contact force enormous hai, isliye yeh total momentum bahut change karti hogi."

Galat — contact force do-body system ke liye internal hai; uska action–reaction partner ise exactly cancel karta hai. Badi ho ya choti, internal forces momentum ko idhar-udhar move karti hain, total kabhi change nahi karti (dekho Figure 2).

Error hunt: "System inelastic hit mein KE lose karta hai, isliye momentum bhi lose karni chahiye."

Galat — energy aur momentum independent hain. Lost KE heat/sound banti hai; momentum ka aisa koi "leak" nahi hota kyunki internal forces phir bhi pairs mein cancel hoti hain.

Figure 2 — Internal action–reaction force pair har instant par equal-and-opposite hai, isliye unke impulses (areas) cancel ho jaate hain aur total momentum steady rehti hai.

Why questions

Internal forces total momentum kyun kabhi change nahi karti?

Kyunki woh action–reaction pairs mein hoti hain

F_{ij} = - F_{j i}

(force on

i

from

j

, minus the force on

j

from

i

); poore system par sum karne par yeh opposite forces zero ho jaati hain, sirf external forces

P

badal sakti hain.

Internal force pair ka impulse bhi kyun cancel hota hai, sirf instantaneous force nahi?

Impulse–momentum theorem kehta hai

\int F d t = Δ p

. Pair ke liye,

\int F_{ij} d t + \int F_{j i} d t = \int (F_{ij} + F_{j i}) d t = \int 0 d t = 0

, kyunki woh har instant par equal-and-opposite hain. Isliye

Δ p_{i} + Δ p_{j} = 0

—

i

jo momentum gain karta hai,

j

utni hi lose karta hai (dekho Impulse–Momentum Theorem aur Figure 2).

2D aur 3D mein har coordinate axis alag kyun conserve karna padta hai?

Kyunki

p = m v

ek vector hai, aur

d P / d t = F^{ext}

component-by-component hold karta hai;

x

direction ki force

y

-momentum change nahi kar sakti.

Worked example mein gun ki recoil velocity negative kyun hai?

Minus sign ka matlab hai "bullet ke opposite direction." Yeh recoil ki physics hai, sirf bookkeeping accident nahi — ise drop karna zero starting momentum violate kar dega.

Collision ke after-state ko predict karne ke liye har instant par force jaane ki zaroorat kyun nahi?

Kyunki conservation sirf before aur after ke totals compare karta hai; internal force ka messy time-history equal-and-opposite impulses mein integrate hota hai jo cancel ho jaate hain (dekho Impulse–Momentum Theorem).

Two-body proof bahut saari bodies ke system ke liye bhi kyun kaam karta hai?

N

bodies ke liye,

d P / d t = \sum_{i} F_{i}^{ext} + \sum_{i \neq = j} F_{ij}

. Har internal term

F_{ij}

ka ek partner

F_{j i} = - F_{ij}

hota hai, isliye internal forces ka poora double sum pairs mein cancel hota hai — sirf external forces bachti hain, exactly two-body case ki tarah.

Isolated system ka centre of mass constant velocity par kyun move karta hai?

Kyunki

P = M v_{c m}

(jahan

M

total mass hai), aur agar

P

constant hai aur

M

fixed hai, toh

v_{c m}

bhi constant hona chahiye (elaborated in Centre of Mass Motion).

Collision analyse karte waqt centre-of-mass frame special kyun hai?

Yeh woh frame hai jismein total momentum exactly zero hoti hai, isliye dono bodies hamesha equal-and-opposite momenta ke saath approach aur (elastic hits ke liye) leave karti hain — algebra symmetric aur simple ho jaata hai. Conservation phir bhi hold karti hai; yeh frame sirf bookkeeping sabse clean karta hai.

Rocket empty space mein kuch push kiye bina accelerate kyun karta hai?

Yeh apne khud ke ejected exhaust ko push karta hai: rocket + fuel system isolated hai, isliye forward gain ki momentum backward throw ki momentum ke barabar hoti hai (dekho Rocket Equation).

Is derivation mein Newton's second law

F = d p / d t

kyun likha jaata hai na ki

F = m a

Momentum form zyada fundamental hai aur tab bhi valid rehti hai jab mass change ho; yeh hume do particle equations add karne aur sum ko

d (p_{1} + p_{2}) / d t

ki tarah pehchanne deta hai, jahan

p_{1}, p_{2}

bodies 1 aur 2 ki momenta hain.

Edge cases

Edge case: Rest par rakhe system ki total momentum kya hai?

Exactly zero, aur tab tak zero rehti hai jab tak koi external force na lage. Yahi woh key hai jo explosion aur recoil problems unlock karta hai.

Edge case: Ek particle ki velocity zero hai — kya yeh total momentum mein contribute karta hai?

Nahi, uski momentum

m v = 0

hai; lekin yeh mass contribute karta hai, jo sticky collision ke baad common velocity ke liye matter karta hai.

Edge case: Gravity sabhi cheezoon par lagti hai — kya Earth par koi bhi collision truly isolated hai?

Bahut short collision time mein gravitational impulse (

\int m g d t

) negligible hota hai, isliye momentum approximately conserved hoti hai. Lambe time mein gravity external hai aur

P

change karti hai.

Edge case: Do identical objects equal speeds par ek doosre ki taraf move karte hain, collide karte hain aur chipak jaate hain — kya hota hai?

Unki momenta equal aur opposite hain, sum hokar zero ho jaati hain, isliye chipka hua pair baad mein rest par hota hai. Saari kinetic energy lost ho jaati hai, lekin momentum (

= 0

) perfectly conserved hai.

Edge case: Kya collision total kinetic energy increase kar sakti hai?

Haan — "super-elastic" event (jaise contact ke dauran explosion, ya compressed spring release) stored internal energy se KE add karta hai, phir bhi momentum conserved rehti hai.

Edge case: Kya fast-moving observer par switch karna kabhi momentum conservation tod sakta hai?

Nahi — kisi bhi inertial frame mein total phir bhi constant hai (sirf numerical value frame-velocity

u

par move karne wale frame ke liye

M u

se shift hoti hai). Conservation ek statement hai "time mein constant," jo koi constant-velocity viewpoint change undo nahi kar sakta.

Connections

Newton's Third Law — internal forces pairs mein kyun cancel hoti hain
Newton's Second Law (momentum form) — $F = d p / d t$ jin par yeh traps based hain
Elastic vs Inelastic Collisions — KE-conserved / KE-lost distinction
Centre of Mass Motion — isolated systems ke liye constant $v_{c m}$
Impulse–Momentum Theorem — sirf before/after totals kyun matter karte hain
Rocket Equation — apne exhaust ko push karna