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

1.4.3 · Physics › Momentum & Collisions

HUM KYA PROVE KAR RAHE HAIN?

HUM KYO CARE KARTE HAIN? Kyunki yeh hume collision/explosion ke baad ka result predict karne deta hai, bina contact ke dauran ki messy forces ki details jaane. Hume kabhi nahi jaanna padta ki force microsecond-by-microsecond kaise vary hui — sirf pehle aur baad ke totals.

KAISE: scratch se derive karo (two-body case)

Do particles lo, 1 aur 2, jo sirf ek doosre se interact karte hain (koi bahari forces nahi). Particle 2, particle 1 par force ${\vec{F}}_{12}$ se push karta hai, aur 1, particle 2 par ${\vec{F}}_{21}$ se push karta hai.

Step 1 — Newton's 2nd law dono ke liye, momentum form mein. ${\vec{F}}_{12}=\frac{d{\vec{p}}_1}{dt},{\vec{F}}_{21}=\frac{d{\vec{p}}_2}{dt}$ Yeh step kyun? Newton's 2nd law actually $\vec{F}=d\vec{p}/dt$ hai (force = momentum ki rate of change). Har particle ka momentum sirf uस par lagne wali force ki wajah se badalta hai.

Step 2 — Newton's 3rd law. ${\vec{F}}_{12}=-{\vec{F}}_{21}$ Yeh step kyun? Action–reaction pair ki forces magnitude mein equal aur direction mein opposite hoti hain. Yahi pure proof ka engine hai.

Step 3 — Step 1 ki dono equations ko add karo. ${\vec{F}}_{12}+{\vec{F}}_{21}=\frac{d{\vec{p}}_1}{dt}+\frac{d{\vec{p}}_2}{dt}=\frac{d}{dt}({\vec{p}}_1+{\vec{p}}_2)$ Yeh step kyun? Add karne se action–reaction pair saath aata hai taaki hum use cancel kar sakein.

Step 4 — Left side par Step 2 use karo. ${\vec{F}}_{12}+{\vec{F}}_{21}={\vec{F}}_{12}+(-{\vec{F}}_{12})=0$

Step 5 — Conclusion. $\frac{d}{dt}({\vec{p}}_1+{\vec{p}}_2)=0⟹{\vec{p}}_1+{\vec{p}}_2=\text{constant}$

N particles tak generalise karna (Forecast-then-Verify)

Total momentum: $\vec{P}={\sum }_i{\vec{p}}_i$. Differentiate karo: $\frac{d\vec{P}}{dt}={\sum }_i\frac{d{\vec{p}}_i}{dt}={\sum }_i{\vec{F}}_i^{\text{net}}$ Har ${\vec{F}}_i^{\text{net}}=\underset{\text{internal}}{\underbrace{\sum _{j\ne i}{\vec{F}}_{ij}}}+\underset{\text{external}}{\underbrace{{\vec{F}}_i^{\text{ext}}}}$.

Har internal force ${\vec{F}}_{ij}$ ka ek partner ${\vec{F}}_{ji}=-{\vec{F}}_{ij}$ hota hai, isliye internal forces ka double sum zero hota hai. Bacha rehta hai: $\frac{d\vec{P}}{dt}={\sum }_i{\vec{F}}_i^{\text{ext}}={\vec{F}}_{\text{net}}^{\text{ext}}$

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Connections

• Newton's Third Law — is proof ka action–reaction core
• Newton's Second Law (momentum form) — $\vec{F}=d\vec{p}/dt$
• Elastic vs Inelastic Collisions — KE kab conserved hai/ nahi
• Centre of Mass Motion — $\vec{P}=M{\vec{v}}_{cm}$, isliye agar isolated hai toh ${\vec{v}}_{cm}$ constant
• Impulse–Momentum Theorem — integral version, $\vec{J}=\Delta \vec{p}$
• Rocket Equation — is principle ka variable-mass application

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