Foundations — Impulse-momentum theorem — derivation

1.4.2 · D1 · Physics › Momentum & Collisions › Impulse-momentum theorem — derivation

Is page ke end tak tum yeh line padh paoge aur samjhoge ki iske har ek mark ka kya matlab hai: $\vec{J}={\int }_{t_1}^{t_2}\vec{F}dt=\Delta \vec{p}$ Lekin abhi isse padhne ki koshish mat karo — iske koi bhi symbols define nahi hue hain. Hum inhe kuch se build karenge, ek-ek karke, har ek ko ek picture se anchor karke. Yahan yeh assume nahi kiya gaya ki tumne pehle koi arrow-over-a-letter ya $\int$ sign dekha ho. Hum Section 7 mein wापas aate hain aur us line ko symbol by symbol assemble karte hain.

1. Upar ka arrow: vector ka matlab kya hota hai

Figure 1 dekho neeche. Dono pushes "5 units strong" hain, lekin ek right taraf dhakelta hai aur ek left taraf. Ek plain number jaise $5$ inhe alag nahi kar sakta — usne direction bhool gayi.

• Plain words: "ek amount jo kisi taraf point kare."
• Picture: Figure 1 — lamba arrow matlab bada; isse rotate karo aur direction badal jaati hai.
• Topic ko iski zaroorat kyun hai: ek ball wall se takraake wापas bounce kare toh uski speed same hoti hai lekin direction opposite. Agar hum sirf plain numbers use karte toh sochte "kuch nahi badla," jo galat hai. Arrow hi hume kehne deta hai ki motion reverse ho gayi.

2. $m$, $\vec{v}$, aur product $m\vec{v}$: momentum

Ab inhe multiply karo. Ek heavy slow truck aur ek light fast pebble dono "rokne mein mushkil" ho sakte hain. Woh combined stubbornness kya capture karta hai?

• Picture: Figure 2 — velocity arrow, mass se stretch (ya shrink) hua. Same direction, nayi length.
• Topic ko iski zaroorat kyun hai: poora theorem momentum badalne ke baare mein hai. Kisi cheez mein change ki baat karna tab tak possible nahi jab tak usका naam na ho. $\vec{p}$ woh "motion-currency" hai jo theorem spend karta hai.

3. $\Delta$: "change in" symbol

• Picture: do arrows, "before" aur "after." $\Delta \vec{p}$ woh arrow hai jo tum purane arrow ki tip se naye arrow ki tip tak draw karte — woh correction jo "before" ko "after" mein badalne ke liye chahiye.
• Topic ko iski zaroorat kyun hai: theorem ka poora right-hand side $\Delta \vec{p}$ hai. Jab ball bounce karti hai, ${\vec{p}}_i=+mv$ aur ${\vec{p}}_f=-mv$, toh $\Delta \vec{p}=-mv-(+mv)=-2mv$ — woh famous "doubling" sirf signs ke saath carefully ki gayi subtraction hai.

4. Force $\vec{F}$, net force ${\vec{F}}_{\text{net}}$, aur rate-of-change idea $\frac{d\vec{p}}{dt}$

Real objects ko rarely sirf ek single force milti hai. Ek thrown ball gravity feel karta hai neeche, air push karti hai peeche, aur bat aage slam karti hai — sab ek saath. Kaunsa uska momentum change karta hai? Jawaab hai: unka combined effect.

• Topic ko iski zaroorat kyun hai: sirf net push momentum ko change karta hai. ${\vec{F}}_{\text{net}}$ likhna (akele $\vec{F}$ ki jagah) hamaara promise hai ki momentum change kehne se pehle object pe act karne wali har cheez add karenge.

Newton ka sabse deep statement $F=ma$ nahi balki yeh hai: net force wahi hai jo momentum badalta hai, aur jitna bada woh hai, utna tez momentum badalta hai. "Kitni tez $\vec{p}$ change ho rahi hai" likhne ke liye hume ek aur symbol chahiye.

• Yeh tool kyun aur sirf $\Delta p/\Delta t$ kyun nahi? $\Delta p/\Delta t$ poore interval mein average change deta hai. Lekin collision ke dauran force moment-to-moment change hoti hai. Derivative $d\vec{p}/dt$ instantaneous version hai — average itne chote slice pe liya gaya ki force ke paas andar badalne ka waqt hi nahi tha. Yeh jawaab deta hai "ठीक is exact instant pe force kya hai?"
• Picture: Figure 3 — momentum-vs-time graph pe, $d\vec{p}/dt$ ek point pe curve ki steepness (slope) hai — steep matlab momentum tez badal raha hai, toh net force badi hai.

Toh Newton ka law, uske original honest form mein: ${\vec{F}}_{\text{net}}=\frac{d\vec{p}}{dt}$ "Net force equals kitni tez momentum change ho rahi hai." (Newton's Second Law dekho kyun yeh $F=ma$ se zyada fundamental hai.)

5. Integral $\int$: infinitely many tiny pushes ko add karna

Derivative ne time ko slices mein toda. Total effect paane ke liye hume slices ko wापas jodna hoga. Woh joining operation integral hai.

• Picture: Figure 4 — force-vs-time graph pe, har $\vec{F}dt$ ek patli rectangle ka area hai. Inhe sab add karne se force curve ke neeche ka total area milta hai. Woh area hi impulse hai.
• Simple multiplication kyun nahi? Agar force constant hoti toh bas $F\times \Delta t$ (ek bada rectangle) kar sakte. Lekin real collision forces spike aur fade karti hain — jab number baar baar change ho raha ho toh ek single number se multiply nahi kar sakte. Integral "varying quantities ke liye multiplication" hai. (Force–Time Graphs dekho.)

6. Theorem assemble karna: Newton's law ko integrate karna

Ab har symbol earn ho chuka hai, toh hum dekh sakte hain inhe combine hote. Hum Newton's law (Section 4) se shuru karte hain aur dono sides ko collision time pe integrate karte hain — kuch naya nahi, bas tools use karna.

Step 1 — Newton's second law se shuru karo. ${\vec{F}}_{\text{net}}=\frac{d\vec{p}}{dt}$ KYA: net force equals instantaneous rate jisme momentum change hota hai. KYUN: yeh law ka sabse honest form hai (Section 4). KAISA LAGTA HAI: Figure 3 — force har instant pe $p$–$t$ curve ki slope hai.

Step 2 — Dono sides ko tiny time slice $dt$ se multiply karo. ${\vec{F}}_{\text{net}}dt=d\vec{p}$ KYA: left pe ek sliver ka push isolate kiya, right pe ek sliver ka momentum change. KYUN: hum ek interval pe total chahte hain, toh slivers add karne ki preparation karte hain. KAISA LAGTA HAI: Figure 4 mein left pe ek thin rectangle; Figure 3 mein right pe arrow-tip ka ek tiny nudge.

Step 3 — $t_1$ se $t_2$ tak har sliver ko sum karo (dono sides integrate karo). ${\int }_{t_1}^{t_2}{\vec{F}}_{\text{net}}dt={\int }_{t_1}^{t_2}\frac{d\vec{p}}{dt}dt={\int }_{{\vec{p}}_1}^{{\vec{p}}_2}d\vec{p}$ KYA: left side saare force rectangles sum karta hai; right side saare tiny momentum changes sum karta hai. KYUN: summing (integrating) exactly wahi hai jisse hum instantaneous facts ko total mein glue karte hain (Section 5). KAISA LAGTA HAI: Figure 4 ka poora shaded area left pe.

Step 4 — Right-hand sum evaluate karo (Fundamental Theorem of Calculus). Start value ${\vec{p}}_1$ se end value ${\vec{p}}_2$ tak har tiny change $d\vec{p}$ add karne se sirf overall change bachta hai — chote steps telescope ho jaate hain: ${\int }_{{\vec{p}}_1}^{{\vec{p}}_2}d\vec{p}={\vec{p}}_2-{\vec{p}}_1=\Delta \vec{p}$ KYA: saare tiny momentum changes sum karne se final-minus-initial milta hai. KYUN: yahi $\Delta$ ka matlab hai (Section 3); change ko integrate karne se total change wapas milti hai. KAISA LAGTA HAI: Section 3 ki picture ka single "before-to-after" arrow.

Step 5 — Left side ko impulse recognize karo. Left integral exactly hamari impulse $\vec{J}$ ki definition hai (Section 5). Dono sides ko saath rakhne se:

7. Symbols ko order mein rakhna

Upar ka har symbol pehle wale ki zaroorat rakhta hai. Neeche ka prerequisite map build order dikhata hai, physics flow nahi. Padhne ka tarika: top boxes se shuru karo (pehle milne wali raw ideas) aur arrows follow karo neeche ki taraf — har arrow ka matlab hai "is idea ki zaroorat hai agla banane ke liye." Saare streams neeche theorem pe converge karte hain.

Top se bottom padho: arrows aur mass se momentum banta hai; force net force mein gather hokar plus rate-of-change idea se Newton's law banta hai; delta plus momentum se $\Delta \vec{p}$ banta hai; integral plus net force se impulse banta hai; aur theorem woh jagah hai jahan teeno streams milte hain.

Equipment checklist

Apne aap ko test karo — right side cover karo. Agar koi bhi jawaab fuzzy lage, derivation se pehle us section ko dobara padho.

Connections

• इसी को Hinglish में पढ़ो → — parent topic, Hinglish mein.
• Newton's Second Law — jahan se ${\vec{F}}_{\text{net}}=d\vec{p}/dt$ aata hai.
• Force–Time Graphs — integral ke peeche ki picture (impulse = area).
• Conservation of Momentum — $\Delta \vec{p}=0$ kahaan le jaata hai.
• Collisions — Elastic and Inelastic — jahan bouncing-vs-sticking sign work kaam aata hai.
• Work–Energy Theorem — same ideas lekin force over distance instead of time.
• Center of Mass Motion — poore systems ke liye momentum bookkeeping.