Systems with variable mass — rocket equation derivation preview: Foundations

1.4.12 · D1 · Physics › Momentum & Collisions › Systems with variable mass — rocket equation derivation prev

Pehle tum rocket derivation padho, usse pehle usmein likhi har cheez ka matlab tumhare liye kuch hona chahiye. Neeche, har symbol ko zero se build kiya gaya hai: plain words → ek picture → kyun yeh topic uske bina nahi chal sakta. Upar se neeche padho; har rung apne neeche wale par tikti hai.

1. Mass $m$ — "kitna stuff hai"

Topic ko yeh kyun chahiye: rocket ki poori trick yahi hai ki yeh number time ke saath ghatata hai jaise jaise fuel jalta hai. Isliye $m$ koi fixed label nahi hai — yeh ek aisi quantity hai jisko hum change hote dekhenge.

2. Velocity $v$ — "kitna fast, aur kaun si direction mein"

Figure — Systems with variable mass — rocket equation derivation preview

Upar number line dekho: rocket $v$ par baitha hai (blue arrow, right ki taraf). Exhaust $v - u$ par uda ja raha hai (pink arrow) — aksar abhi bhi positive lekin chhota, kabhi kabhi negative. Hum sab kuch ek fixed frame mein measure karte hain (ground), jaise ek camera jo launch pad par bolted ho.

Topic ko yeh kyun chahiye: thrust $v$ badalne ke baare mein hai. Momenta ko sahi se add karne ke liye hume har piece ki velocity same frame mein jaanni chahiye.

3. Relative velocity & exhaust speed $u$ — "kiski tulna mein fast?"

$u$ exhaust speed relative to the rocket hai — ek engine property (lagbhag $2000$ – $4500$ m/s), ground speed nahi.

Top row (blue) dono objects ko ground se dekhna hai. Bottom row (yellow) wahi pal rocket se dekhna hai: ab rocket still hai aur gas bilkul $u$ speed se peeche stream ho raha hai. Woh $u$ kabhi rocket ki ground speed ke saath nahi badalta — yeh engine mein built-in hai.

Topic ko yeh kyun chahiye: poori force (thrust) $u$ par depend karti hai, gas ki ground-frame velocity par nahi. Frame-shift trick ke liye Relative Velocity dekho.

4. Momentum $p = m v$ — "motion ki quantity"

Bar chart $p$ ko ek $m$ -by- $v$ rectangle ke area ke roop mein dikhata hai: ise wider karo (zyada mass) ya taller karo (zyada speed) aur area (momentum) badh jaata hai.

Topic ko yeh kyun chahiye: momentum woh currency hai jo matter ki ek closed batch ke liye conserved hoti hai — Conservation of Linear Momentum dekho. Rocket derivation pure momentum bookkeeping hai.

5. Infinitesimal $d$ — "itni patli slice jiske square ko hum zero round kar dete hain"

Ek important aadat: do tiny slivers ka product, jaise $d m d v$ , tiny-times-tiny hota hai — negligibly small, isliye hum ise drop kar dete hain. (Ek sliver ka sliver invisible hota hai.)

Topic ko yeh kyun chahiye: hum rocket ko ek instant $d t$ mein study karte hain, jismein woh ek tiny lump $d m_{ej}$ eject karta hai aur ek tiny speed $d v$ gain karta hai. Slivers ke saath kaam karna hume exact instantaneous physics pakadne deta hai.

6. Rate of change $\frac{d ( \cdot )}{d t}$ — "koi cheez per second kitni fast change hoti hai"

Topic ko yeh kyun chahiye: Newton ka honest law $F = \frac{d p}{d t}$ hai — force woh rate hai jis par momentum change hota hai. Rates se bachna namukin hai.

7. Newton's second law, honest form — $F_{ext} = \frac{d p}{d t}$

Topic ko yeh kyun chahiye: jaana-maana $F = ma$ ek assumption chupaata hai — ki $m$ constant hai. Jab mass nikalta hai, woh shortcut toot jaata hai. Honest $F = \frac{d p}{d t}$ , matter ki ek fixed batch par apply kiya, wahi hamari madad karta hai. $F d t = d p$ ke liye Impulse–Momentum Theorem dekho.

m

ka matlab bata sakta hoon aur yahan kyun change hota hai

Mass in kg = kitna matter hai; yeh fuel jalne ke saath ghatata hai, isliye

\frac{d m}{d t} < 0

Number line par ek signed velocity padh sakta hoon

Positive

v

matlab forward, negative matlab backward, sab ek fixed ground frame mein measure kiya gaya.

Jaanta hoon ki

u

kiske relative hai

Exhaust speed rocket ke relative (engine property); ground-frame gas velocity

v - u

hai.

Momentum compute kar sakta hoon

p = m v

— mass times velocity.

Jaanta hoon kyun

d m d v

drop ho jaata hai

Yeh do tiny slivers ka product hai — negligibly small.

Rocket ke liye

d m

ka sign jaanta hoon

Negative — mass ja raha hai.

Newton ka law honestly likh sakta hoon

F_{ext} = \frac{d p}{d t}

, ek closed system par apply kiya gaya.

Jaanta hoon ki aage ka push kahan se aata hai

Newton's third law — gas rocket ko aage dhakelta hai jab rocket gas ko peeche dhakelta hai.

Jaanta hoon kyun

ln

aata hai

Har kg hamesha halke hote baaki mass ko push karta hai;

\frac{d m}{m}

sum karne par

ln

milta hai.

Jaanta hoon kyun hum poori batch track karte hain

Ek closed batch momentum conserve karti hai; sirf external forces uske center of mass ko move kar sakti hain.

Foundations — Systems with variable mass — rocket equation derivation preview

1. Mass $m$ — "kitna stuff hai"

2. Velocity $v$ — "kitna fast, aur kaun si direction mein"

3. Relative velocity & exhaust speed $u$ — "kiski tulna mein fast?"

4. Momentum $p = m v$ — "motion ki quantity"

5. Infinitesimal $d$ — "itni patli slice jiske square ko hum zero round kar dete hain"

6. Rate of change $\frac{d ( \cdot )}{d t}$ — "koi cheez per second kitni fast change hoti hai"

7. Newton's second law, honest form — $F_{ext} = \frac{d p}{d t}$

8. Newton's third law — "har push ka ek equal backward push hota hai"

9. Natural log $ln$ — "compounding ka jawaab"

10. Center of mass — "woh ek point jo simple law follow karta hai"

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