Foundations — Tsiolkovsky rocket equation — full first-principles derivation from momentum

3.3.1 · D1 · Physics › Rocket Propulsion › Tsiolkovsky rocket equation — full first-principles derivati

Yeh parent topic ke liye D1 Foundations page hai. Derivation se pehle ise padho. Hum assume karte hain tumne kuch bhi nahi dekha — yahan tak ki vector ke upar wala arrow bhi nahi. Main note mein use hone wala har ek symbol yahan banaya gaya hai, ek aisi order mein jahan har ek sirf pehle waalon par dependent hai.

0. Yahan symbol hota kya hai

Ek symbol ek chhota naam hai kisi physical cheez ko describe karne wale number ke liye. Jab hum $m$ likhte hain, hum kehna chahte hain "rocket mein abhi kitna stuff hai" — ek akela number, kilograms mein. Isse zyada mysterious kuch nahi — bas ek box par label. Neeche humara kaam yeh hai ki ensure karein ki equation mein aane se pehle har label ek real picture se attached ho.

1. Mass, $m$ — "kitna stuff"

Ek bathroom scale imagine karo. Jo number woh dikhata hai woh (proportional to) $m$ hai. Ek rocket ke liye, $m$ rocket ka shell plus jo bhi fuel is waqt board par bacha hai, dono hain.

2. Velocity, $v$ — "kitni tezi se, aur kis direction mein"

Figure dekho: rocket ek seedhi line par right move kar raha hai. Amber arrow $v$ hai. Lamba arrow matlab tez; agar arrow ulta left ki taraf ho jaaye, toh $v$ negative ho jaayega. Kyunki sab kuch ek hi line par hota hai, hum upar-arrow notation chhod sakte hain aur $v$ ko ek plain signed number ki tarah treat kar sakte hain.

3. Change aur symbol $\Delta$ (delta)

Do snapshots imagine karo: pehle aur baad mein. Agar rocket $v_i=0$ se shuru hua aur $v_f=4000$ m/s par khatam hua, toh $\Delta v=4000-0=4000$ m/s. Delta jawaab deta hai "needle kitna hila?"

4. Momentum, $p$ — "throwing power"

Figure mein, momentum ko ek arrow ki tarah draw kiya gaya hai jiska length $m\times v$ hai. Notice karo do tarikay ek lamba arrow paane ke: bahut zyada mass (bada block) ya high speed (lamba velocity arrow). Momentum dono ko ek number mein bundle karta hai.

5. Conservation of momentum — "total kabhi nahi badalta"

Figure mein skateboarder-in-space idea dikhaya gaya hai. Throw se pehle, person + ball still baithe hain: total $p=0$. Throw ke baad, ball left fly karta hai kuch negative momentum ke saath, isliye person ko right fly karna padega equal positive momentum ke saath — dono arrows cancel ho jaate hain total ko $0$ par rakhne ke liye. Kisi ne bahar se push nahi kiya; unhone sirf ek dusre par push kiya.

6. Newton's law momentum form mein — "push momentum ko badalta hai"

7. Relative exhaust velocity, $v_e$

Do frames draw kiye hain. Upar: rocket par sawaar hoke, tum hamesha gas ko same speed $v_e$ par apni tail se jaate dekho ge — chahe poora rocket kitni bhi tezi se ja raha ho. Neeche: ground se, gas ki speed rocket ki speed $v$ minus $v_e$ hai, likha jaata hai $v-v_e$, kyunki exhaust peeche ki taraf point karta hai.

8. Infinitesimal $d$ aur derivative $\frac{d}{dt}$

Ek movie ki frames socho. $dt$ do neighbouring frames ke beech ka gap hai — almost zero. Us gap mein mass $dm$ se badlta hai aur velocity $dv$ se.

9. Natural logarithm, $\ln$

10. Mass ratio, $R$

Prerequisite map

Map ko top-down padho: mass aur velocity momentum banate hain; momentum plus Newton's law conservation dete hain; conservation plus exhaust speed plus calculus derivation produce karte hain; derivation ek logarithm deta hai; log plus mass ratio final formula deta hai.

Equipment checklist

Khud test karo — right side cover karo aur reveal karne se pehle zor se jawaab do.