Foundations — Thrust equation F = ṁv_e + (P_e − P_a)A_e — derivation

3.3.6 · D1 · Physics › Rocket Propulsion › Thrust equation F = ṁv_e + (P_e − P_a)A_e — derivation

Is page ka ek hi kaam hai: thrust equation ka har letter zero se build karna, ek ek karke, jahan har cheez agli cheez ko earn karti hai. Hum poori formula sirf end mein likhenge, jab uska har symbol ek meaning aur ek picture le chuka hoga. Yahan kuch bhi yeh assume nahi karta ki tumne parent derivation dekhi hai.

Pehle, ek promise direction ke baare mein. Is poore page par hum agree karte hain ki forward (jis taraf rocket travel karta hai) positive count hoga, aur backward negative. Isse fix rakho — yeh ek hi rule hai jo decide karta hai ki koi term thrust mein add karta hai ya ushe khaata hai.

0 — "Force" aur "thrust" ka matlab kya hai

Thrust bas ek engine ke forward force ka special naam hai. Toh hamara poora target ek sawaal hai: rocket engine jo forward arrow draw karta hai, woh kitna lamba hai? Kyunki forward hamari positive direction hai, ek thrust jo rocket ki help kare woh ek positive $F$ hai.

Figure s01 (neeche) exactly yahi dikhata hai: ek hi rocket ek chhote mint arrow aur ek lamba coral arrow ke saath, taaki tum dekh sako ki "zyada force" ka matlab simply "lamba forward arrow" hai — woh length hi woh number $F$ hai jise hum dhundh rahe hain.

Yeh kyun chahiye: poora topic ek number compute karta hai — us forward arrow ka size. Baki sab kuch ise sahi tarike se paane ki bookkeeping hai.

1 — Velocity $v$ aur do frames

Yahan woh subtle part hai jis par parent baar baar lean karta hai: velocity depend karti hai ki kaun dekh raha hai.

• Ek ground observer exhaust gas ko kisi speed par move karte dekhta hai.
• Rocket khud gas ko bahut faster jaate dekhta hai (rocket usse door bhaag raha hai).

"Kaun dekh raha hai" ke baare mein baat karne ke liye humein rocket ki apni motion ka ek naam chahiye.

Figure s02 (neeche) dono viewpoints ko ek scene mein rakhta hai: ek mint arrow rocket ko $v$ par forward jaate dikhata hai; ek coral arrow gas puff ko $v_e$ par backward shoot karte dikhata hai jaise rocket use dekhta hai. Yahi woh picture hai jo frame confusion ke peeche hai — callout se pehle ise padho.

Yeh kyun chahiye: thrust depend karta hai ki gas rocket se kitni fast jaati hai, isliye $v_e$ (ground speed $v$ nahi) "kitna fast throw karte hain" ka honest measure hai.

2 — Mass $m$ aur mass flow rate $\dot{m}$

Rocket apni saari gas ek saath nahi phenkata; woh ek steady stream phenkta hai. Ek stream describe karne ke liye humein chahiye "per second kitna".

Figure s03 (neeche) ek lavender trickle (chhota $\dot{m}$) aur ek coral fire-hose (bada $\dot{m}$) ke beech contrast karta hai taaki abstract symbol flow rate ki picture ban sake — har second nozzle ke muh se kitne kilograms cross karte hain.

Yeh kyun chahiye: "zyada gas phenko" matlab bada $\dot{m}$. Product $\dot{m}{v}_e$ = (per second kitna) × (kitna fast) = per second throw kiya gaya momentum = thrust ka main part.

3 — Momentum $p$ aur hum $\dot{m}$ ko $v_e$ se kyun multiply karte hain

Newton ka insight, jo is poore chapter ko power karta hai, yeh hai ki force equals per second throw kiya gaya momentum. Momentum ko peeche steady rate par bahar phenko aur tumhe aage equal, opposite force feel hogi — yeh exactly Newton's Third Law aur Conservation of Momentum ka combination hai.

$\text{momentum per second}=\dot{m}{v}_e=\left(\frac{\text{kg}}{\text{s}}\right)\left(\frac{\text{m}}{\text{s}}\right)=\frac{\text{kg}\cdot \text{m}}{{\text{s}}^2}=\text{N}$

Units newtons ke roop mein aate hain — ek force. Yeh hamara sanity check hai ki $\dot{m}{v}_e$ sach mein ek thrust hai, aur (upar ke sign argument se) ek positive, forward wala.

Yeh kyun chahiye: momentum "gas phenkne" se "force feel karne" tak ka bridge hai. Iske bina, $\dot{m}{v}_e$ sirf letters multiply kiye hue hain.

4 — Pressure $P$ aur area $A$

Ab key link: agar pressure $P$ area $A$ par push kare, toh total force hai

$F=P\times A\text{(force = pressure}\times \text{area).}$

Isliye pressure kabhi equation mein force banta hai: $P$ akela push nahi hai, lekin $P$ exit hole $A_e$ par act karta hua hai.

Figure s04 (neeche) exit plane par tug-of-war dikhata hai: coral arrows exhaust hain jo baahir push kar rahe hain ($P_e$), mint arrows air hai jo andar push kar rahi hai ($P_a$), dono makkhan-rang ke area $A_e$ par act karte hue. Picture subtraction ko visible banati hai — net force wahi hai jo do opposing pushes ke ladhne ke baad bachta hai.

Yeh kyun chahiye: equation mein "bonus" term aur kuch nahi balki $\text{pressure difference}\times \text{area}$ hai, aur uska sign seedha hamare forward-positive rule se aata hai.

Equation Assemble Karna

Ab — aur sirf ab — har symbol ka ek meaning, ek picture, aur ek sign hai. Hum finally poori cheez likh sakte hain:

Pieces kaise equation ko feed karte hain

Is map ko top se bottom tak padho, arrows follow karte hue: har box is page ka ek idea hai, aur arrow ka matlab hai "yeh idea us box ko build karne ke liye chahiye jis par yeh point karta hai". Sab kuch "Thrust equation" wale single box mein funnel hota hai daayein taraf. Neeche-left ke do grey ideas, Newton's Third Law aur Conservation of Momentum, woh physical laws hain jo "momentum per second" step ko justify karte hain.

Koi bhi arrow "Thrust equation" mein follow karo aur tum $F=\dot{m}{v}_e+(P_e-P_a)A_e$ mein ek term ka meaning padh rahe ho.

Yeh aage kahan le jaata hai

Jab yeh symbols solid ho jayein, toh same pieces chapter mein dobara dikhai dete hain:

• $v_e$ aur $\dot{m}$ Tsiolkovsky Rocket Equation ko drive karte hain (rocket kabhi kitna fast ja sakta hai).
• Combination $F/\dot{m}$ Specific Impulse ban jaata hai (fuel-efficiency score).
• Woh nozzle jo $P_e$ aur $A_e$ set karta hai woh De Laval Nozzle hai, jo Bernoulli & Compressible Flow se govern hota hai.

Equipment Checklist

Apne aap ko test karo — har line ka right side cover karo.