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

3.3.6 · Physics › Rocket Propulsion

WHY kyon zaroorat hai is equation ki?

WHAT humein chahiye: net forward force $F$ (thrust) jo ek rocket engine produce karta hai.

WHY sirf $\dot{m}{v}_e$ kyon nahin hai: Ek rocket nozzle gas ko kabhi-kabhi perfectly ambient pressure tak expand nahin kar paata. Exit plane par gas abhi bhi pressure $P_e\ne P_a$ par ho sakta hai. Ye bacha hua pressure exit area par kaam karta hai aur real force contribute karta hai. Ise ignore karne par real engines ke liye galat answer aayega.

HOW — first principles se derivation (control-volume / momentum)

Hum Newton ka 2nd law momentum form mein use karte hain, ek moving control volume par jo rocket ke around hai, phir uski surface par forces ko track karte hain.

Step 1 — Variable-mass system ke liye Conservation of Momentum

Rocket + uske instantaneous fuel ko hum apna system maante hain time $t$ par.

Ye step kyon? Rocket ki mass change hoti hai, isliye hum seedha $F=ma$ use nahin kar sakte; hum ek fixed chunk of matter ka total momentum $dt$ ke over track karte hain.

• Time $t$ par: mass $m$, velocity $v$. Momentum $p_1=mv$.
• Time $dt$ mein mass $d{m}_{ex}=\dot{m}dt$ exhaust hoti hai exhaust velocity (rocket ke relative) $v_e$ backward.
• $t+dt$ par: rocket mass $m-d{m}_{ex}$, velocity $v+dv$.

$p_2=(m-d{m}_{ex})(v+dv)+d{m}_{ex}(v-v_e)$

Ye step kyon? Ground frame mein exhaust ki velocity hai (rocket velocity) minus (rocket ke relative exhaust speed) $=v-v_e$.

Step 2 — Momentum mein change

$dp=p_2-p_1=mdv-v_{e}d{m}_{ex}+(\text{2nd-order terms})$

Ye step kyon? Expand karo aur $mv$ cancel karo; $d{m}_{ex}dv$ drop karo (do tiny quantities ka product).

Toh $\frac{dp}{dt}=m\frac{dv}{dt}-v_e\dot{m}$

Step 3 — Ye $dp/dt$ kis external force se hota hai?

Newton ke 2nd law se, $\frac{dp}{dt}=\sum F_{ext}$. Control volume surface par external forces pressure forces hain. Ab engine ko isolate karte hain.

Ye step kyon? Hum chahte hain ki engine jo force deliver karta hai usse pata karein, isliye internal momentum change ko external surface pressure forces se equate karo.

Agar hum rocket ke frame mein kaam karein aur thrust $F$ ko wo net force define karein jo engine rocket ko accelerate karne ke liye exert karta hai: $F=\dot{m}{v}_e+\underset{\text{Step 4 se}}{\underbrace{\text{(pressure force term)}}}$

Step 4 — Pressure term (the "bonus")

Control surface ko rocket ke around tightly wrap karo. Ambient pressure $P_a$ poori outer surface par press karta hai. Ye forces har jagah cancel ho jaate hain siway nozzle exit area $A_e$ ke, kyunki wahan koi rocket skin nahin hoti — balki wahan pressure $P_e$ par exhaust gas hai.

• Exhaust gas engine par forward force $P_e{A}_e$ se push karta hai.
• Agar hole band hota, ambient air $P_a{A}_e$ se push back karta; wo back-push missing/replaced hai.

Exit plane par net forward pressure force: $F_{pressure}=(P_e-P_a)A_e$

Ye step kyon? Ek closed surface par sab ambient pressure integrate hokar zero ho jaata hai (ek closed body ka equilibrium). Sirf exit par surface "open" hai, isliye sirf wahan net pressure force bachta hai.

Key consequences (WHY ye matter karte hain)

Worked examples

Common mistakes (Steel-manned)

Flashcards

Connections

• Newton's Third Law — momentum thrust action–reaction hai.
• Conservation of Momentum — variable-mass systems.
• Tsiolkovsky Rocket Equation — thrust integrate karke $\Delta v$ nikalta hai.
• Specific Impulse — $I_{sp}=v_{eff}/g_0$, effective exhaust velocity use karta hai.
• De Laval Nozzle — $P_e$, $v_e$, $A_e$ set karta hai expansion ke through.
• Bernoulli & Compressible Flow — chamber se exit conditions relate karta hai.

Concept Map