Effective exhaust velocity c = v_e + (P_e − P_a)A_e - ṁ: Question bank

3.3.8 · D5 · Physics › Rocket Propulsion › Effective exhaust velocity c = v_e + (P_e − P_a)A_e - ṁ

Pehle: woh formula banao jise ye traps test karte hain

Ek quantity ka naam lete hain jiske baare mein hum care karte hain. ==Thrust $F$ == woh forward push hai (newtons mein) jo engine rocket ko deta hai. Yeh nozzle exit par do dhakkon se banta hai, aur neeche ki picture dono dikhati hai.

Figure — Effective exhaust velocity c = v_e + (P_e − P_a)A_e - ṁ

Dhakka 1 — momentum thrust. Ek second mein engine $\overset{m}{˙}$ kilogram gas ko real speed $v_{e}$ par peeche fenkta hai. Mass ko peeche fenkne se rocket aage push hota hai (Newton ka 3rd law), aur us push ki size hai $F_{mom} = \overset{m}{˙} v_{e} .$

Dhakka 2 — pressure thrust. Figure mein exit ring dekho. Andar ki gas $P_{e}$ pressure se bahar ki taraf dabaati hai; aas-paas ki hawa $P_{a}$ se andar ki taraf dabaati hai. Khuli exit area $A_{e}$ par koi wall nahi hai jo bacha hua dabav cancel kar sake, isliye net push hai $F_{pres} = (P_{e} - P_{a}) A_{e} .$

Jodo aur repackage karo. Total thrust hai $F = \overset{m}{˙} v_{e} + (P_{e} - P_{a}) A_{e}$ . Hum ek khayal ki speed $c$ — effective exhaust velocity — define karte hain taaki $F = \overset{m}{˙} c$ dono dhakkon ko ek saath reproduce kare. Total ko $\overset{m}{˙}$ se divide karo: $c = v_{e} + \frac{( P _{e} - P _{a} ) A _{e}}{m ˙}$

Neeche har cheez ke liye symbol key: $c$ effective exhaust velocity, $v_{e}$ real gas exit speed, $P_{e}$ exit-plane gas pressure, $P_{a}$ aas-paas ki hawa ka pressure, $A_{e}$ exit area, $\overset{m}{˙}$ propellant jo per second nikal raha hai, $F = \overset{m}{˙} c$ thrust. Doosra figure teen expansion regimes ka preview deta hai jinka traps baar baar zikar karte hain.

True ya false — justify karo

TF1. "

c

hamesha

v_{e}

se bada hota hai."

False. Sirf tab jab underexpanded ho (

P_{e} > P_{a}

). Agar overexpanded ho (

P_{e} < P_{a}

) toh correction negative hai aur

c < v_{e}

; agar matched ho toh

c = v_{e}

TF2. "Agar nozzle optimally expanded hai, toh thrust zero ho jaata hai kyunki pressure term khatam ho jaati hai."

False. Sirf pressure term khatam hoti hai; momentum term

\overset{m}{˙} v_{e}

bacha rehta hai, isliye

F = \overset{m}{˙} v_{e} = \overset{m}{˙} c

— aksar thrust ka sabse bada hissa.

TF3. "Perfect vacuum mein rocket koi thrust nahi banata kyunki push karne ke liye koi hawa nahi hoti."

False. Thrust mass eject karne se aata hai (Newton ka 3rd law), hawa ko push karne se nahi. Vacuum mein

P_{a} = 0

, isliye

F = \overset{m}{˙} v_{e} + (P_{e} - P_{a}) A_{e}

mein pressure term apna maximum

+ P_{e} A_{e}

ban jaata hai, aur zyada thrust milta hai, na ki zero.

TF4. "

c

rocket ke chadhte waqt constant rehta hai, kyunki

v_{e}

A_{e}

aur

\overset{m}{˙}

nahi badalte."

False.

P_{a}

altitude ke saath girta hai, isliye

- P_{a} A_{e} / \overset{m}{˙}

kam negative hota jaata hai aur

c

ascent ke dauran badhta hai — dekho Atmospheric Pressure vs Altitude.

TF5. "Kyunki sirf difference

(P_{e} - P_{a})

matter karta hai, tum dono ke liye gauge pressures use kar sakte ho."

True — agar tum consistent ho. Gauge dono se same reference subtract karta hai, isliye difference unchanged rehta hai. Khatre ki baat yeh hai ki ek absolute aur ek gauge mix karo, jo isliye safe habit yeh hai ki dono ke liye absolute use karo.

TF6. "Overexpansion se thrust kam hota hai, isliye yeh hamesha bura hai aur kabhi nahi hona chahiye."

False. High altitude ke liye optimised nozzle sea level par deliberately overexpanded hota hai; neeche ka chhota penalty accept kiya jaata hai taaki bada vacuum performance mile. Asli khatre mein flow separation hai, sirf sign nahi — dekho Nozzle Expansion (Under/Over/Optimal).

TF7. "

I_{s p}

aur

c

alag physics carry karte hain, isliye ek mein badlaav doosre ko zaruri affect nahi karta."

False.

I_{s p} = c / g_{0}

jahan

g_{0}

ek fixed constant hai; ye dono alag units mein same quantity hain, isliye jo bhi

c

badhata hai woh

I_{s p}

ko proportionally badhata hai — dekho Specific Impulse.

TF8. "Tsiolkovsky ka

Δ v = c ln (m_{0} / m_{f})

mein

v_{e}

, real gas speed, use karni chahiye."

False. Yeh effective

c

use karta hai, kyunki

c

pehle se hi momentum aur pressure thrust dono ko us number mein bundle kar chuka hai jo actually rocket ko accelerate karta hai — dekho Tsiolkovsky Rocket Equation.

Error dhundho

SE1. "Ek student

F = \overset{m}{˙} v_{e}

likhta hai aur ise complete thrust equation kehta hai."

Incomplete. Poora Thrust Equation hai

F = \overset{m}{˙} v_{e} + (P_{e} - P_{a}) A_{e}

; unhone pressure thrust chhod diya, jo sirf tab valid hai jab

P_{e} = P_{a}

ho.

SE2. "Kyunki

c = v_{e} + (P_{e} - P_{a}) A_{e} / \overset{m}{˙}

hai, isliye bada

\overset{m}{˙}

hamesha

c

ko bada banata hai."

Galat.

\overset{m}{˙}

correction ke denominator mein hai, isliye bada

\overset{m}{˙}

pressure contribution ko chhhota karta hai aur

c

v_{e}

ki taraf le jaata hai; yeh

c

ko scale up nahi karta.

SE3. "Sea level par

P_{a} = 101.3

kPa aur

P_{e} = 70

kPa hai, isliye engine underexpanded hai."

Error.

P_{e} < P_{a}

matlab overexpanded hai (exit pressure ambient se neeche). Underexpanded

P_{e} > P_{a}

hota hai. Sign ka verdict tabhi sahi hoga jab dono ko sahi se compare karo.

SE4. "Pressure thrust isliye exist karta hai kyunki bahari hawa nozzle walls par dabaati hai."

Galat jagah. Wall pressures pehle se

\overset{m}{˙} v_{e}

ke andar count ho chuki hain (woh flow ko shape karti hain). Extra pressure thrust woh uncancelled net push hai

(P_{e} - P_{a})

khuli exit ring par, jahan use balance karne ke liye koi wall nahi hai.

SE5. "Engineer vacuum model karne ke liye

P_{e} = P_{a} = 0

set karta hai, aur pure momentum thrust

c = v_{e}

milta hai."

Error. Vacuum mein

P_{a} = 0

hota hai lekin

P_{e}

zero nahi hota — gas abhi bhi real static pressure ke saath nikalta hai. Isliye term hai

+ P_{e} A_{e} / \overset{m}{˙} > 0

aur

c > v_{e}

, equal nahi.

SE6. "Space mein thrust boost karne ke liye, bas exit area

A_{e}

ko koi doosra badlaav kiye bina bahut bada kar do."

Oversimplified. Bada

A_{e}

pressure-thrust term ko tabhi badhata hai jab

P_{e}

positive rahe; flow ko aur expand karna actually

P_{e}

ko girata hai (aur atmosphere mein overexpansion cause kar sakta hai). Tum

A_{e}

ko akele inflate nahi kar sakte — dekho Nozzle Expansion (Under/Over/Optimal).

Why questions

WHY1. "Hum

c

kyun invent karte hain, do alag terms rakhne ki bajay?"

Taaki har downstream formula —

F = \overset{m}{˙} c

I_{s p} = c / g_{0}

Δ v = c ln (m_{0} / m_{f})

— ek single number use kare, aur pressure bookkeeping ek effective speed ke andar chhup jaaye.

WHY2. "Usi engine ka

I_{s p}

vacuum mein sea level se zyada kyun hota hai?"

Kyunki

c

badhta hai jab

P_{a} \to 0

hota hai (back-push

- P_{a} A_{e} / \overset{m}{˙}

khatam ho jaata hai), aur

I_{s p} = c / g_{0}

, isliye zyada

c

matlab zyada

I_{s p}

WHY3. "Momentum thrust surrounding air se independent kyun hai, lekin pressure thrust nahi?"

Momentum thrust ejected mass ka reaction hai (ek Newton's-3rd-law effect jo exhaust ke andar hai), jabki pressure thrust ambient

P_{a}

ke against ek difference hai, isliye environment directly isme aata hai.

WHY4. "Overexpansion physically flow ko damage ya destabilise kyun kar sakta hai?"

P_{e} < P_{a}

hone par bahari hawa andar push karti hai aur exhaust ko nozzle wall se alag kar sakti hai (flow separation), jo asymmetric, unsteady side-loads produce karta hai — yeh ek real structural risk hai, sirf ek thrust number nahi.

WHY5. "Standard statement mein pressure term absolute pressures kyun use karta hai, gauge nahi?"

Ambiguity se bachne ke liye: jab tak dono absolute hain, difference unmistakable hai. Conventions mix karna silently

(P_{e} - P_{a})

ko ek atmosphere se shift kar deta hai aur

c

corrupt ho jaata hai.

WHY6. "'Perfectly expanded' fixed geometry ke liye maximum thrust ka point kyun hai?"

P_{e} = P_{a}

par flow ne sabse zyada pressure energy ko velocity mein convert kar liya hai aur bahari hawa push back nahi kar rahi; aage jao (overexpand) aur ambient air thrust cheen leta hai, isliye matched pressure us altitude par us nozzle ke liye peak hai.

Edge cases

EC1. "Us ek altitude par

c

exactly kya hai jahan

P_{e} = P_{a}

ho?"

Correction exactly zero hai, isliye

c = v_{e}

. Yeh ek specific "design altitude" par hota hai; uske upar aur neeche term nonzero hoti hai.

EC2. "Jab altitude

\to

orbit (

P_{a} \to 0

) hoti hai,

c

ki limiting value kya hai?"

Yeh apne maximum

c_{m a x} = v_{e} + P_{e} A_{e} / \overset{m}{˙}

tak pahunchta hai, poori vacuum effective velocity, jisme subtract karne ke liye koi ambient back-push nahi bachta.

EC3. "Agar

P_{e} = 0

ho (gas itni expand ho gayi ki apna pressure khatam) atmosphere mein, toh kya hoga?"

Correction

- P_{a} A_{e} / \overset{m}{˙} < 0

ban jaata hai, ek pure penalty — strongly overexpanded, atmosphere mein physically unrealistic limit jahan sirf ambient hawa pushback karti hai; real flow pehle separate ho jaata.

EC4. "Kya

c

kabhi negative ya zero ho sakta hai?"

Sirf ek degenerate overexpanded extreme mein jahan negative pressure term

\overset{m}{˙} v_{e}

se zyada ho; ek kaam karne wala engine

c > 0

rakhta hai, kyunki non-positive

c

ka matlab hai koi net forward thrust nahi.

EC5. "Agar

\overset{m}{˙} \to 0

ho jabki chhota sa pressure imbalance baki rahe toh pressure term ka kya hoga?"

Term

(P_{e} - P_{a}) A_{e} / \overset{m}{˙}

blow up ho jaata hai (bahut chhoti number se divide karo), isliye

c

ill-defined ho jaata hai — yeh reminder hai ki

c

tab hi meaningful hai jab propellant actually flow kar raha ho.

EC6. "Throttle-down ke dauran

\overset{m}{˙}

girta hai lekin

P_{e}

bhi aksar girta hai — kya

c

zaruri badhega?"

Zaruri nahi. Chhota

\overset{m}{˙}

correction ko magnify karta hai, lekin girta hua

P_{e}

(P_{e} - P_{a})

ko chhhota karta hai;

c

par net effect depend karta hai ki kaunsa zyada tezi se badalti hai — tum sirf

\overset{m}{˙}

se judge nahi kar sakte.

Connections

Thrust Equation — poora $F = \overset{m}{˙} v_{e} + (P_{e} - P_{a}) A_{e}$ jis par yahan har trap tika hai.
Specific Impulse — $I_{s p} = c / g_{0}$ ; $c$ ka twin.
Tsiolkovsky Rocket Equation — $c$ use karta hai, $v_{e}$ nahi.
Nozzle Expansion (Under/Over/Optimal) — sign aur separation risk decide karta hai.
Conservation of Momentum — isliye vacuum thrust real hai.
Atmospheric Pressure vs Altitude — isliye $c$ rocket ke saath upar chadha hai.