Rocket staging — series staging, parallel staging: Question bank

3.3.45 · D5 · Physics › Rocket Propulsion › Rocket staging — series staging, parallel staging

True or False — justify karo

Staging ka poora point yeh hai ki rocket ko overall halka banaya jaye taaki use kam fuel chahiye.

False — staging se launch mass kam nahi hoti; yeh mid-flight mein dead structure ko remove karti hai taaki bacha hua fuel kam mass ko accelerate kare. Overall tum heavier bhi ho sakte ho phir bhi zyada

Δ v

gain kar sakte ho.

Series staging mein total

Δ v

har stage ke mass ratio ka product hota hai.

False — $Δ v$ values add hote hain (

\sum Δ v_{i}

); logs ke andar mass ratios multiply hote hain, aur logs us product ko sum mein convert kar dete hain.

Ek single stage ke propellant ko double karne se uska

Δ v

double ho jaata hai.

False —

Δ v

badhta hai

ln (m_{0} / m_{f})

ke saath, toh yeh logarithmically badhta hai; har extra tonne fuel pehle se kam speed kharidta hai. Yahi diminishing return exactly kyun hum ek giant tank banane ki jagah staging karte hain.

Zyada (higher) exhaust velocity wala stage hamesha better mass ratio wale identical stage se behtar hota hai.

False —

Δ v = v_{e} ln (m_{0} / m_{f})

donon ko multiply karta hai; akele

v_{e}

se judge nahi kar sakte. Dead structure se bhara high-

v_{e}

engine, lean structural fraction wale ek ordinary engine se haar sakta hai.

Parallel staging mein boosters core ko lift karte hain, toh core ko upar jaane ka free ride milta hai.

False — ascent ke dauran core aur boosters ek doosre ka dead weight uthate hain; boosters core ke empty tanks ko "payload" ki tarah carry nahi karte. Yeh co-lifting exactly wahi reason hai kyun parallel staging series se thodi kam

Δ v

-efficient per mass hoti hai.

Two-stage series rocket ke liye, stage 1 ko enormous exhaust velocity dene se eventually stage 2 drop ki ja sakti hai.

False — stage 2 exist karti hai taaki stage 1 ki structure jettison hone ke baad fresh, favourable mass ratio carry kar sake; stage 1 par koi bhi

v_{e}

stage 1 ke empty tank ko orbit tak haul karne ki penalty nahi hataata.

Single-stage rocket principle mein impossible hai.

False — yeh principle mein possible hai (SSTO) lekin practice mein brutal hai: real structural fractions (~5–10%) ke saath empty-tank mass almost poora

Δ v

budget kha jaata hai, payload ke liye bahut kam bachta hai.

Parallel staging mein booster separation

Δ v

ka source hai, jaise ek chhota rocket kick.

False — separation khud essentially koi

Δ v

nahi deta; faayda yeh hai ki suddenly $n m_{s, b}$ dead mass drop ho jaati hai, jo remaining core burn ke liye mass ratio improve karta hai.

Error dhundo

"Stage 2 ka mass ratio

m_{0} / m_{f} = (S_{2} + payload) / (m_{p, 2})

hai."

Final mass propellant nahi hota — yeh empty stage plus payload hota hai:

m_{f} = m_{s, 2} + m_{p a y l o a d}

. Propellant woh hai jo nikalta hai, toh yeh

m_{0} - m_{f}

mein belong karta hai, kabhi bhi

m_{f}

ke roop mein nahi.

"Parallel staging ke phase 2 mein core poora

m_{p, c}

jaala deta hai, toh uska mass ratio

\frac{m _{s, c} + m _{p, c} + m _{u pp er}}{m _{s, c} + m _{u pp er}}

hai."

Galat — core apna kuch propellant booster phase ke dauran pehle hi jala chuka hai. Sirf bacha hua propellant

m_{p, c}^{(2)} = m_{p, c} - m_{p, c}^{(1)}

available hai, toh wahi numerator mein aata hai, poora

m_{p, c}

nahi.

"Parallel boosters ke liye, effective exhaust velocity plain average

\frac{v _{e, c} + v _{e, b}}{2}

hai."

Yeh thrust-weighted hona chahiye:

\overset{v}{ˉ}_{e} = \frac{F _{c} v _{e, c} + n F _{b} v _{e, b}}{F _{c} + n F _{b}}

, kyunki har engine group apne mass flow (thrust) ke proportion mein momentum contribute karta hai, equally nahi.

"Two-stage rocket ka payload fraction

ϵ_{1} \times ϵ_{2}

hai."

Payload fraction structural fractions, exhaust velocities aur required

Δ v

ko mix karta hai — yeh structural fractions ka product nahi hai. Structural fraction sirf yeh batata hai ki ek stage ka kaunsa portion dead weight hai, yeh nahi ki launch mass ka kaunsa portion orbit tak pahunchta hai.

"Kyunki mass ratio ke logs add hote hain, maximum total log ke liye saara propellant ek bade pehle stage mein daalna chahiye."

Saara propellant ek stage mein dalne se uska khud ka empty tank bhi badhta hai, toh

m_{f}

badhta hai aur log shrink ho jaata hai har added tonne ke saath. Stages mein split karne se har stage apni structure shed kar sakta hai aur ek fresh favourable ratio pe reset ho sakta hai.

"Staging design ke liye gravity losses matter nahi karte — yeh cancel out ho jaate hain."

Yeh cancel nahi hote: Gravity losses slow, low-thrust ascents ko penalise karte hain, isi liye high-thrust parallel staging (sab engines liftoff par) exist karta hai — taaki tezi se climb karo aur

g

se ladne mein kam time bitao.

Why questions

Staging same total fuel aur structure wale single stage se zyada

Δ v

kyun deta hai?

Kyunki ek single stage apni saari empty structure burnout tak carry karta hai,

m_{f}

inflate karta hai; staging spent structure ko jettison karta hai toh baad ka fuel choti mass ke khilaaf kaam karta hai, har log term improve hota hai.

Total series

Δ v

sum kyun hai, kuch aur complicated kyun nahi?

Har stage previous stages ne jo speed di hai usse start karta hai, aur Tsiolkovsky ek independent additive

Δ v_{i}

deta hai jab tum us stage ke khud ke

m_{0}

aur

m_{f}

plug in karte ho; velocities simply accumulate hoti hain.

Series stages size karte waqt hum payload se backward kyun kaam karte hain?

Kyunki har lower stage ka payload uske upar ki saari cheez hai (upper stages + actual payload). Upper mass pata hona zaroori hai pehle, tabhi stage 1 ke

m_{0}

aur

m_{f}

define bhi hote hain.

Falcon Heavy apne side boosters recover kyun karta hai lekin centre core hamesha nahi?

Centre core zyada der tak jaalta hai aur zyada tezi se aur zyada upar separate hota hai, toh boostback ke liye bahut zyada fuel reserve chahiye — aksar use expend karna zyada economical hota hai. Side cores pehle/dheeray separate hote hain aur kam propellant cost mein wapas land karte hain.

Parallel staging solid boosters ko liquid core ke saath mix kar sakta hai, jabki series stages usually ek hi stage ke andar same propellant family kyun hoti hain?

Parallel staging mein har unit ek independent engine group hai jo simultaneously fire karta hai — ek solid ek liquid ke saath reh sakta hai kyunki hum sirf thrusts sum karte hain. Specific impulse differences ko thrust-weighted

\overset{v}{ˉ}_{e}

se handle kiya jaata hai.

Higher Specific impulse upper stage aksar higher-

I_{s p}

first stage se zyada valuable kyun hoti hai?

Upper stage ka

Δ v

sabse choti remaining mass par act karta hai aur final velocity ke sabse kareeb hota hai, toh uski efficiency improve karna final payload tak sabse kam dead weight ke saath propagate hota hai.

Edge cases

Agar kisi stage ka structural fraction

ϵ = 1

hai (saari structure, koi fuel nahi), toh uska

Δ v

kya hai?

Zero — koi propellant nahi hai, toh

m_{0} = m_{f}

, isliye

ln (m_{0} / m_{f}) = ln 1 = 0

. Jo stage pure structure hai woh sirf neeche wale stages mein dead weight add karta hai.

Agar ek stage fixed structure ke saath infinite propellant carry kare, toh kya uska

Δ v

infinite ho jaata hai?

Nahi — mass ratio

m_{0} / m_{f} \to \infty

, toh

Δ v \to \infty

mathematically hota hai, lekin sirf logarithmically slowly, aur physically infinite fuel ka matlab hai infinite tank mass, toh structure bhi badhega. Asli limit finite structural fraction hai.

Agar boosters zero thrust produce karein (

F_{b} = 0

) toh parallel-staging

\overset{v}{ˉ}_{e}

ka kya hoga?

Yeh core ki value par collapse ho jaata hai:

\overset{v}{ˉ}_{e} = \frac{F _{c} v _{e, c}}{F _{c}} = v_{e, c}

. Thrust-weighted average theek se un engines ko ignore karta hai jo koi momentum contribute nahi karte.

Parallel staging mein, agar core phase 1 ke dauran apna koi bhi propellant nahi jaalaata (

m_{p, c}^{(1)} = 0

), kya phase 2 phir bhi valid hai?

Haan — tab

m_{p, c}^{(2)} = m_{p, c}

(poora load bacha hai) aur phase 2 sirf ek normal core burn hai jisme boosters ka dead mass already drop ho chuka hai. Formula phir bhi hold karta hai; kuch bhi zero se divide nahi hota.

n

-stage series rocket ke liye, realistic structural fractions ke saath

n \to \infty

hone par payload fraction ka kya hota hai?

Yeh ek limit ki taraf badhta hai lekin diminishing benefit ke saath: har extra stage separation hardware, joints aur failure points add karta hai, toh ~3–4 stages (jaise Saturn V mein) ke baad reliability aur mass cost shrinking

Δ v

gain se zyada ho jaata hai.

Agar booster aur core exhaust velocities equal hoon (

v_{e, b} = v_{e, c} = v_{e}

), toh kya parallel staging phir bhi help karta hai?

Haan thrust/gravity losses ke liye — tum phir bhi tezi se climb karte ho aur booster structure jaldi drop karte ho — lekin

\overset{v}{ˉ}_{e}

exactly

v_{e}

ban jaata hai, toh extra-

v_{e}

benefit khatam ho jaata hai; faayda purely mass-drop aur high initial thrust ka hota hai.

Recall Sabse fast self-test

Total series $Δ v$ $v_{e, i} ln (m_{0, i} / m_{f, i})$ ka ek sum hai ::: kyunki har stage previous speed ke upar ek independent kick add karta hai. Parallel $\overset{v}{ˉ}_{e}$ thrust se weighted hai ::: kyunki momentum contribution mass flow ke saath scale karta hai, aur thrust $= \overset{m}{˙} v_{e}$ . Stage ka $m_{f}$ structure + payload hota hai, kabhi propellant nahi ::: propellant woh hai jo nikalta hai, $m_{0} - m_{f}$ mein aata hai.