Foundations — Trajectory optimization — minimum gravity loss, minimum drag loss

3.4.15 · D1 · Physics › Rocket Flight Mechanics › Trajectory optimization — minimum gravity loss, minimum drag

Yeh page — bilkul zero se — har letter, symbol, aur idea build karta hai jis par parent topic depend karta hai. Isko upar se neeche padho: har item sirf unhi cheezoon ka use karta hai jo usse pehle define ho chuki hain.

1. Speed $v$ aur uska change $\Delta v$

Pehle, sabse simple symbol: $v$ rocket ki instantaneous speed hai — abhi is waqt yeh kitni tezi se move kar raha hai, metres per second mein ($\text{m/s}$). Ek frozen instant mein speedometer ki needle imagine karo.

Greek letter $\Delta$ (delta) "the change in" ka shorthand hai. Toh $\Delta v$ ka matlab hai "speed $v$ mein change". Agar ek rocket $0$ se $3000\text{m/s}$ tak speed badhata hai, toh $\Delta v=3000\text{m/s}$ — yeh bhi $\text{m/s}$ mein hai.

Topic ko iska kya zaroorat hai? Kyunki rocket engine seedha tumhe altitude ya orbit nahi deta — yeh tumhe ek speed budget deta hai. Har task aur har loss ek hi unit, $\text{m/s}$, mein measure hoti hai, toh inhe paise ki tarah add aur subtract kiya ja sakta hai.

2. Mass symbols: $m_0$, $m_f$, $m(t)$, aur ${\dot{m}}_p$

Rocket fuel jaalaane par halka hota jaata hai — yahi poora reason hai ki rockets kaam karte hain. Toh hume alag-alag moments par uski mass ke liye naam chahiye.

${\dot{m}}_p$ mein chhota dot per second rate of change ka Newton ka notation hai. Toh ${\dot{m}}_p$ (padho "m-dot-p") yeh hai ki har second nozzle se kitne kilograms fuel nikalte hain.

Neeche di gayi figure padho taaki yeh symbols abstract na rahen. Purple curve hai $m(t)$ jo time ke saath neeche jaati hai. Teen cheezein notice karo jo figure point kar raha hai: upar mint dashed line hai $m_0$ (full mass jisse hum shuru karte hain); neeche coral dashed line hai $m_f$ (khali shell jis par hum khatam hote hain); aur purple slope ki steepness exactly ${\dot{m}}_p$ hai — jitna tez girta hai, utna tez fuel ja raha hai. Curve aur $m_f$ ke beech ka shaded band kharch hone wala propellant $m_p$ hai.

Topic ko inki zaroorat kyun hai? Tsiolkovsky equation $m_0$ ko $m_f$ se compare karti hai: rocket ka speed budget full-to-empty mass ke ratio par depend karta hai, dono mein se kisi ek par akele nahi.

3. Integral sign $\int$ — badlti cheez ko add karna

Isse pehle ki hum speed budget ya koi bhi loss build karein, hume notation ka ek piece chahiye jo is topic mein har jagah aata hai: integral sign $\int$.

Hum ise do forms mein use karenge: mass par summing (jaise fuel $m_0$ se $m_f$ tak drain hota hai, agla section) aur time par summing (launch se burnout tak, §8). Iss ek tool ke saath, neeche ke derivations honest rehte hain — har $\int$ jo tum dekhte ho ab defined hai.

4. Exhaust speed $v_e$, natural logarithm $\ln$, aur Tsiolkovsky kahan se aata hai

Ab ek nayi tool aati hai: $\ln$, the natural logarithm.

Word ideal yeh flag karta hai ki yeh perfect speed budget hai — bina gravity, bina air, bina kisi bhi wasted pointing ke. Reality isse ghataati hai; woh ghataav hi "losses" hain.

5. Local gravity $g$ (aur $g(t)$ ke baare mein sachchi baat)

Gravity altitude ke saath kamzor hoti hai, toh jab hum ek lambi climb par honest rehna chahte hain toh hum $g(t)$ likhte hain — time ke function ke roop mein gravity.

Topic ko iska kyun zaroorat hai: gravity woh force hai jisse rocket climb karte waqt ladhta hai, aur yeh woh free force hai jo baad mein rocket ko dheere se palta hai. Dono roles $g$ ke through measure hote hain. Gravity exactly kaise speed-stealing part aur turning part mein split hoti hai, yeh flight-path angle (§6) aur sine/cosine split (§7) ka kaam hai — aur wahi jagah hai jahan hum finally gravity-loss integral mein real numbers plug karenge.

6. Flight-path angle $\gamma$ — show ka star

Yeh poore topic mein sabse zaroori symbol hai. $\gamma$ (Greek "gamma") rocket ki velocity aur local horizon (flat ground direction) ke beech ka angle hai.

Neeche di gayi figure mein walk karo. Horizontal grey arrow local horizon hai — "flat ground" direction. Coral arrow rocket ki velocity hai. Unke beech ka purple wedge, $\gamma$ se marked, flight-path angle hai: jaise yeh $90^{\circ }$ ki taraf khulta hai coral arrow seedha upar swing karta hai; jaise yeh $0^{\circ }$ ki taraf band hota hai coral arrow flat ho jaata hai. Mint arrow gravity dikhata hai, hamesha seedha neeche ki taraf regardless of $\gamma$ — woh fixed downward direction exactly wahi hai jisse $\gamma$ akela control karta hai ki gravity motion se kaise relate karti hai.

Yahi reason hai ki is topic ki har loss formula mein $\gamma$ hota hai. Dekho bhi Flight-path angle and equations of motion.

7. Sine aur cosine — WHY yeh aate hain, triangle par

Gravity seedha neeche kheenchti hai, lekin rocket apni velocity ke along angle $\gamma$ par move karta hai. Yeh poochhhne ke liye ki "gravity kitna motion ke against ladhti hai?" hume downward pull ko do pieces mein split karna hoga: ek velocity ke along, ek perpendicular to it. Ek seedhe-neeche wale arrow ko angle use karke split karna exactly wahi kaam hai jiske liye sine aur cosine banaye gaye the.

Neeche di gayi figure ko dhyaan se study karo — yeh poore topic ka mechanical heart hai. Mint arrow seedha neeche point karti poori gravity $g$ hai. Hum ise do dashed pieces mein todh dete hain. Coral dashed arrow, velocity direction ke along rakkha gaya, $g\sin \gamma$ hai — woh part jo motion ke against peeche kheenchta hai aur speed churaata hai. Lavender dashed arrow, velocity ke perpendicular, $g\cos \gamma$ hai — woh part jo sirf path ko curve karta hai bina ise slow kiye. Extremes dekhte hain:

• $g\sin \gamma$ (speed-stealer): seedha upar ($\gamma =90^{\circ }$), $\sin 90^{\circ }=1$, toh $g$ ka sab kuch tumse ladhta hai; flat ($\gamma =0^{\circ }$), $\sin 0^{\circ }=0$, toh kuch bhi tumhari speed se nahi ladhta.
• $g\cos \gamma$ (path-curver): seedha upar, $\cos 90^{\circ }=0$, koi turning nahi; flat, $\cos 0^{\circ }=1$, maximum turning.

Yeh split hume $g\sin \gamma$ (→ gravity loss) aur $g\cos \gamma$ (→ gravity-turn steering) deta hai. Dono seedhe isi ek figure se aate hain.

Ab ki $\gamma$, sine, aur integral sab defined hain, hum ek gravity-loss integral real numbers ke saath padh sakte hain:

8. Burn time $t_b$ aur burn par time-integrals

Losses ko ${\int }_0^{t_b}(\dots )dt$ ke roop mein likha jaata hai — wahi integral tool §3 se, ab time par summing karte hue.

$t_b$ (subscript $b$ = "burnout") simply woh clock time hai jab engine band hota hai — har loss sum ka upper edge. Yeh Thrust-to-weight ratio and burn time se set hota hai. Kyunki $\gamma$, $g$, $v$, aur $m(t)$ sab un seconds ke dauran drift karte hain, sirf integral (plain multiplication nahi) sach mein loss deta hai.

9. Force symbols: thrust $T$ aur drag $D$

Us formula ko symbol by symbol unpack karte hain:

• $\rho$ (Greek "rho") — air density, kilograms of air per cubic metre. Ground ke paas thick, space mein near-zero. Imagine karo rocket ko kitna "stuff" push karke nikaalna padta hai.
• $v$ — air mein rocket ki speed (wahi $v$ §1 se).
• $v^2$ — speed squared: speed double karne par drag chaar guna ho jaata hai. Yeh squared term hi reason hai ki thick air mein tez rehna itna punishing hai.
• $C_D$ — drag coefficient, ek shape number ($\sim 0.3$ ek sleek rocket ke liye): body kitna streamlined hai.
• $A$ — frontal area, woh cross-section jo rocket wind ke samne present karta hai, ${\text{m}}^2$ mein.

Topic ko inki zaroorat kyun hai: drag-loss integrand $D/m$ hai, toh in sab numbers ki feed hoti hai us loss mein jise tum minimize karna chahte ho. $\rho v^2$ ke dangerous peak ko max-Q kehte hain — dekho Aerodynamic Drag and Max-Q.

10. Har loss ko naam dena, phir unhe jodna

Formulas se pehle, har $\Delta v$ term ka ek-line meaning yahan hai jo tum meet karoge — har ek speed ka ek chunk hai, $\text{m/s}$ mein:

Steering term ka angle $\alpha$ apne picture-anchored definition ka haqdaar hai:

Ab har symbol defined hai, toh loss formulas saaf padhte hain (note karo ki inke andar bare $m$ ka matlab hai §2 se instantaneous $m(t)$):

Grand bookkeeping phir yeh hai: $\Delta v_{\text{ideal}}=\Delta v_{\text{orbit}}+\Delta v_{\text{grav}}+\Delta v_{\text{drag}}+\Delta v_{\text{steer}}$ jahan $\Delta v_{\text{orbit}}$ Orbital insertion and required orbital velocity se set hota hai.

11. Yeh foundations topic ko kaise feed karte hain

Yeh map kaise padhen: har box upar se ek foundation hai, aur ek arrow ka matlab hai "feeds into". Flow ko scattered starting symbols (upar) se neeche single goal tak follow karo: mass, exhaust speed aur $\ln$ milkar ideal budget banate hain; gravity aur angle $\gamma$ sine/cosine se speed-stealing aur path-curving parts mein split hote hain; air density aur speed drag force banate hain; aur integral gravity aur drag pieces ko losses mein wrap karta hai. Sab kuch phir ek node "minimize total loss" par converge hota hai.

Equipment checklist

Har sawaal padho, zyubaan par jawab do, phir reveal karo. Agar koi atkaaye, toh uska section upar se dobara padho.