Foundations — Payload fraction as function of Δv and Isp

3.3.47 · D1 · Physics › Rocket Propulsion › Payload fraction as function of Δv and Isp

Is page par assume kiya gaya hai ki aapne kuch nahi dekha. Hum har letter, ratio, aur symbol ek ek karke banayenge, aur parent note Payload Fraction jo foundations use karta hai unhe ek ek picture ke saath samjhayenge. End tak, master formula ka har symbol obvious lagega.

1. Rocket actually hai kya (ek mass jo do mein split hoti hai)

Kisi bhi symbol se pehle, ek rocket ko ek mass ka block samjho jo khud ko do dher mein baantne wala hai: ek dher jo woh phenkta hai (peeche se nikalti jali hui gas) aur ek dher jo woh rakhta hai (khali shell aur uska cargo).

Poora topic is ek bar ki bookkeeping hai. Isliye hume iske slices ke naam chahiye.

2. Bar ke teen slices

Figure mein bar dekho. Usse teen labelled pieces mein kaata gaya hai:

Yeh teen hi kyun, aur zyada nahi? Kyunki ek burn ke baad, ek kilogram ke saath sirf do cheezein ho sakti hain: ya toh woh chali gayi (propellant) ya rahi (payload + structure). Koi chautha category nahi hota.

3. Pehle aur baad mein: $m_0$ aur $m_f$

Ek burn ek movie hai jisme ek start frame aur ek end frame hota hai. Hume har frame ki total mass ke liye ek symbol chahiye.

Subscript $0$ ka matlab hai "time zero par" (shuruaat); $f$ ka matlab hai "final". Picture mein lamba bar ($m_0$) chhota bar ($m_f$) ban jaata hai jab fuel pile gayab ho jaati hai.

4. Ratios: "kitna" se "kitna efficient" tak

Physicists ko raw kilograms ki parwah kam hoti hai — ek bada rocket aur ek chhota rocket utne hi achhe ho sakte hain. Jo matter karta hai woh hai fractions: poore ka kitna hissa kaam ka hai? Ratio bas ek bar ko doosre se divide karna hai; iske koi units nahi hote, yeh ek plain number hai. Kuch ratios 0 aur 1 ke beech rehne par majboor hain (poore ka ek hissa, jaise neeche payload fraction); doosre 1 se bade ho sakte hain (ek badi cheez ki chhoti se comparison, jaise neeche mass ratio). Hamesha poochho "kya upar hai?" range assume karne se pehle.

5. Speed kyun aata hai: $\Delta v$

Orbit mein jaane ke liye aapko apni velocity change karni hoti hai. Us change ka symbol hai $\Delta v$.

$\Delta v$ mission ka "cost" kyun hai, distance ya time nahi? Kyunki space ke near-vacuum mein koi friction nahi hota — rocket jo kharcha karta hai kisi destination tak pahunchne mein woh velocity change hai, miles nahi. $\Delta v$ fuel-budget ki currency hai. (Jab aap gravity well mein deep fire karte ho toh bonus energy milti hai — woh hai Oberth Effect — lekin accounting unit phir bhi $\Delta v$ hai.)

6. Engine ki quality: $g_0$, $v_e$, aur $I_{sp}$

Do engines dono fuel jala sakte hain, lekin ek use zyada tez phenkta hai. Tez exhaust = har kg fuel se zyada push = better engine.

7. Woh tool jo sab ko jodta hai: natural log aur $e$

Parent formula mein $\ln$ aur $e$ hain. Yeh logon ko dara dete hain, isliye hum inhe zero se banate hain.

$\ln$ actually kahan se aata hai

Chalo rocket ko ek tiny bit mass khoते hue dekhen aur $\ln$ ko khud appear hote dekhen.

HUM KYA KARTE HAIN: rocket ko kisi instant par track karo jab uski mass $m$ hai aur speed $v$ hai. Ek tiny slice of time mein woh gas ka ek tiny mass $d{m}_{\text{ex}}$ exhaust speed $v_e$ par (peeche) phenkta hai. Conservation of momentum kehta hai rocket ko jo push milti hai woh us gas ke carried momentum ke barabar hai:

$mdv=v_e(-dm)$

Yahan $dm$ rocket ki apni mass mein (negative) change hai — woh mass khota hai, isliye $-dm$ ejected positive lump hai. YEH EQUATION KYUN: yeh bas ek tiny puff ke liye "momentum in = momentum out" hai.

AAGE HUM KYA KARTE HAIN: do variables alag karo taaki har side par sirf ek letter ho, phir full state se empty state tak har tiny puff ko add (integrate) karo:

$dv=-v_e\frac{dm}{m}⟹{\int }_0^{\Delta v}dv=-v_e{\int }_{m_0}^{m_f}\frac{dm}{m}$

$m$ SE DIVIDE KYUN KARTE HAIN: kyunki har puff ek rocket ko push karta hai jo pichli baar se halka hota hai — ek puff ka effect current mass par depend karta hai, isliye $dm/m$ (fractional mass loss) natural quantity hai. Kai $dm/m$ pieces ko jodna exactly wahi hai jo ek logarithm produce karta hai.

INTEGRAL KYA DETA HAI: $dm/m$ ka running total $\ln m$ hai, isliye

$\Delta v=-v_e[\ln m{]}_{m_0}^{m_f}=-v_e(\ln m_f-\ln m_0)=v_e\ln \left(\frac{m_0}{m_f}\right).$

$v_e=I_{sp}{g}_0$ swap karne par woh form milti hai jo parent use karta hai:

Toh $\ln$ decoration nahi hai — yeh ek hamesha halke hote rocket ko, ek puff at a time, push karne ki accumulated cost hai.

8. Jab ek rocket kaafi nahi hota: staging

Agar ek bar kaam nahi kar sakta (parent ke Example 1 mein aap dekhoge woh negative ho jaata hai), toh beech mein dead weight drop karo aur halke hokar aage bado.

Foundations kaise chain karte hain ek saath

Ise ek dependency ladder ki tarah padho — har rung ko sense banane ke liye neeche wala rung chahiye:

• $\lambda$ (payload fraction, goal) chahiye → $m_0$, $m_{\text{payload}}$, aur $ϵ$.
• $ϵ$ chahiye → $m_{\text{structure}}$ aur $m_{\text{propellant}}$ (slices).
• Mass ratio $m_0/m_f$ chahiye → $m_0$, $m_f$, aur $\zeta$ se link $m_0/m_f=1/(1-\zeta )$ ke through.
• Tsiolkovsky equation chahiye → mass ratio, $\Delta v$, $I_{sp}$, $g_0$, aur $\ln$/$e$ pair.
• Tsiolkovsky ko $ϵ$ ke saath solve karna produce karta hai → master payload-fraction formula.
• Staging escape hatch hai jab kisi single $\lambda$ ka result negative aata hai.

Equipment checklist

Khud test karo — right side cover karo aur jawab do: