Specific impulse Isp = v_e - g₀ — definition, physical meaning, units

3.3.4 · Physics › Rocket Propulsion

Specific impulse KYA hai?

Aise kyun define karte hain? Engineers chahte the ki ek tiny model rocket aur giant Saturn V ko compare karne ke liye ek number ho, engine ki size se independent. Akela Impulse ( $\int F d t$ ) bade engines ko reward karta hai. Propellant ke weight consumed se divide karne par size hat jaata hai aur sirf efficiency measure hoti hai.

$I_{s p} = v_{e} / g_{0}$ scratch se KAISE derive karein

Definition of impulse se shuru karo ek rocket ke liye jo mass exhaust karta hai.

Step 1 — Thrust from momentum. Time $d t$ mein engine mass $d m = \overset{m}{˙} d t$ eject karta hai effective exhaust speed $v_{e}$ par (rocket ke relative). Har second us gas ko diya gaya momentum thrust hai: $F = \overset{m}{˙} v_{e}$ Yeh step kyun? Newton ka 3rd law: gas ko $v_{e}$ par peeche push karna rocket ko aage push karta hai; force = rate of momentum ejection.

Step 2 — Burn ke dauran total impulse. $v_{e}$ constant maano: $J = \int_{0}^{t_{b}} F d t = \int_{0}^{t_{b}} \overset{m}{˙} v_{e} d t = v_{e} = m_{p} \int_{0}^{t_{b}} \overset{m}{˙} d t = v_{e} m_{p}$ jahan $m_{p}$ burn kiya gaya total propellant mass hai. Yeh step kyun? $v_{e}$ integral se bahar aa jaata hai; baaki mass-flow ka integral bas use ki gayi mass hai.

Step 3 — Propellant weight se divide karo. Us propellant ka weight $W_{p} = m_{p} g_{0}$ hai. Definition se: $I_{s p} = \frac{J}{W _{p}} = \frac{v _{e} m _{p}}{m _{p} g _{0}} = \frac{v _{e}}{g _{0}}$ Yeh step kyun? Propellant mass $m_{p}$ cancel ho jaata hai — yahi exactly reason hai ki $I_{s p}$ size-independent hai. Jo bachta hai woh exhaust speed aur ek fixed constant ka ratio hai.

Units — KYUN seconds?

$I_{s p} = \frac{impulse}{weight}$ par dimensional check karo: $[I_{s p}] = \frac{N\cdots}{N} = s$ Newtons cancel ho jaate hain aur tum seconds ke saath bacha rahte ho.

Note: kyunki $g_{0}$ ek fixed constant hai, $I_{s p}$ seconds mein har jagah same number hai — Earth par, Mars par, ya deep space mein. Kabhi kabhi engineers $g_{0}$ skip kar dete hain aur directly m/s mein $v_{e}$ quote karte hain (ise specific impulse by mass kehte hain, units m/s ya N·s/kg). Dono same engine ko describe karte hain.

Figure — Specific impulse Isp = v_e - g₀ — definition, physical meaning, units

Worked examples

Forecast-then-Verify

Recall Answer padhne se pehle forecast karo

Q: Chemical rockets max $I_{s p} \approx 450 s$ tak pohonchte hain; ion thrusters $I_{s p} \approx 3000 s$ tak pohonchte hain. Kaun zyada thrust deta hai, aur kaun zyada fuel-efficient hai? Apna guess likho, phir open karo.

Verify: Ion thrusters ka $I_{s p}$ bahut zyada hai ⇒ bahut zyada efficient (bada $v_{e}$ , tiny propellant use). Lekin unka $\overset{m}{˙}$ bahut kam hai, isliye $F = \overset{m}{˙} v_{e}$ bahut chhota hai (millinewtons). Chemical rockets: low $I_{s p}$ , lekin enormous $\overset{m}{˙}$ ⇒ liftoff ke liye meganewtons of thrust. High $I_{s p}$ ≠ high thrust.

Common mistakes

Feynman

Recall Ek 12-saal ke bachche ko samjhao

Socho ki rocket fuel ka har kilogram ek coin hai. Jab tum ek coin kharch karte ho, rocket ko kuch waqt ke liye ek certain push milta hai. Specific impulse bas woh number hai jo batata hai ki har coin kitna acha hai — ek coin kitne der aur kitni strongly push kar sakta hai. Ek fancy engine har coin ke badle lambi, strong push deta hai (bada number); ek weak engine chhoti push deta hai. Yeh nahin batata ki tumhare paas kitne coins hain ya tum unhe kitni tezi se kharch karte ho — sirf yeh batata hai ki har coin kitna acha hai.

Active recall

Specific impulse kya hai words mein?

Consume kiye gaye propellant ke unit weight per deliver kiya gaya total impulse. :::

Constant exhaust velocity ke liye

I_{s p}

ka formula?

I_{s p} = v_{e} / g_{0}

. :::

I_{s p}

ke SI units kya hain aur kyun?

Seconds; kyunki impulse (N·s) ÷ weight (N) = s. :::

I_{s p}

formula mein

g_{0}

ki value aur meaning?

9.80665 m/s^{2}

, ek fixed standard-gravity constant jo weight↔mass convert karne ke liye use hota hai, local gravity NAHIN. :::

Kya

I_{s p}

change hoga agar rocket Mars par ho?

Nahin —

g_{0}

ek constant hai, isliye

I_{s p}

ek fixed engine property hai. :::

I_{s p}

v_{e}

recover karo?

v_{e} = I_{s p} g_{0}

. :::

I_{s p}

use karke thrust express karo?

F = \overset{m}{˙} v_{e} = \overset{m}{˙} g_{0} I_{s p}

. :::

Kya higher

I_{s p}

higher thrust guarantee karta hai?

Nahin; thrust ke liye mass-flow

\overset{m}{˙}

bhi chahiye. High-

I_{s p}

ion drives ki thrust tiny hoti hai. :::

Propellant mass se total impulse?

J = I_{s p} g_{0} m_{p}

. :::

Derivation mein propellant mass kyun cancel hota hai?

Impulse

= v_{e} m_{p}

, weight

= g_{0} m_{p}

; divide karne par

m_{p}

cancel hota hai,

I_{s p}

ko size-independent banata hai. :::

Connections

Rocket Propulsion
Tsiolkovsky Rocket Equation — $I_{s p}$ woh $v_{e}$ set karta hai jo $Δ v = v_{e} ln (m_{0} / m_{f})$ drive karta hai
Thrust and Mass Flow Rate — $F = \overset{m}{˙} v_{e}$
Effective Exhaust Velocity
Impulse-Momentum Theorem
Standard Gravity g0