Foundations — Electric propulsion — thrust, power, Isp trade-off

3.3.40 · D1 · Physics › Rocket Propulsion › Electric propulsion — thrust, power, Isp trade-off

Is page pe assume kiya gaya hai ki tumne kuch nahi dekha. Pehle tum parent topic padh sako, uske liye tumhare paas har ek letter hona chahiye. Hum ek ek karke build karenge, har ek pichle ke upar.

1. Skateboard wali picture (poora subject ek image mein)

Neeche jo bhi hai, woh sab ek mental picture mein hai: ek person frictionless skateboard pe baith ke balls throw kar raha hai.

2. Symbols ko build karna, ek ek karke

2.1 Mass — $m$

Topic ko iske liye kyun chahiye: propellant (balls) hi mass hai. Har equation count karti hai ki ship se kitni mass jaati hai.

2.2 Time aur tiny slice — $t$, $dt$

$dt$ ko slow-motion video ke ek frame ki tarah socho. Hum poochhenge "ek frame mein kitna hua?" aur isi se neeche wale har rate ka janam hoga.

2.3 Speed aur velocity — $v$

Velocity speed mein ek direction add karta hai. Hamare convention ke hisaab se, forward velocity $+v$ hai aur backward $-v$ hai. Balls backward fly karti hain, isliye unki velocity mein minus sign aata hai; ship forward move karti hai, isliye uski velocity positive hai. Is topic mein yeh opposite directions hi poori baat hain.

2.4 Exhaust speed — $v_e$

Topic ko iske liye kyun chahiye: $v_e$ electric propulsion mein sabse important number hai. Chemical rockets $v_e\approx 4500\text{m/s}$ ke paas stuck hain; electric wale $v_e\approx 30,000\text{m/s}$ tak pahuncht jaate hain. Poora subject yeh hai ki "bade $v_e$ ke liye tumhe kya milta hai?"

2.5 Mass flow rate — $\dot{m}$

Ise har second kitni balls throw karte ho ke roop mein socho. Har second ek 0.5 kg ball throw karo → $\dot{m}=0.5\text{kg/s}$.

2.6 Momentum — $p$

Topic ko iske liye kyun chahiye: is subject ka sabse gehra law yeh hai ki momentum conserved hota hai — balls ka backward momentum $(-)$ bilkul ship ke forward momentum $(+)$ se cancel hona chahiye. Yeh exactly Newton's Third Law ka number form hai.

2.7 Force aur thrust — $F$

Key idea, upar ki pictures se build hua:

"Per second" kyun? Force ek continuous push hai. Ek ball phenkna ek single dhakka deta hai; balls har second phenkna ek steady push deta hai — aur steady push per second exactly force hai.

2.8 Kinetic energy aur power — $E$, $P$

2.9 Jet power $P_{jet}$ aur $F$, $v_e$, aur $P$ ke beech ka seesaw

Pehle, us specific power ka naam rakho jis ki hume parwah hai.

Ab iska formula build karo, ek ek honest step se.

Step A — ek second mein throw ki gayi mass ki energy. Ek second mein hum $\dot{m}$ mass throw karte hain (yahi $\dot{m}$ ka matlab hai). Iska har kilogram $v_e$ speed se move karta hai, toh $E=\frac{1}{2}m{v}^2$ se uski kinetic energy $\frac{1}{2}\dot{m}{v}_e^2$ hai.

Step B — energy-per-second ko power mein convert karo. Woh energy har second nikalti hai, toh $P=dE/dt$ (energy per second) se jet power bas wohi number hai: $P_{jet}=\frac{dE}{dt}=\frac{1}{2}\dot{m}{v}_e^2.$ Yeh kyun allowed hai: humne per-mass energy $\frac{1}{2}{v}_e^2$ ko mass flow $\dot{m}$ (mass per second) se multiply kiya, aur "energy per mass × mass per second = energy per second = power." Kuch assume nahi kiya gaya.

Step C — $\dot{m}{v}_e$ ko $F$ se replace karo. §2.7 dekho: thrust $F=\dot{m}{v}_e$ hai. Toh formula ke andar hum $\dot{m}$ ke saath ek $v_e$ ko group kar sakte hain: $P_{jet}=\frac{1}{2}\dot{m}{v}_e^2=\frac{1}{2}\underset{=F}{\underbrace{(\dot{m}{v}_e)}}{v}_e=\frac{1}{2}F{v}_e.$ $\frac{1}{2}$ kahan se aaya: yeh wahi $\frac{1}{2}$ hai jo $E=\frac{1}{2}m{v}^2$ mein hai — energy, momentum nahi, woh factor carry karta hai. Thrust (momentum se) mein $\frac{1}{2}$ nahi hota; power (energy se) mein hota hai. Substitution bas us original $\frac{1}{2}$ ko untouched carry karta hai.

2.10 Efficiency — $\eta$

Topic ko iske liye kyun chahiye: real thrusters power waste karte hain, aur waste hissa heat ban jaata hai jise spacecraft ko Thermal Control (radiators) se dump karna padta hai.

2.11 Standard gravity aur specific impulse — $g_0$, $I_{sp}$

2.12 Economy law — $\Delta v=v_e\ln (m_0/m_f)$

3. Prerequisite map

Equipment checklist

Connections

• 3.3.40 Electric propulsion — thrust, power, Isp trade-off (Hinglish) — parent topic jiske liye yeh page prepare karta hai.
• Newton's Third Law — momentum peeche phenkna = ship aage push hona.
• Specific Impulse — fuel-economy score $I_{sp}=v_e/g_0$.
• Tsiolkovsky Rocket Equation — jahan $\Delta v$, $m_0$, $m_f$, $\ln$ aate hain.
• Chemical vs Electric Propulsion — chemical rockets ke liye $v_e$ cap kyun hai.
• Ion and Hall Thrusters — hardware jo high $v_e$ tak pahunchta hai.
• Spacecraft Power Systems — available power $P$ set karta hai.
• Thermal Control (radiators) — $(1-\eta )P$ waste heat dump karta hai.