Systems with variable mass — rocket equation derivation preview

1.4.12 · Physics › Momentum & Collisions

WHY ordinary $F = ma$ kaafi nahi hai

Seedha equation $F = m \frac{d v}{d t}$ yeh assume karta hai ki $m$ constant hai. Newton ke 2nd law ka sahi statement yeh hai:

$F_{ext} = \frac{d p}{d t}$

lekin tumhe ise aise system pe apply karna hoga jiska membership $d t$ ke dauran change na ho. Agar tum laparwaahi se "rocket" ke liye $\frac{d ( m v )}{d t} = m \overset{v}{˙} + \overset{m}{˙} v$ likhte ho, to tum galat sign/term paate ho, kyunki woh $v$ ejected mass ki velocity honi chahiye, rocket ki nahi. Hum ise fix karte hain is cheez pe dhyan dekar ki kya nikalta hai aur kitni tez move karta hai.

HOW derive karo — ek closed system ko $d t$ ke over track karo

Figure — Systems with variable mass — rocket equation derivation preview

Setup (sab ek fixed inertial frame mein):

Time $t$ pe: rocket ka mass $m$ hai, velocity $v$ hai. Total momentum $p (t) = m v$ .
Interval $d t$ ke dauran, rocket ek tiny mass $d m_{ej} > 0$ exhaust eject karta hai.
Rocket mass lose kare: $\frac{d m}{d t} < 0$ , isliye $d m_{ej} = - d m$ .
Exhaust rocket ke relative $u$ speed se peeche jaata hai, isliye ground frame mein uski velocity $v - u$ hai.

Time $t + d t$ pe:

Rocket: mass $m + d m$ (dhyan raho $d m < 0$ ), velocity $v + d v$ .
Abhi release hua exhaust: mass $- d m$ , velocity $v - u$ .

Ideal rocket equation (Tsiolkovsky)

Gravity-free, drag-free space lo: $F_{ext} = 0$ . Tab

$m d v = - u d m \Rightarrow d v = - u \frac{d m}{m}$

Ratio $\frac{m _{0}}{m _{f}}$ mass ratio hai. Kyunki yeh $ln$ ke andar baitha hai, bada $Δ v$ paana brutally expensive hai — tumhe exponentially zyada fuel carry karna padta hai. Yeh hai tyranny of the rocket equation.

Worked examples

Common mistakes

Recall Feynman: ek 12-saal ke bachche ko explain karo

Socho tum ek frictionless skateboard pe ho aur tumhare haath mein bhaari bricks ka stack hai. Jab bhi tum ek brick peeche fekते ho, tum thoda aage roll karte ho — yeh Newton ka "push back" rule hai. Rocket same hai: woh hot gas peeche bahut fast fenkta hai, isliye aage shoot karta hai. Twist yeh hai ki bricks (fuel) tumhara hissa hain, isliye jaise tum unhe fenkte ho tum halke hote ho — aur halke tum pe har baar speed zyada badhti hai. Isliye speed logarithm ki tarah add hoti hai: baad ke throws zyada count karte hain kyunki tumhara less hi bachta hai push karne ke liye.

Flashcards

Rocket pe directly

F = ma

kyun apply nahi kar sakte?

Kyunki uska mass change hota hai;

F = \overset{p}{˙}

ek closed system (rocket + ejected gas) pe apply hona chahiye, sirf shrinking rocket pe nahi.

General variable-mass equation kya hai?

m \frac{d v}{d t} = F_{ext} - u \frac{d m}{d t}

, jahan

u

body ke relative exhaust speed hai.

Rocket ka thrust kya hota hai?

F_{thrust} = - u \frac{d m}{d t} = u ∣ \overset{m}{˙} ∣

Tsiolkovsky rocket equation state karo.

Δ v = u ln \frac{m _{0}}{m _{f}}

Δ v

mein logarithm kyun aata hai?

Har kg burn hone se ever-smaller remaining mass push hota hai, isliye per kg gain compound hota hai —

d m / m

integrate karne se

ln

milta hai.

Rocket equation mein

u

physically kya mean karta hai?

Exhaust speed rocket ke relative, ground speed nahi.

Derivation mein kaunsa term cancel hota hai aur kyun?

Ejected mass ke ground-frame velocity terms

v d m

aur

- v d m

cancel ho jaate hain, sirf relative speed

u

bachta hai.

Earth se rocket lift off karne ki condition kya hai?

Thrust > weight:

u ∣ \overset{m}{˙} ∣ > m_{0} g

(thrust-to-weight ratio > 1).

High

Δ v

itna costly kyun hai?

Mass ratio ek

ln

ke andar hai, isliye required

Δ v

ke saath fuel exponentially badhta hai ("tyranny of the rocket equation").

Mass ratio kya hota hai?

m_{0} / m_{f}

— initial (fuelled) mass divided by final (burnout) mass.

Connections

Conservation of Linear Momentum — closed-system principle jis pe poori derivation depend karti hai.
Newton's Third Law — exhaust rocket ko push karta hai; rocket exhaust ko push karta hai.
Impulse–Momentum Theorem — $F_{ext} d t = d p$ har interval pe use hota hai.
Center of Mass Motion — (rocket+exhaust) ka CoM $F_{ext}$ follow karta hai chahe rocket accelerate kare.
Relative Velocity — isliye $u$ rocket ke w.r.t. measure hota hai.
Logarithms and Exponential Growth — mass ratio ke peeche ka math.