Vis-viva equation v² = GM(2 - r − 1 - a) — derivation

3.2.10 · Physics › Orbital Mechanics & Astrodynamics

YE HAI KYA?

Circle: $r = a \Rightarrow v^{2} = GM / r$ (constant speed).
Ellipse: $a$ finite hai, $v$ $r$ ke saath badhta-ghatta hai (perigee par fast, apogee par slow).
Parabola (escape): $a \to \infty \Rightarrow v^{2} = 2 GM / r$ .
Hyperbola: $a < 0 \Rightarrow v^{2} > 2 GM / r$ (excess speed).

TOTAL ENERGY SIRF $a$ PAR HI KYU DEPEND KARTI HAI?

DERIVE KAISE KAREIN — First Principles se

Step 1 — Total energy likho

Ek orbiting body ki specific energy (per unit mass): $ε = kinetic \frac{1}{2} v^{2} - potential \frac{GM}{r}$

Ye step kyun? Gravitational PE per unit mass $- GM / r$ hoti hai (infinity par zero, bound hone par negative). KE per unit mass $\frac{1}{2} v^{2}$ hai. Inका sum conserved hai kyunki gravity koi non-conservative work nahi karta.

Step 2 — Perigee aur apogee par $ε$ evaluate karo

Apsides par (perigee aur apogee) velocity purely tangential hoti hai ( $r$ ke perpendicular), isliye $v = v_{p}$ ya $v = v_{a}$ cleanly milta hai. Distances: $r_{p} = a (1 - e), r_{a} = a (1 + e)$

Ye step kyun? Energy har jagah same hai, lekin do apsides do aasaan equations dete hain jo hum saath solve kar sakte hain. Inhe choose karne se radial velocity component hat jaata hai.

$\varepsilon = \frac12 v_p^2 - \frac{GM}{r_p} = \frac12 v_a^2 - \frac{GM}{r_a} \tag{1}$

Step 3 — Conservation of angular momentum use karo

Apsides par $v ⊥ r$ hai, isliye specific angular momentum simply ye hai: $h = r_p v_p = r_a v_a \;\Rightarrow\; v_a = v_p\frac{r_p}{r_a} \tag{2}$

Ye step kyun? Angular momentum bhi conserved hai. Do conserved quantities (energy + $L$ ) aur do unknowns ( $v_{p}, v_{a}$ ) ke saath hum sab kuch solve kar sakte hain.

Step 4 — $v_{p}^{2}$ solve karo

(2) ko (1) mein substitute karo: $\frac{1}{2} v_{p}^{2} - \frac{GM}{r _{p}} = \frac{1}{2} v_{p}^{2} \frac{r _{p}^{2}}{r _{a}^{2}} - \frac{GM}{r _{a}}$

$v_{p}^{2}$ terms aur $GM$ terms group karo: $\frac{1}{2} v_{p}^{2} (1 - \frac{r _{p}^{2}}{r _{a}^{2}}) = GM (\frac{1}{r _{p}} - \frac{1}{r _{a}})$

Left bracket ko difference of squares ki tarah factor karo aur right side ko common denominator par laao: $\frac{1}{2} v_{p}^{2} \cdot \frac{( r _{a} - r _{p} ) ( r _{a} + r _{p} )}{r _{a}^{2}} = GM \cdot \frac{r _{a} - r _{p}}{r _{p} r _{a}}$

$(r_{a} - r_{p})$ cancel karo: $\frac{1}{2} v_{p}^{2} \frac{r _{a} + r _{p}}{r _{a}^{2}} = \frac{GM}{r _{p} r _{a}}$

Toh $v_{p}^{2} = \frac{2 GM r _{a}}{r _{p} ( r _{a} + r _{p} )}$

Ye step kyun? Pure algebra hai, lekin difference-of-squares cancellation hi hai jo ise kuch clean tak collapse karti hai.

Step 5 — $r_{a} + r_{p} = 2 a$ plug in karo

Kyunki $r_{p} = a (1 - e)$ aur $r_{a} = a (1 + e)$ hain, unka sum $2 a$ hai. Tab $v_{p}^{2} = \frac{2 GM r _{a}}{r _{p} \cdot 2 a} = \frac{GM r _{a}}{a r _{p}}$

Step 6 — Energy nikalo, phir generalize karo

Ab perigee par $ε$ compute karo: $ε = \frac{1}{2} v_{p}^{2} - \frac{GM}{r _{p}} = \frac{GM r _{a}}{2 a r _{p}} - \frac{GM}{r _{p}} = \frac{GM}{r _{p}} (\frac{r _{a}}{2 a} - 1)$

$r_{a} = 2 a - r_{p}$ use karo: $ε = \frac{GM}{r _{p}} (\frac{2 a - r _{p}}{2 a} - 1) = \frac{GM}{r _{p}} (- \frac{r _{p}}{2 a}) = - \frac{GM}{2 a}$

Ye hai bada result: $ε = - \frac{GM}{2 a}$ — energy sirf $a$ par depend karti hai. ✅

Step 7 — Vis-viva padho

Kyunki $ε$ constant hai, kisi bhi point $r$ par speed $v$ ke saath: $\frac{1}{2} v^{2} - \frac{GM}{r} = - \frac{GM}{2 a}$

2 se multiply karo aur rearrange karo: $v^{2} = GM (\frac{2}{r} - \frac{1}{a})$

Figure — Vis-viva equation v² = GM(2 - r − 1 - a) — derivation

Worked Examples

Common Mistakes

Recall Feynman: 12-saal ke bachche ko explain karo

Imagine karo ek ball ek stretchy string par tumhare around oval mein ghoom rahi hai. Jab wo paas aati hai toh fast ghoomti hai; jab door drift karti hai toh slow jaati hai — lekin total "oomph" (speed-energy + height-energy) kabhi nahi badlta. Vis-viva equation bas ek recipe hai: mujhe batao ball abhi kitni door hai aur uska poora loop kitna bada hai, aur main tumhe exactly bataoonga wo kitni fast ja rahi hai. "Loop ki size" hi sirf wo cheez hai jo total oomph set karti hai.

Flashcards

Vis-viva equation mein

a

kya represent karta hai?

Orbit ka semi-major axis (poori orbit ki ek fixed property).

Vis-viva equation mein

r

kya represent karta hai?

Central body ke center se orbiting body tak ki instantaneous distance.

a

ke terms mein specific orbital energy batao.

ε = - GM / (2 a)

— sirf orbit size par depend karta hai.

Vis-viva se escape speed derive karo.

a \to \infty

set karo toh

1/ a \to 0

v^{2} = 2 GM / r

, isliye

v_{esc} = 2 GM / r

Circular orbit ke liye vis-viva reduce hokar kya banta hai?

v = GM / r

, kyunki

r = a

Perigee aur apogee par energy same kyun hoti hai?

Gravity conservative hai; total mechanical energy orbit par har jagah conserved hoti hai.

Derivation mein kaun se do conservation laws use hote hain?

Conservation of energy aur conservation of angular momentum.

Hyperbolic orbit ke liye

a

ka sign kya hota hai?

Negative, jo

v^{2} > 2 GM / r

deta hai (excess hyperbolic speed).

Same radius par escape aur circular speed ka relationship kya hai?

v_{esc} = 2 v_{c i r c}

GM

ko kya kehte hain aur aksar kaun sa symbol use hota hai?

Standard gravitational parameter,

μ

Connections

Conservation of Energy — poori derivation ka engine.
Conservation of Angular Momentum — perigee/apogee speeds ko link karta hai.
Kepler's Laws — third law bhi period ko $a^{3}$ se jodta hai.
Hohmann Transfer Orbit — $Δ v$ nikalne ke liye vis-viva do baar use karta hai.
Escape Velocity — $a \to \infty$ limit.
Orbital Elements — $a$ , $e$ , $r_{p}$ , $r_{a}$ ki definitions.