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

3.2.10 · D1 · Physics › Orbital Mechanics & Astrodynamics › Vis-viva equation v² = GM(2 - r − 1 - a) — derivation

Derivation parent page (topic note) par padhne se pehle, vis-viva equation ke har letter ka kuch picture jaisa matlab hona chahiye, sirf ek symbol nahi. Ye page unhe ek ek karke banata hai, ek aisi order mein jahan har ek sirf usse pehle wale par depend karta hai. Hum poori equation tab tak likhenge bhi nahi jab tak usme har letter "earn" nahi ho jaata.

0. Characters ki cast (hum kis cheez ki baat kar rahe hain)

Do cheezein involved hain:

Ek bada body — ek planet ya star — jo almost hilta nahi. Hum iska mass $M$ kehte hain (capital M matlab "massive").
Ek chhota body — ek satellite, moon, ya spacecraft — jo bade ke around chakkar lagata hai. Iska mass $m$ hai (chhota m).

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

Chhota body ek closed loop trace karta hai. Wo loop, aur jis point ke around wo loop karta hai, baaki sab kuch ka stage hai.

$m$ ko yaad rakhna — §6 mein tum dekhoge ki ye har equation se cancel out ho jaata hai, aur yahi wajah hai ki final formula mein satellite ka mass kabhi mention nahi hota.

1. $r$ — tum abhi kahan ho

Neeche figure mein arrow dekho: $r$ planet ke center se satellite tak ek rubber band hai. Jaise jaise satellite move karta hai, rubber band stretch aur shrink hota hai — isliye $r$ har second badalta hai.

Picture: do dots ko join karne wali line ki length.
Topic ko iske zaroorat kyun hai: vis-viva answer karta hai "main yahaan kitni fast ja raha hoon?" — aur "yahaan" exactly wohi hai jo $r$ measure karta hai.

2. $v$ — tum kitni fast ja rahe ho

Picture: satellite ke saath chalte chhote velocity arrow ki length, jo us direction mein point karta hai jahan wo ja raha hai.
Topic ko iske zaroorat kyun hai: $v$ vis-viva ka poora output hai. Baaki sab kuch isi ek number ko pin down karne ke liye exist karta hai.

Notice karo ki jis equation ko hum bana rahe hain wo $v^{2}$ , speed ki square, ke liye solve karta hai, kabhi $v$ ke liye nahi. Ye ek hint hai ki ye equation energy se aati hai (energy mein $v^{2}$ hota hai), jo hum §6 mein dekhenge.

3. $G$ aur $GM$ — gravity ki strength

Kisi bhi energy formula se pehle, hum wo number chahiye jo bataye ki gravity kitni strong hai. Ye yahan se har step mein dikhai deta hai, isliye hum ise ab earn karte hain.

Picture: poore planet ke liye ek single "gravity strength" dial.
Topic ko iske zaroorat kyun hai: $GM$ vis-viva mein appear hone wala ek maatra planet-property hai; ye sab kuch scale karta hai. Astronomers ise directly ek lump mein measure karte hain kyunki $G$ akela poorly known hai aur kisi planet ka $M$ weigh karna mushkil hai — lekin $GM$ seedha orbits dekhne se mil jaata hai.

Toh $GM$ symbol jo tum neeche har formula mein dekhoge iska matlab sirf "gravity strength" hai.

4. Ellipse — ek bound orbit ki shape

Orbits circles nahi hote; ye ellipses hote hain — halke squashed circles. Ek ko describe karne ke liye, hum teen ideas chahiye: focus, semi-major axis, aur eccentricity.

Picture: oval ka sabse lamba diameter ka aadha (figure mein mint line).
Topic ko iske zaroorat kyun hai: derivation ka poora punchline ye hai ki orbit ki energy sirf $a$ par depend karti hai — na uski shape par, na tum kahan ho. Isliye $a$ poore loop ka single "size knob" hai.

Hum story bound ellipse ( $0 \leq e < 1$ ) par banate hain kyunki ye wo hai jo tum draw kar sakte ho — lekin §8 dikhata hai ki wohi vis-viva formula escape aur fly-by cases tak bhi stretch hota hai. Yahi wajah hai ki $e$ matter karta hai chahe wo final equation mein appear na ho.

5. Perigee aur apogee — sabse close aur sabse door ke points

Kyunki $r$ orbit karte waqt change hota hai, ek sabse chhota $r$ aur ek sabse bada $r$ hota hai.

Ek fact jo tumhe aage zaroor yaad rakhna hai:

Picture: figure mein long axis ke do tips — ek planet ke paas, ek door flung.
Topic ko iske zaroorat kyun hai: in do points par velocity purely sideways hoti hai (andar/bahar koi motion nahi), jo energy aur angular momentum likhna easy bana deta hai. Ye derivation ki "easy chairs" hain.

6. Energy — wo quantity jo kabhi nahi badlti (aur jahan $m$ gayab ho jaata hai)

Ye poore topic ka engine hai. Orbiting body ki total energy ke do flavours hain:

$E = kinetic \frac{1}{2} m v^{2} - potential \frac{GM m}{r}$

Dono pieces mein satellite ka mass $m$ hai. Dekho kya hota hai jab hum poori cheez ko $m$ se divide karte hain taaki energy per kilogram mile:

$ε = \frac{E}{m} = \frac{1}{2} v^{2} - \frac{GM}{r}$

$m$ har jagah cancel ho jaata hai — ek 1-tonne satellite aur ek 1-gram bolt ek hi orbit par same $ε$ rakhte hain. Yahi wajah hai ki hum in "specific" (per-unit-mass) quantities ke saath kaam karte hain: ye orbit ko describe karte hain, object ko nahi.

Magic fact (parent page par prove kiya gaya) ye hai ki in donon ka sum frozen hai:

Picture: ek marble ek smooth bowl mein roll kar rahi hai — neeche jaane par speed up hoti hai, upar jaane par slow down, lekin total (motion + height) hamesha same rehta hai kyunki kuch energy nahi churata.
Topic ko iske zaroorat kyun hai: "har jagah same" hi wo cheez hai jo tumhe kisi bhi $r$ par $v$ calculate karne deti hai jab constant pata ho. Literally yahi vis-viva hai.

Ye tumhare prerequisite Conservation of Energy ka payoff hai.

7. Angular momentum — wo "spin" jo kabhi nahi badlta

Ek doosra frozen quantity hai, sirf do apside equations ko saath mein solve karne ke liye use hota hai. Pehle, do speed labels jo hume chahiye honge:

Picture: ek figure skater apni arms andar kheench raha hai — jaise $r$ shrink hota hai, $v$ grow karna padta hai taaki product $r \times v$ fixed rahe. Yahi wajah hai ki perigee fast hai aur apogee slow.
Topic ko iske zaroorat kyun hai: do conserved quantities (energy aur angular momentum) do equations dete hain, jo do unknown speeds $v_{p}$ aur $v_{a}$ ko pin down karte hain. Dekho Conservation of Angular Momentum.

8. Key collapse — energy sirf $a$ par depend karti hai

Jab tum §6 aur §7 ke do conserved quantities ko do apside equations mein daalo aur algebra grind karo (parent page har line karta hai), sab kuch shape-related cancel ho jaata hai aur tum ek astonishingly simple result ke saath bache rehte ho:

Ab $ε$ ke do forms ko equal set karo: $\frac{1}{2} v^{2} - \frac{GM}{r} = - \frac{GM}{2 a}$ 2 se multiply karo aur rearrange karo — ab har letter earned hai, isliye hum finally equation likh sakte hain: $v^{2} = GM (\frac{2}{r} - \frac{1}{a})$

Equation ki picture, aakhirkar

$kitna fast v^{2} = gravity strength GM tum kahan ho \frac{2}{r} - loop ki size \frac{1}{a}$

$\frac{2}{r}$ huge ho jaata hai jab tum paas dive karo (chhota $r$ ) → fast.
$\frac{1}{a}$ orbit ki size se ek baar set hone wala fixed toll hai.
Unhe subtract karo, gravity strength se multiply karo, to tumhara speed-squared milta hai.

Edge cases (same formula, charo regimes)

Prerequisite map

Equipment checklist

Khud ko test karo — daayni taraf cover karo aur har cheez zabanaan bolo reveal karne se pehle.

r

kya measure karta hai, aur kahan se?

Bade body ke center se orbiting body tak ki instantaneous distance — ye har moment change hoti hai.

v

kya represent karta hai, aur equation

v^{2}

kyun use karti hai?

Current speed;

v^{2}

isliye aata hai kyunki vis-viva energy se aati hai, aur kinetic energy mein

\frac{1}{2} v^{2}

hota hai.

G

kya hai, aur

GM

kya hai?

G

universal gravitational constant hai;

GM

ise planet ke mass ke saath ek "gravity strength" number mein bundle karta hai, jo directly orbits se measure hota hai.

Bada mass

M

ellipse par kahan baithta hai?

Ek focus par, kabhi center par nahi.

Semi-major axis

a

ko ek sentence mein define karo.

Ellipse ke across sabse lambi line ki half-length — orbit ki overall size.

e

ke chaar ranges ka kya matlab hai?

e = 0

circle,

0 < e < 1

ellipse,

e = 1

parabola (escape,

a \to \infty

e > 1

hyperbola (fly-by,

a < 0

v_{p}

aur

v_{a}

kya denote karte hain?

Orbiting body ki speed perigee par (sabse fast) aur apogee par (sabse slow), respectively.

Hum "per unit mass" (specific) quantities ke saath kyun kaam karte hain?

Energy aur angular momentum ko

m

se divide karne par satellite ka mass cancel ho jaata hai, isliye numbers orbit ko describe karte hain, object ko nahi.

Derivation ke liye perigee aur apogee mein kya special hai?

Wahaan velocity purely sideways hoti hai, jo energy aur angular momentum likhna easy banata hai; saath hi

r_{p} + r_{a} = 2 a

bhi hota hai.

a

ke terms mein constant specific energy batao.

ε = - GM / (2 a)

— key collapse; energy sirf orbit size par depend karti hai.

Vis-viva escape aur hyperbolic paths ko kaise handle karta hai?

a \to \infty

v^{2} = 2 GM / r

milta hai (escape);

a < 0

v^{2} > 2 GM / r

milta hai (excess speed) — same formula, sirf

a

ka alag sign.

Connections

Conservation of Energy — wo frozen $ε$ jo derivation ko power karta hai.
Conservation of Angular Momentum — $v_{p}$ aur $v_{a}$ ko $h$ ke zariye link karta hai.
Orbital Elements — $a$ , $e$ , $r_{p}$ , $r_{a}$ ki formal definitions.
Kepler's Laws — ellipses aur periods ki geometry.
Escape Velocity — inhi ideas ka $a \to \infty$ limit.
Hohmann Transfer Orbit — jahan tum actually ye sab use karte ho.