Foundations — Combined maneuvers — optimal split between plane change and velocity change

3.2.23 · D1 · Physics › Orbital Mechanics & Astrodynamics › Combined maneuvers — optimal split between plane change and

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1. Velocity ek arrow ki tarah (sirf number nahi)

Topic ko yeh kyun chahiye: ek plane change velocity ke arrow ki direction badalta hai jabki (aksar) uski length wahi rehti hai; ek burn jo circularize karta hai uski length badalta hai. Agar tum velocity ko sirf ek number ki tarah sochoge, toh literally tum in dono ke beech ka difference dekh hi nahi sakte — tumhe arrow chahiye.

2. Do arrows ko subtract karna: ek maneuver asal mein hota kya hai

Arrows subtract karna dikhta kaise hai? ${\vec{v}}_1$ aur ${\vec{v}}_2$ ko tail-to-tail rakho (unhe ek hi point se shuru karo). Arrow jo ${\vec{v}}_1$ ki tip se ${\vec{v}}_2$ ki tip tak drawn hai, woh $\overrightarrow{\Delta v}$ hai — woh missing side jo triangle close karti hai.

3. Do arrows ke beech angle $\theta$ (aur total tilt $i$)

Picture: $\theta =0$ matlab dono arrows ek hi direction mein point kar rahe hain (koi turn nahi, sirf possible length change). $\theta =90^{\circ }$ matlab naya arrow purane ke sideways point kar raha hai. Bada $\theta$ = sharper turn = gap bridge karne ke liye lamba $\Delta v$ arrow.

Topic ko yeh kyun chahiye: $\theta$ (aur uska total $i$) woh dial hai jise tum optimize kar rahe ho. Poora "optimal split" ka sawaal yeh hai: ek required total tilt $i$ diya ho, kitna turn burn 1 par karo aur kitna burn 2 par? Dekho Plane Change Maneuvers.

4. Cosine aur Law of Cosines — angle length mein kyun aata hai

5. Dot product — formula ke peeche ki machine

Topic ko yeh kyun chahiye: yeh woh algebra hai jo hamare arrows ke liye Law of Cosines derive karta hai. $\overrightarrow{\Delta v}$ ki length paane ke liye hum isse khud se dot karte hain: $\Delta v^2=\overrightarrow{\Delta v}\cdot \overrightarrow{\Delta v}=({\vec{v}}_2-{\vec{v}}_1)\cdot ({\vec{v}}_2-{\vec{v}}_1)=v_2^2-2{\vec{v}}_1\cdot {\vec{v}}_2+v_1^2.$ Phir ${\vec{v}}_1\cdot {\vec{v}}_2=v_1{v}_2\cos \theta$ middle term ko angle mein convert kar deta hai — exactly boxed formula reproduce ho jaata hai. Dot product woh machine hai jo arrows ko us formula mein convert karta hai.

6. Pure-turn case: $2v\sin (\theta /2)$ kahan se aata hai

Ab woh special case lo jahan burn speed nahi badalta, sirf direction. Matlab before aur after arrows ki length same hai: $v_1=v_2=v$. Yeh crucial assumption hai — isse drop karo toh neecha wala tidy formula hold nahi karta.

$v_1=v_2=v$ boxed formula mein substitute karo: $\Delta v=\sqrt{v^2+v^2-2vv\cos \theta }=v\sqrt{2-2\cos \theta }.$

Half-angle kahan se aata hai? Kyunki dono sides equal length $v$ hain, velocity triangle isosceles hai (do equal sides). Tip se seedha line beech mein neeche daalo: yeh tip-angle $\theta$ ko do equal halves $\theta /2$ mein split karta hai, aur base ($\Delta v$) ko bhi do equal halves mein split karta hai — woh even split is liye forced hai kyunki dono sides equal hain, sirf convenient nahi hai. Figure s05 dekho: har half ek right triangle hai jiska hypotenuse $v$ hai aur jiska "opposite" side $\frac{1}{2}\Delta v$ hai. Sine ki definition se, $\sin \left(\frac{\theta }{2}\right)=\frac{\frac{1}{2}\Delta v}{v}⟹\boxed{\Delta v=2v\sin \left(\frac{\theta }{2}\right)}$

Check: $\theta =0$ par, $\sin 0=0$, toh $\Delta v=0$ — koi turn nahi, koi cost nahi. Jaise $\theta$ badhta hai cost badhti hai, aur crucially yeh $v$ ke proportional hai: jitna tez jaa rahe ho, utna zyada turn cost karta hai.

7. Orbital speeds: $v$ ek jagah bada kyun, doosri jagah chhota kyun

Topic ko yeh kyun chahiye: pure-turn cost $2v\sin (\theta /2)$ turn ke waqt speed $v$ ke proportional hai. Toh fast near-point par turning brutal hai, slow far-point par cheap hai. Yahi ek fact hai jo optimal split ko apoapsis pe almost saara plane change dhekelne par majboor karta hai. Do circular orbits ke beech transfer jo yeh fast/slow burns set up karta hai woh Hohmann Transfer Orbit hai; far point ko aur bhi door push karna taaki aur bhi saste mein turn ho sake woh Bi-elliptic Transfer hai.

8. Derivative $\frac{d}{ds}$ — "optimal" kaise find hota hai

Topic ko yeh kyun chahiye: total cost $\Delta v_{\text{tot}}(s)$ ko split $s$ ke against plot karo aur yeh ek lowest point tak dip karta hai — sabse cheap split. Woh lowest point exactly wahan hai jahan slope zero hai, isliye parent $\frac{d\Delta v_{\text{tot}}}{ds}=0$ set karta hai. Woh condition zor se padhne par milta hai "ek aur degree turn ki marginal cost dono burns par equal hai" — ek degree us burn ki taraf shift karo jo sasta ho jab tak woh balance na ho jayein.

Prerequisite map

Equipment checklist

Khud ko test karo — sirf tab reveal karo jab zor se answer de chuke ho.