Foundations — Pitch program — open-loop pitch-over

3.4.14 · D1 · Physics › Rocket Flight Mechanics › Pitch program — open-loop pitch-over

$\dot{\gamma }=-\frac{g\cos \gamma }{v}$ padhne se pehle, iske har letter ka matlab kuch aisa hona chahiye jo tum dekh sako. Yeh page har ek cheez ko scratch se build karta hai, us order mein jis order mein woh ek doosre par depend karte hain.

0. Woh ek picture jis par sab kuch tika hai

Is topic mein sab kuch ek flat vertical slice of the world mein hota hai: ground ek horizontal line hai, "upar" vertical line hai, aur rocket is slice mein move karta hua ek single dot hai. Hum sideways-out-of-page motion ko bilkul ignore karte hain.

Point kyun? Kyunki trajectory shape ke liye hum sirf yeh care karte hain ki dot kahan jaata hai, rocket kitna lamba hai yeh nahi. Uski shape sirf drag aur steering ke liye matter karti hai, jo hum alag se handle karte hain.

1. Velocity arrow $\vec{v}$ aur uski speed $v$

Rocket move kar raha hai. Kisi bhi instant par, dot se ek arrow draw karo jo dikhaye ki woh kis taraf ja raha hai aur kitni tezi se.

Picture: dot se nikalta ek blue arrow. Lamba arrow = tez; chhota arrow = slow. Is arrow ka tilt agla symbol hai.

Topic ko yeh kyun chahiye: orbit is arrow par ek demand hai — yeh eventually lamba (~7.8 km/s) aur flat (horizontal) hona chahiye. Poora pitch program is ek arrow ko steer karne ke baare mein hai.

2. Flight-path angle $\gamma$

Velocity arrow dekho. Woh kitni steeply climb kar raha hai? Woh tilt, horizontal se upar measure kiya gaya, $\gamma$ ("gamma") hai.

Horizontal se kyun measure karein (vertical se nahi)? Kyunki "orbit" ka matlab hai $\gamma =0$. Horizontal ko zero choose karne se target ek clean number ban jaata hai, aur poora ascent $\gamma$ ke $90^{\circ }$ se $0^{\circ }$ ki taraf shrink hone ki story hai.

3. Sine aur cosine — ek arrow ko up-aur-along mein split karna

Tilted velocity arrow ke saath physics karne ke liye humein ise (aur gravity ko) ek horizontal part aur ek vertical part mein todna hoga. Yeh splitting karne ke tools sine aur cosine hain.

Koi bhi arrow lo length $L$ ka jo horizontal se $\gamma$ angle par tila hua ho. Ise ek right triangle mein dalho:

Yeh do hi kyun, kuch aur kyun nahi? Ek right triangle mein exactly do legs hote hain; $\cos$ tumhe woh deta hai jo angle ke paas hai (ground ke along) aur $\sin$ woh deta hai jo uske saamne hai (upar). Yeh unique dial hai jo "ek arrow $\gamma$ se tila hua" ko "kitna sideways vs kitna upar" mein convert karta hai.

Extremes check karo:

• $\gamma =90^{\circ }$ par (seedha upar): $\cos 90^{\circ }=0$ (horizontal nahi), $\sin 90^{\circ }=1$ (sab vertical). ✓
• $\gamma =0^{\circ }$ par (horizontal): $\cos 0^{\circ }=1$ (sab horizontal), $\sin 0^{\circ }=0$ (vertical nahi). ✓

4. Gravity $g$ aur uske do components

$g$ woh force hai jisse planet har kilogram ko neeche ki taraf kheenchta hai: Earth ki surface ke paas $g\approx 9.8{\text{m/s}}^2$. Hamare picture mein yeh hamesha seedha neeche point karta hai.

Lekin velocity arrow tila hua hai, isliye gravity ke arrow par do effects hain: ek jo climb ko slow karta hai ($\vec{v}$ ke along) aur ek jo arrow ko bend karta hai ($\vec{v}$ ke across).

Velocity arrow ke against kyun split karein (ground ke against nahi)? Kyunki arrow ke along force sirf uski length (speed) badalta hai, jabki arrow ke across force sirf uska direction (angle) badalta hai. Is tarah split karne se woh do sawaal alag ho jaate hain jo topic poochhta hai: "kitni tezi se?" aur "kis taraf?"

5. Mass $m$, thrust $T$, drag $D$, aur force

Picture: dot par, aage ek green arrow = thrust, peechhe ek red arrow = drag, neeche ek yellow arrow = weight $mg$.

Mass har jagah kyun aata hai: Newton kehta hai push = mass × (velocity ke change ki rate). Zyada mass → same push arrow ko kam bend ya speed karta hai. Isliye $m$ turn law se cancel ho jaata hai (dono sides par aata hai) — direction ka turning mass se nahi, sirf speeding up se care karti hai.

6. Dot — "rate of change" notation

Parent likhta hai $\dot{v}$ aur $\dot{\gamma }$. Upar wala dot ek shorthand hai.

Picture: agar $\dot{\gamma }$ negative hai, toh velocity arrow neeche tilt ho raha hai (horizontal ki taraf) jaise seconds guzarte hain. Number jitna bada, arrow utni tezi se swing karta hai.

Topic $\dot{\gamma }$ par kyun rehta hai: poora pitch program ek statement hai arrow kitni tezi se tilt hota hai ke baare mein. $\dot{\gamma }<0$ exactly hai "nose pitch over ho raha hai."

7. Radians — angles ek strange unit mein kyun measure hote hain

$\dot{\gamma }$ radians per second mein nikla, degrees mein nahi. Radians angle measure karne ka "natural" tarika hai jab woh lengths aur speeds se multiply ho.

Yahaan radians kyun? Turn equation $\dot{\gamma }=-g\cos \gamma /v$ mein ek angle-rate speeds aur accelerations ke saath mix hoti hai. Woh algebra tabhi clean hoti hai jab angle radians mein ho (taaki arc = angle × radius bina extra factors ke hold kare). Human intuition ke liye hum end mein hi degrees mein convert karte hain.

8. Thrust-to-weight ratio $n$

Parent set karta hai $a=T/m$ aur $n=a/g$. Ise build karo:

Picture: $n=1$ → engine weight exactly cancel karta hai (rocket hover karta hai). $n=1.3$ → $0.3g$ ka net upward push; rocket upar accelerate karta hai. Dekho Thrust-to-weight ratio.

Yeh kyun matter karta hai: $n$ decide karta hai speed kitni tezi se badhti hai, jo (turn law mein $v$ ke through) decide karta hai turn kitni tezi se slow down karta hai. Isliye $n$ aur kick angle milkar poora ascent shape fix karte hain.

Yeh topic ko kaise feed karte hain

Yeh poori chain the pitch program mein terminate hoti hai, aur aage Gravity turn trajectory se connect hoti hai, aur jab feedback add ho jaata hai toh Closed-loop ascent guidance (PEG / IGM) se.

Equipment checklist

Right side cover karo aur parent note padhne se pehle har ek ka jawab do.