Visual walkthrough — Forces on a rocket in flight — thrust, aerodynamic (normal, axial), gravity

3.4.3 · D2 · Physics › Rocket Flight Mechanics › Forces on a rocket in flight — thrust, aerodynamic (normal,

Hum sirf itna assume karte hain: ek arrow ek force represent kar sakta hai (lamba = zyada strong, jis direction mein push karta hai usi taraf pointing), aur forces tip-to-tail add hote hain jaise ek walk.

Step 0 — Do arrows jinhein tum kabhi confuse mat karna

KYA: hum ek picture mein nose direction aur travel direction mark karte hain, $\alpha$ ka positive sense ek curved arrow ke roop mein draw karke. KYU: neeche ka har force in dono lines mein se ek ke relative measure hota hai, to hum inhe — aur inke sign convention ko — pehle fix karna padega. PICTURE: neeche ke do arrows; curved orange arrow $\alpha$ ka positive (counter-clockwise) sense dikhata hai, $\hat{v}$ se upar $\hat{b}$ tak.

Dekho Angle of Attack and Stability kyun $\alpha$ yeh bhi decide karta hai ki rocket seedha rehna chahta hai ya tumble karna chahta hai.

Step 1 — Hawa ek HI force se peeche dhakelta hai

KYA: single aerodynamic resultant $\vec{R}$ ko rocket par acting draw karo. KYU: agar hum honest single force se shuru karein, to charon component names ($L,D,A,N$) same arrow ke bas do alag shadows ban jaate hain — aur conversion formulas seedhe nikal aate hain. PICTURE: ek plum arrow $\vec{R}$ tilted body par push karta hua.

Step 2 — Pehla shadow: lift aur drag (WIND frame)

Apna ruler velocity $\hat{v}$ ke saath align karo. Hume ek doosri direction chahiye uske right angles par, aur hume fix karna hoga kaunsa right angle — warna lift ka sign ambiguous rahega.

Ab single force $\vec{R}$ ko $\hat{v}$ aur $\hat{n}$ par drop karo.

KYA: hum $\vec{R}$ ko do arrows mein split karte hain velocity line ke off measure karke, perpendicular side aur signs pin down karke. KYU yeh do aur koi nahi? Kyunki $D$ aur $L$ airspeed par cleanly depend karte hain: har ek $\frac{1}{2}\rho v^{2}S$ times ek coefficient hai. Yeh aerodynamics ke liye natural frame hai. PICTURE: teal $\hat{v}$ line, orange $D$ uske along backward, teal $L$ uske square $+90^{\circ }$ side par.

Step 3 — Doosra shadow: axial aur normal (BODY frame)

Ab apna ruler $\hat{v}$ ki jagah body axis $\hat{b}$ ke saath rakho. Body ka apna perpendicular unit arrow $\hat{p}$ wahi $+90^{\circ }$ counter-clockwise rule use karke banao: $\hat{p}$ woh $\hat{b}$ hai jo left mein $90^{\circ }$ rotate hua ho. ($\hat{n}$ aur $\hat{p}$ ke liye identical turning rule use karna hi Step 4 mein rotation ko clean banata hai.) Wohi force $\vec{R}$ inpar drop karo.

KYA: wohi arrow $\vec{R}$, nose direction par re-shadow kiya gaya. KYU: structure — fins, skin, stages ke beech ke joints — tube ke along aur across force feel karta hai, wind ke along nahi. $A$ rocket ko compress karta hai; $N$ use bend aur rotate karne ki koshish karta hai. Yeh woh frame hai jis mein airframe size karne wala engineer kaam karta hai. PICTURE: wohi plum $\vec{R}$, ab orange $A$ body ke down aur teal $N$ across.

Dono frames ek arrow describe karte hain: $-D\hat{v}+L\hat{n}=A\hat{b}+N\hat{p}$. Frames exactly angle $\alpha$ se differ karte hain — kyunki body velocity se $\alpha$ tilt hai. Yeh ek fact poora conversion hai.

Step 4 — Ek frame ko doosre par rotate karna

Chaar building blocks (sab "$\hat{b}$ woh $\hat{v}$ hai jo $+\alpha$ se turn hua hai" se). Do unit arrows jo ek angle se alag hain, unka dot product us angle ke cosine ke barabar hota hai. Har pair work out karte hain: $\hat{b}\cdot \hat{v}=\cos \alpha ,\hat{b}\cdot \hat{n}=-\sin \alpha ,\hat{p}\cdot \hat{v}=\sin \alpha ,\hat{p}\cdot \hat{n}=\cos \alpha .$

KYA: hum vector sum ko tilted body arrows par project karte hain dot products le kar. KYU trig, aur exactly $\cos$ aur $\sin$ kyun? Kyunki "ek unit arrow ka kitna hissa doosre ke along point karta hai" cosine hi hai unke beech ke angle ka; ek perpendicular pair sine deta hai. Yeh exactly woh sawaal hai "kitna fraction of $L$ ya $D$ body axis par land karta hai?"

Axial — $\vec{R}$ ko $\hat{b}$ par project karo, ek term at a time:

Yeh axial component hai $A$ positive nose $+\hat{b}$ ke down measure karke. Kyunki hawa ka resultant actually backward push karta hai (yeh forward flight ka virodh karta hai), uska along-body part is convention mein negative aata hai. Parent note retarding axial load ka magnitude report karta hai — sign flip karo (i.e. positive $A$ aft point karna define karo, woh direction jahan force actually point karta hai) aur familiar mil jaata hai

i.e. hawa kitni strongly tube ke along backward press karti hai. Dono terms yahan positive hain kyunki drag aur tilted lift dono airframe ko body-along-aft push karte hain.

Normal — $\vec{R}$ ko $\hat{p}$ par project karo, ek term at a time:

Yahan signs seedhe bina kisi flip ke aate hain: lift ka cross-body share add hota hai, drag ka cross-body share subtract hota hai. Woh minus real hai — tilt drag ko normal force se thoda rob kar leta hai.

PICTURE: tilted-rectangle projection, har dashed drop-line apne $\cos$ ya $\sin$ leg se labelled.

Step 5 — Edge cases: $\alpha =0$, negative $\alpha$, aur large $\alpha$

Case A — $\alpha =0$ (nose wahan point karta hai jahan tum ja rahe ho). Body ruler aur velocity ruler coincide karte hain. $\cos 0=1$, $\sin 0=0$: $A=D(1)+L(0)=D,N=L(1)-D(0)=L.$ Axial barabar drag aur normal barabar lift sirf yahan.

Case B — $\alpha <0$ (nose velocity se neeche). Kyunki $\sin (-\alpha )=-\sin \alpha$, har $\sin$ term sign flip karta hai: $A=D\cos \alpha -L\sin |\alpha |,N=L\cos \alpha +D\sin |\alpha |.$ Lift ab body ke doosri taraf across pull karti hai, to uska axial share subtract karta hai aur drag term $N$ mein add karta hai. Kuch nayi memorise nahi karna — sine bas sign badal leta hai.

Case C — large $\alpha$ ($90^{\circ }$ se past). Jab $\alpha >90^{\circ }$, $\cos \alpha$ negative ho jaata hai: $A=D\cos \alpha +L\sin \alpha$ negative ho sakta hai, matlab hawa ab body ke along forward push kar rahi hai hamare aft-positive convention ke relative (rocket almost sideways fly kar raha hai — ek tumbling ya re-entry attitude). Formula tab bhi hold karta hai; woh bas reversed axial load honestly report karta hai.

KYA: general formulas specialise aur sign-check kiye gaye. KYU: taaki koi bhi reader kisi attitude se — nose-down, ya tumbling — na mile jise algebra pehle se cover na kar chuka ho. PICTURE: do rulers coincident ($\alpha =0$), ek small inset ke saath nose-below-velocity ($\alpha <0$) mirror dikhata hua.

Step 6 — Thrust aur gravity add karna: do equations of motion

Ab flight-path axes par (along $\hat{v}$ aur across $\hat{n}$) SAARE forces assemble karo. Hume ek aur angle aur do aur symbols chahiye:

Gravity $-mg\sin \gamma$ aur $-mg\cos \gamma$ mein kyun split hoti hai (ek projection, exactly Step 4 jaisi). Weight ek arrow hai length $mg$ ka jo seedha neeche point karta hai. Ise flight-path axes mein use karne ke liye hume wahi do sawaal poochne padte hain jaise pehle: kitna "seedha neeche" $\hat{v}$ ke along hai, aur kitna uske across ($\hat{n}$ ke along)?

Geometry setup karo: $\hat{v}$ horizontal se $\gamma$ angle upar point karta hai, isliye downward direction $\hat{v}$ se $\gamma$ neeche retarding side par baithti hai. Down-arrow ko project karo:

Length $mg$ se multiply karo:

• Path ke along: $-mg\sin \gamma$. $\gamma =90^{\circ }$ par (seedha upar) yeh $-mg$ hai — gravity climb se sabse hard fight karti hai; $\gamma =0$ par (level) yeh $0$ hai — gravity tumhari speed ke liye kuch nahi karti. Dive ke liye ($\gamma <0$), $\sin \gamma <0$ to term positive ho jaati hai — gravity ab speed up karti hai, exactly sahi.
• Path ke across: $-mg\cos \gamma$. $\gamma =0$ par (level) yeh $-mg$ hai — gravity poori path ko neeche bend karti hai; $\gamma =90^{\circ }$ par yeh $0$ hai — seedha upar, gravity tumhe curve nahi karti. Yeh cross-path term hi hai jo ek rocket ko Gravity Turn Trajectory mein over tip karta hai.

PICTURE (gravity projection): down-arrow $mg$, $\gamma$-tilted $\hat{v},\hat{n}$ cross par drop kiya gaya, apne $\sin \gamma$ aur $\cos \gamma$ legs ke saath.

Ab dusre players add karo, har ek $\hat{v}$ aur $\hat{n}$ par project karke:

• Thrust $T$ body $\hat{b}$ ke along point karta hai, $\hat{v}$ se $\alpha$ tilt — path ke along $T\cos \alpha$ aur path ke across $T\sin \alpha$ deta hai (same $\cos /\sin$ split jaise Step 4).
• Aerodynamics wind axes mein already aligned hai: path ke along $-D$, path ke across $+L$.
• Gravity: path ke along $-mg\sin \gamma$, path ke across $-mg\cos \gamma$ (abhi derive kiya).

KYA: $\vec{F}=m\vec{a}$ ke do projections. KYU do? Velocity size mein change ho sakti hai (pehli line — speed up: yahan $v$ speed hai, $\vec{v}$ ki length, aur $dv/dt$ woh rate hai jis par woh length badhti hai) ya direction mein (doosri line — turning: $vd\gamma /dt$ woh rate hai jis par path bend hota hai). Change-karne-ke-do-tarike ke liye ek-ek equation.

(Yaad raho har term mein mass $m$, instantaneous mass $m(t)=m_0-\dot{m}t$ hai — dekho Variable Mass Systems aur Tsiolkovsky Rocket Equation.)

Ek-picture summary

Ek tilted arrow $\vec{R}=-D\hat{v}+L\hat{n}=A\hat{b}+N\hat{p}$, do rulers ($\hat{v}$ aur $\hat{b}$) $\alpha$ apart, thrust body ke down, gravity seedha neeche — is page ka har formula is ek drawing ka shadow hai.

Quick self-test