Foundations — Equations of motion — 3DOF point mass (trajectory analysis)

3.4.4 · D1 · Physics › Rocket Flight Mechanics › Equations of motion — 3DOF point mass (trajectory analysis)

Is page par assume kiya gaya hai ki aapne parent page ki koi bhi notation nahi dekhi. Hum har symbol ko ek picture se banate hain, ek aisi order mein jahan har ek symbol sirf unhi symbols pe depend karta hai jo pehle aa chuke hain. End tak aap parent equations ko left se right padh sakte hain bina kisi unexplained mark ke mile.

0. Woh picture jisme hum rehte hain

Sab kuch ek flat vertical plane mein hota hai: ek floor (zameen) aur ek wall (seedha upar). Hum ek dot — rocket, ek point tak chhota hua — ko is plane mein move karte hue dekhte hain.

1. Position: dot kahan hai — $x$ aur $h$

Yeh batane ke liye ki dot kahan hai, hume do numbers chahiye, kyunki plane two-dimensional hai.

Kyun topic ko inki zaroorat hai: trajectory analysis ka final product $h$ ka $x$ ke against ek curve hai — flight path jo wall par bana ho. Baaki sab kuch is liye exist karta hai ki $(x,h)$ kaise change hota hai yeh compute kiya ja sake.

2. Velocity arrow: $V$ aur $\gamma$

Dot move karta hai, toh uski ek velocity hoti hai — ek arrow jiska length speed hai aur jiska direction woh jagah hai jahan woh ja raha hai. Velocity ko "horizontal part" aur "vertical part" mein split karne ki jagah, parent note arrow ko directly uski length aur uski tilt se describe karta hai. Yeh sab se important idea hai jiske saath comfortable hona chahiye.

3. Angle padhna: velocity triangle par sine aur cosine

$\sin \gamma$ aur $\cos \gamma$ symbols parent note mein har jagah aate hain. Yeh zero se kya hain.

Velocity arrow ko ek right triangle mein daalein: uski horizontal chhaya, uski vertical rise, aur arrow khud slanted side (the hypotenuse, length $V$) ke roop mein.

Yeh seedha parent ki do boxed kinematic equations explain karta hai: $\dot{x}=V\cos \gamma (\text{horizontal part}),\dot{h}=V\sin \gamma (\text{vertical part}).$

4. Upar ka dot: $\dot{V}$, $\dot{\gamma }$, $\dot{x}$, $\dot{h}$, $\dot{m}$

Parent note mein bahut se symbols hain jo upar dot pehne hue hain. Yeh decoration nahi hai.

Ek equation jisme left par dotted variable ho ("$\dot{V}=\dots$") use ODE kehte hain — ek equation ek rate ke liye. Parent ki chaar boxed equations chaar aisi rate-rules hain.

5. Chaar forces aur unke symbols

Arrow ki length aur tilt ki rate forces se set hoti hai. Chaar forces, har ek ka apna symbol.

6. Aakhri do symbols: $I_{sp}$ aur $g_0$

Mass-rate rule $\dot{m}=-T/(I_{sp}{g}_0)$ mein do naye symbols aate hain.

Prerequisite map

Ise aise padhein: dot hamein position aur velocity deta hai; velocity length ($V$) aur tilt ($\gamma$) deti hai; trig arrow ko split karta hai; trig se split ki gayi forces rates set karti hain; saari rates milke hain 3DOF equations. $\gamma$ ki geometry ke liye Flight-Path Angle and Velocity Frame bhi dekhein, aur jo hum jaanbujhkar chhod gaye uske liye 6DOF Rigid-Body Dynamics dekhein.

Equipment checklist

Parent note kholne se pehle answers chhupao aur khud ko test karo.