Foundations — Relative motion — 1D and 2D; river-boat problems

1.1.21 · D1 · Physics › Measurement, Vectors & Kinematics › Relative motion — 1D and 2D; river-boat problems

Ye page assume karta hai ki tumne kuch bhi nahi dekha. Hum har arrow, letter, aur subscript build karte hain jis par parent note river-boat topic rely karta hai, ek aise order mein jahan har idea pehle wale par tika ho. §7 ke end tak tum famous subtraction rule khud derive kar loge; hum ise symbols mein tab tak nahi likhte jab tak tum har piece ke malik nahi ban jaate.

1. "Vector" kya hota hai? (ek arrow jisme length aur direction dono ho)

Kisi bhi physics se pehle, hume wo object chahiye jo velocity carry kare: vector.

Hum vector ko upar chhoti arrow ke saath likhte hain: $\vec{v}$. Plain letter $v$ (bina arrow ke) sirf uski length batata hai, jise magnitude kehte hain — hamesha ek positive number ya zero.

Figure dhyan se padho. Bottom-left ke paas solid burnt-orange arrow $\vec{v}$ hai; uske saath teal double-headed label uski length $=$ magnitude $v$ mark karta hai, aur plum note uska tilt $=$ direction mark karta hai. Ab faded orange arrow upar aur right mein dekho: wo solid wale ko slide karke banaya gaya hai — same length, same tilt, naye spot par draw kiya. Figure ka poora point ye hai ki faded aur solid dono arrows ek hi vector $\vec{v}$ hain: page par position ka koi matlab nahi, sirf length aur direction matter karti hai.

Is topic ko ye kyun chahiye: ek boat ki velocity, current ki velocity, aur ground observer kya dekhta hai — ye sab "kitna + kidhar" hain — exactly arrows. Aur gehre jaane ke liye dekho Vectors — addition, components, unit vectors.

2. Components — ek arrow ko across-part aur along-part mein katna

Tilted arrow ke saath compute karna awkward hai. Trick: ise do right angle par arrows mein split karo. Lekin "kiske right angle par?" — pehle hume axes aur unke positive directions nail down karne honge.

Figure mein do black axis arrows label hain: $+x$ across (rightward) aur $+y$ downstream (yahan upar draw kiya gaya hai sirf visibility ke liye). Plum arrow $\vec{v}$ hai. Iske tip se $x$-axis tak straight line girjao: axis ke along teal arrow $v_x$ hai, positive kyunki ye $+x$ direction mein point karta hai. $y$-axis ki taraf across drop karo: orange arrow $v_y$ hai, positive kyunki ye $+y$ direction mein point karta hai. Chhota square $90^{\circ }$ corner mark karta hai — do shadows ek right angle par milte hain, ek right triangle banate hain jiska diagonal $\vec{v}$ hai.

3. Vectors add karna — nose to tail

Relative-velocity rule arrows ko add aur subtract karne se bana hai, isliye hume dono define karne honge.

Figure ko apni finger se trace karo: origin se shuru karo, teal arrow $\vec{a}$ ke along uske tip tak chalo; us tip se orange arrow $\vec{b}$ ke along chalo (notice karo $\vec{b}$ wahan se start hota hai jahan $\vec{a}$ khatam hua). Tum jahan finish karte ho wo plum arrow $\vec{a}+\vec{b}$ ka tip hai, origin se tumhari final position tak seedha drawn hai. Plum arrow net trip hai. "Vectors add karna" bas isi ka naam hai.

"Tip to tail" kyun? Kyunki ye literally hai "pehli trip karo, phir doosri trip" — total displacement. River problem mein, "paani mein tairo" (trip 1) phir "paani ground ke upar drift kare" (trip 2) exactly isi tarah chain hota hai: ${\vec{v}}_{bg}={\vec{v}}_{bw}+{\vec{v}}_w$.

Wo "$\vec{b}$ ke tip se $\vec{a}$ ke tip tak" picture exactly "B se shuru karo, A tak jao" hai — relative-velocity rule ka seed jo hum §7 mein assemble karenge.

4. Signs aur number line — 1D sirf ek line par arrows hai

2D se pehle, parent note plain $+$ aur $-$ numbers ke saath 1D motion karta hai. Ye legal kyun hai.

5. Reference frames aur observers — "ground" ka actually matlab kya hai

Poora topic "ground," "water," "observer" ki baat karta hai. Ye sab reference frames hain. Hume precisely kehna hoga ki ek frame kya hota hai, pehle uske upar position build karne se.

6. Position vector $\vec{r}$ aur derivative $\frac{d}{dt}$

Ab jab ek frame ne origin aur axes diye hain, hum cheezein locate kar sakte hain aur unhe move hote dekh sakte hain.

Ye tool kyun, aur sirf "distance ÷ time" kyun nahi? Kyunki velocity moment to moment change ho sakti hai; derivative honest instantaneous rate hai — is bilkul is waqt arrow ki motion, poori trip par average nahi. Is topic mein velocities constant hain, isliye tum $\frac{d\vec{r}}{dt}$ ko safely "steady velocity arrow" padh sakte ho — lekin symbol general tool hai.

7. Subtraction rule build karna — aur ye kyun subtract karta hai

Ab hum wo ek formula assemble karte hain jis par poora topic chalta hai, sirf §3, §5, §6 use karke.

Step 1 — "$A$ ki position as seen by $B$" KYA hai? Ye wo arrow hai jo tum $B$ se $A$ tak draw karte — jahan observer $B$ baitha hai wahan se start karo, jahan $A$ hai wahan point karo. §3 ke subtraction picture se (arrow from tip of ${\vec{r}}_B$ to tip of ${\vec{r}}_A$), wo arrow hai

${\vec{r}}_{AB}={\vec{r}}_A-{\vec{r}}_B.$

Ye ${\vec{r}}_A+{\vec{r}}_B$ kyun nahi? Add karne se do trips chain hoti hain (§3); lekin "$B$ se $A$ tak" ek positions ka difference hai, jo subtraction hai. Pehla index $A$ endpoint hai, doosra index $B$ start hai.

Step 2 — differentiate karo, §5 ka shared clock use karke. Kyunki har frame time $t$ par agree karta hai, hum dono sides ka $\frac{d}{dt}$ le sakte hain aur left aur right par same $t$ ka matlab hai:

$\frac{d{\vec{r}}_{AB}}{dt}=\frac{d{\vec{r}}_A}{dt}-\frac{d{\vec{r}}_B}{dt}.$

Step 3 — velocities padho (§6). Position ka har rate-of-change ek velocity hai:

8. Angle $\theta$ aur trig ratios $\sin \theta ,\cos \theta$ — aim measure karna

"Shortest path" case mein ek aiming angle chahiye. Yahan picture scratch se hai, angle convention explicitly bata ke.

Figure teeno label karta hai: plum hypotenuse ${\vec{v}}_{bw}$, teal horizontal leg "across $=v_{bw}\cos \theta$", orange vertical leg "upstream $=v_{bw}\sin \theta$", aur origin par arc $\theta$. Right triangle trace karo: hypotenuse = engine speed, base = actually kitni tez cross karte ho, height = current se ladne mein kitna throw away karte ho.

$\sin$ aur $\cos$ kyun na ki kuch aur? Kyunki hum ${\vec{v}}_{bw}$ ko across-part aur upstream-part mein split karte hain (ye phir §2 hai). $\cos$ across leg pick karta hai, $\sin$ upstream leg pick karta hai. Upstream leg ko current ke equal set karna, $v_{bw}\sin \theta =v_w$, zero-drift condition hai — iska har symbol ab earn kiya ja chuka hai.

Kyunki $\sin \theta \le 1$ hamesha, $\sin \theta =v_w/v_{bw}$ tabhi solvable hai jab $v_w\le v_{bw}$ — "current ko beat nahi kiya ja sakta" limit seedha is picture se nikalta hai.

Foundations topic ko kaise feed karte hain

Equipment checklist

Connections

• Vectors — addition, components, unit vectors — arrows, components aur tip-to-tail rules jo yahan build hue.
• Frames of reference & Galilean transformation — kyun shared clock aur frame choice ${\vec{v}}_{AB}={\vec{v}}_A-{\vec{v}}_B$ deti hai.
• Newton's laws — inertial frames — jahan "observers" aur "frames" ka physical meaning milta hai.
• Kinematics in 2D — projectile motion — same independent-perpendicular-components idea.
• Rain-man umbrella problem — is page ka har symbol reuse karta hai.