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

1.1.21 · Physics › Measurement, Vectors & Kinematics

Hume relative motion ki ZARURAT kyun hai?

WHAT chahiye: object A ki velocity jaise observer B dekhta hai (ise $v_{A B}$ kaho).

WHY matter karta hai: Ek boat ka engine use paani mein push karta hai, ground mein nahi. Current paani ko le jaata hai. To ground, boat-velocity dekhta hai = (paani mein velocity) + (paani ki velocity). In frames ko mix karna galti number 1 hai.

Relative-velocity rule ko scratch se derive karna

Yeh step kyun? "A ki position relative to B" ka matlab literally hai B se shuru karo, A tak jao — woh arrow hai $r_{A} - r_{B}$ .

Ab time ke saath differentiate karo (frames rotate nahi karte, Newtonian physics mein $t$ universal hai):

$\frac{d r _{A B}}{d t} = \frac{d r _{A}}{d t} - \frac{d r _{B}}{d t}$

Do consequences jo tumhe internalize karni hain:

$v_{A B} = - v_{B A}$ (A, B ko ulti direction mein move karte dekhta hai jaise B, A ko dekhta hai).
Frames ka chain rule: $v_{A C} = v_{A B} + v_{B C}$ . Middle index fractions ki tarah cancel ho jaata hai.

1D relative motion

Ek sign convention chuno (right = +). Phir sab kuch bas signed addition hai.

2D relative motion aur river-boat problem

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

Axes set karo: $x$ = river ke across (width $d$ ), $y$ = downstream. Lo:

$v_{b w}$ = boat ki velocity water ke saath respect mein (engine ka kaam — magnitude motor se fixed).
$v_{w}$ = water ki velocity wrt ground (current, $y$ ke along).
$v_{b g}$ = boat ki velocity ground ke saath respect mein = jo hum actually dekhte hain.

Chain rule se: $v_{b g} = v_{b w} + v_{w}$

Yeh step kyun? Boat paani mein tarti hai; poora paani ka sheet drift karta hai. Dono ko add karo.

Case 1 — Cross karne ka sabse kam time

WHAT: time minimize karo. HOW: boat ko seedha across point karo ( $v_{b w}$ poora $x$ ke along).

Crossing sirf across-component par depend karti hai, aur poori engine speed $v_{b w}$ across hai: $t_{m i n} = \frac{d}{v _{b w}}$ Is time mein downstream drift: $x_{drift} = v_{w} t_{m i n} = \frac{v _{w} d}{v _{b w}}$

Yeh step kyun? Current sirf $y$ -velocity add karta hai; yeh crossing ko kabhi speed up ya slow down nahi karta. To seedha across head karna 100% engine power crossing par spend karta hai → minimum time. Tum phir bhi downstream land karte ho — yahi cost hai.

Case 2 — Sabse chhota path (seedha opposite land karo)

WHAT: downstream drift zero. HOW: angle $θ$ par upstream aim karo taaki $v_{b w}$ ka upstream component current ko exactly cancel kar de.

$v_{b w} sin θ = v_{w} \Rightarrow sin θ = \frac{v _{w}}{v _{b w}}$

Yeh step kyun? Net downstream velocity zero honi chahiye. Sirf boat ka upstream component current se lad sakta hai.

Effective across-speed bacha hua component hai: $v_{across} = v_{b w} cos θ = v_{b w}^{2} - v_{w}^{2}$ $t = \frac{d}{v _{b w}^{2} - v _{w}^{2}}$

Forecast-then-Verify

Recall Padhne se pehle predict karo

Q: Ek swimmer 20 s mein min-time route se river cross karta hai 60 m drift ke saath. Zero drift ke liye (same river, same swimmer), time zyada hoga ya kam?

Forecast: zyada — upstream power divert karne se crossing ke liye kam bachta hai. Verify: $t = d / v_{b w}^{2} - v_{w}^{2} > d / v_{b w}$ kyunki denominator chota ho gaya. ✓

Common mistakes (Steel-man)

Recall Feynman: 12-saal ke bacche ko explain karo

Socho tum ek stream ke across swim kar rahe ho. Paani ek moving carpet ki tarah hai — jahan bhi tum tairo, carpet tumhe sideways bhi slide karti hai. Agar tum seedha doosre kinare tak tairo, tum wahan jaldi pahunchoge, lekin carpet tumhe downstream chhod degi. Agar tum bilkul seedha across land karna chahte ho, tumhe thoda carpet ke khilaf tairna hoga — jaise ek moving walkway par diagonal chalna ek lane mein rehne ke liye. Woh diagonal trick tumhara kuch swimming carpet se larne mein waste karta hai, to zyada time lagta hai. Aur agar carpet tumse zyada tez slide kare jitna tum tairo, tum simply use nahi haar sakte — tum hamesha downstream end karoge.

Flashcards

Relative velocity of A wrt B formula

v_{A B} = v_{A} - v_{B}

Chain rule for relative velocity

v_{A C} = v_{A B} + v_{B C}

(inner indices cancel)

Relation between

v_{A B}

and

v_{B A}

v_{A B} = - v_{B A}

Velocity of boat wrt ground in terms of water

v_{b g} = v_{b w} + v_{w}

Minimum crossing TIME ke liye, boat ko kahan aim karo?

Seedha across;

t_{m i n} = d / v_{b w}

Minimum-time crossing mein drift

x = v_{w} d / v_{b w}

Seedha opposite land karne ke liye (zero drift), steering angle?

sin θ = v_{w} / v_{b w}

upstream from straight-across

Zero-drift route ke liye crossing time

t = d / v_{b w}^{2} - v_{w}^{2}

Seedha opposite point tak pahunchne ki condition

v_{b w} > v_{w}

(warna

sin θ > 1

, impossible)

Ek doosre ki taraf move karte do objects ki closing speed

Unki speeds ka sum (negative ghatana)

Current min-time crossing kyun nahi badalta?

Current across-motion ke perpendicular hai; perpendicular velocity crossing distance/time ko affect nahi karti

Connections

Vectors — addition, components, unit vectors — relative velocity is vector subtraction.
Kinematics in 2D — projectile motion — same "independent perpendicular components" logic.
Frames of reference & Galilean transformation — $v_{A B} = v_{A} - v_{B}$ ka formal basis.
Newton's laws — inertial frames — relative velocity inertial frames ke beech constant rehti hai.
Rain-man umbrella problem — river-boat problem ka seedha cousin.