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

1.1.21 · D5 · Physics › Measurement, Vectors & Kinematics › Relative motion — 1D and 2D; river-boat problems

Shuru karne se pehle, ek reminder notation ka jo poore note mein use hogi, taaki koi bhi symbol unexplained na rahe — isse Figure 1 ke saath padho, jo har quantity ko river par ek arrow ki tarah draw karta hai.

Definition Woh symbols jo yahan milenge

Sabhi velocities mein do frame labels hote hain: pehla letter kya move karta hai, doosra kaun dekh raha hai. "With respect to" hamesha dekhne wale ko name karta hai.

$r_{A}, r_{B}$ ::: position vectors — woh arrow jo ground origin $O$ se object A (ya B) tak draw hoti hai. "Ground se" unka frame hai; position ke baare mein kuch bhi absolute nahi hota jab tak tum $O$ choose nahi karte.
$v_{A}, v_{B}$ ::: A aur B ki velocities dono ground ke saath respect to measured — woh rate jis par $r_{A}$ aur $r_{B}$ change hoti hain. Jab bhi subtraction rule mein bare $v_{A}$ dikhta hai, padho " $A$ with respect to ground."
$v_{A B}$ ::: A ki velocity jaise B dekhta hai — A, B ki aankhon mein kitna fast aur kis direction mein move karta hai.
$v_{b w}$ ::: boat ki velocity water ke saath respect to (jo engine/arms produce karta hai). Iski magnitude motor ya swimmer ke zariye fixed hoti hai.
$v_{w}$ ::: water ki velocity ground ke saath respect to (current).
$v_{b g}$ ::: boat ki velocity ground ke saath respect to (jo bank par khada insaan dekhta hai).
$d$ ::: river ki width — stream ke across ki woh distance jo tumhe cover karni hai.
$t$ ::: crossing time — ek bank se doosre bank tak jane mein kitna waqt lagta hai, hamesha $t = (across distance) / (across speed)$ .
$θ$ ::: steering angle, seedha-across direction se measure kiya gaya, upstream ki taraf tilt kiya hua (dekho Figure 3).

Figure 2 do geometric facts sketch karta hai jis par poora bank tika hai: min-time aur zero-drift routes apne velocity triangles kaise banate hain.

Figure 3 zero-drift triangle ko zoom in karke dikhata hai taaki tum dekh sako ki $sin θ = v_{w} / v_{b w}$ aur $t = d / v_{b w}^{2} - v_{w}^{2}$ kahan se aate hain, kisi cited formula par trust karne ki bajay.

True or false — justify

Har ek ka jawab true/false AUR reason ke saath do — sirf verdict se kuch nahi milega.

Current ka tez hona ek straight-across (min-time) crossing ko zyada time lene wala banata hai

False. Jab tum straight across aim karte ho, current tumhari crossing motion ke perpendicular hoti hai, isliye woh sirf downstream drift add karti hai — crossing time

t = d / v_{b w}

rehta hai, current se independent.

Agar do cars ek doosre ki taraf drive karti hain, toh har ek doosre ko jo speed se approach karte dekhta hai woh unki speeds ka sum hoti hai

True. Right

= +

ke saath, ek car

+ v_{1}

hai aur doosri

- v_{2}

; relative velocity

v_{1} - (- v_{2}) = v_{1} + v_{2}

hai, isliye observer ki negative velocity subtract karne par speeds add ho jati hain.

v_{A B} = v_{B A}

False. Ye exactly opposite hain:

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

. Do cars head-on close hoti hain imagine karo: B ke frame mein A right se aata hai, isliye

v_{A B}

left ki taraf point karta hai; A ke frame mein B left se aata hai, isliye

v_{B A}

right ki taraf point karta hai. Same closing situation, opposite arrows — observer badalne par vector ka direction flip ho jaata hai.

Minimum-time route aur zero-drift route kabhi same time nahi le sakte

True (jab bhi current ho). Zero-drift time

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

strictly min-time

d / v_{b w}

se bada hota hai jab tak

v_{w} > 0

ho; sirf ek bilkul still river (

v_{w} = 0

) unhe equal banati hai.

Tum hamesha boat ko steer karke seedha opposite point par land kar sakte ho

False. Iske liye

v_{b w} > v_{w}

zaroori hai; agar current kam se kam boat jitna fast ho, toh

sin θ = v_{w} / v_{b w} \geq 1

ka koi real solution nahi hota aur current jeet jaata hai.

Relative velocity sirf two dimensions mein meaningful hai

False. Rule

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

1D (signed numbers), 2D, aur 3D mein identically kaam karta hai — kisi bhi dimension mein yeh vector subtraction hai.

Dono boat speed aur current speed ko double karne par min-time crossing ka angle-of-drift unchanged rehta hai

True. Drift direction ratio

v_{w} / v_{b w}

par depend karta hai (resultant ke slope ke zariye), aur dono ko double karne par woh ratio fixed rehta hai — same drift angle, halanki trip ab faster hai.

Ground ke saath respect to boat ki velocity ki magnitude min-time case mein

v_{b w} + v_{w}

hoti hai

False. Min-time mein,

v_{b w}

(across) aur

v_{w}

(downstream) perpendicular hain, isliye ground speed

v_{b w}^{2} + v_{w}^{2}

hai, arithmetic sum nahi.

Spot the error

Har line ek galat reasoning batati hai. Reveal mein error ka naam aur uski repair hai.

"Boat 5 m/s karti hai, current 3 m/s, toh woh ground par 8 m/s se cross karti hai."

Error: perpendicular vectors ko scalars ki tarah add karna. Ye right angles par hain (dekho Figure 2), isliye ground speed

5^{2} + 3^{2} \approx 5.83

m/s hai, aur woh partly downstream point karti hai, seedha across nahi.

"Relative matlab combine karna, isliye

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

Error: galat operation. Relative velocity observer ko subtract karti hai:

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

. Adding sirf chain rule

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

ke liye hai, jahan ek shared middle index cancel hota hai.

"Seedha opposite land karne ke liye, boat ko straight across aim karo aur current average out ho jayegi."

Error: us setup mein kuch bhi current ko cancel nahi karta. Straight across aim karne par poori current uncancelled rehti hai, isliye tum drift karte ho. Tumhe upstream tilt karna hoga taaki

v_{b w}

ka ek component actively current se lad sake (Figure 3).

"Faster current hamesha crossing time badhati hai."

Error: yeh do routes ko confuse kar raha hai. Min-time (straight-across) route ke liye, current sirf drift change karti hai, time nahi. Woh time sirf zero-drift route par badhati hai, jahan tum power upstream ki taraf divert karte ho.

"Kyunki paani boat ko carry karta hai, ground velocity sirf current velocity hai."

Error: engine ko ignore karta hai. Boat paani mein bhi move karti hai; ground velocity sum

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

hai, sirf current nahi.

"Shortest path ke liye hum

v_{b w} sin θ

minimize karte hain, isliye

θ = 0

set karte hain."

Error: shortest path matlab zero drift hai, jiske liye

v_{b w} sin θ = v_{w}

chahiye (current se match karna), ise minimize karna nahi.

θ = 0

set karna maximum drift deta hai, yani sabse lamba path.

"Chain rule

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

mein outer indices match karne chahiye."

Error: inner indices (

B

aur

B

) woh hain jo match karke cancel hote hain, jaise fractions; outer letters

A

aur

C

bachte hain result ko name karne ke liye.

Why questions

Velocity hamesha kisi frame ke "with respect to" kyun batai jaati hai?

Kyunki motion ka koi absolute meaning nahi hota — train mein baithe hue tum nahi bata sakte ki tum move kar rahe ho ya agali train. Ek velocity value tab tak khali hai jab tak tum yeh nahi batate ki isse kis cheez ke against measure kiya gaya hai.

Ek negative velocity subtract karne par closing speed kyun badi ho jaati hai?

Ek negative velocity matlab hai "doosri taraf (tumhari taraf) move karna." Ise subtract karne par,

v_{A} - (- v_{B}) = v_{A} + v_{B}

, magnitudes add ho jaati hain — exactly woh physical "woh sum speed par approach karte hain" intuition.

Min-time route mein current crossing time ko kyun affect nahi karti?

Current purely downstream (

y

) axis ke along hai, across (

x

) motion ke perpendicular. Perpendicular components independent hote hain, isliye current

x

-distance ya use cover karne ke time ko nahi chhu sakti.

Zero-drift crossing time

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

kyun hai,

d / v_{b w}

kyun nahi?

Kyunki tum boat ki speed ka ek hissa upstream point karne mein spend karte ho current ko cancel karne ke liye (Figure 3). Sirf bacha hua across-component

v_{b w}^{2} - v_{w}^{2}

crossing karta hai, aur chhoti across-speed matlab zyada time.

Hum

r_{A B} = r_{A} - r_{B}

ko term-by-term differentiate karke velocity rule kyun nikaal sakte hain?

Yahan

r_{A}, r_{B}

A aur B ki ground-frame positions hain (origin

O

se arrows), aur

r_{A B} = r_{A} - r_{B}

B se A ki taraf arrow hai. Newtonian physics mein time universal hai aur frames rotate nahi karte, isliye

t

sabke liye same hai aur

d / d t

subtraction par distribute ho jaata hai, deta hai

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

Straight-across crossing mein resultant ground velocity downstream kyun point karti hai?

Across-component (

v_{b w}

) aur downstream-component (

v_{w}

) tip-to-tail add hoti hain; sum downstream side ki taraf tilt ho jaata hai, isliye actual path ek diagonal hai, straight across nahi (Figure 2, left).

Current se ladne ka ek hi tarika upstream aim karna kyun hai, downstream nahi?

Sirf ek upstream velocity component current ke against point karta hai; downstream tilt drift mein aur add kar deta. Net zero downstream motion ke liye tumhe use directly oppose karna hoga.

Edge cases

Jab current zero ho (

v_{w} = 0

) toh dono routes ka kya hota hai?

Woh identical ho jaate hain:

sin θ = 0/ v_{b w} = 0

isliye dono mein tum straight across aim karte ho, drift zero hai, aur dono

t = d / v_{b w}

dete hain. Koi current nahi matlab koi trade-off nahi.

Jab current exactly boat speed ke barabar ho (

v_{w} = v_{b w}

) toh steering angle kya hoga?

sin θ = v_{w} / v_{b w} = 1

, isliye

θ = 90°

— tumhe fully upstream aim karna padega, bacha hoga zero across-speed. Tum actually kabhi cross nahi karoge, isliye opposite land karna impossible hai (limiting case).

Jab current boat se faster ho (

v_{w} > v_{b w}

) toh min-time drift kya hogi?

Min-time crossing tab bhi kaam karti hai — tum hamesha

t = d / v_{b w}

mein cross karte ho — lekin drift

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

ab width

d

se zyada ho jaati hai. Tum cross karte ho, lekin far downstream land karte ho; tum bas opposite land nahi kar sakte.

Agar A aur B identical velocity se move karein toh

v_{A B}

kya hoga?

Zero:

v_{A} - v_{B} = 0

. Ek doosre ke frame mein woh at rest dikhte hain — isliye co-moving frames (dono inertial) apni shared motion detect nahi kar sakte.

Limit

v_{w} \to v_{b w}

neeche se aane par, zero-drift crossing time ka kya hoga?

Across-speed

v_{b w}^{2} - v_{w}^{2} \to 0

, isliye

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

. Crossing unboundedly lamba time leta hai jab current boat ki speed ke paas pohonchti hai — route smoothly impossible ho jaata hai.

Agar boat straight upstream aim kare (

θ = 90°

) aur

v_{b w} > v_{w}

ho, toh uski ground motion kya hogi?

Woh ground speed

v_{b w} - v_{w}

par upstream move karti hai aur river ke across koi progress nahi karta. Saari engine power current se ladne mein jaati hai aur crossing ke liye kuch nahi bachta — ek degenerate, useless heading.

Recall One-line self-test

Upar ke answers cover karo. Agar tum har True/False justify kar sako aur har error ko bina dekhte name kar sako, toh tum is topic ke malik ho. Zyaatar students ki sabse weak link ::: "current min-time crossing nahi badhalti" — kyunki feel hota hai ki river tumhe slow karni chahiye.

Connections

Relative motion — 1D and 2D; river-boat problems — woh parent note jise ye traps stress-test karti hain.
Vectors — addition, components, unit vectors — yahan har "spot the error" actually ek vector-vs-scalar slip hai.
Frames of reference & Galilean transformation — "kisi frame ke saath respect to kyun" wale questions yahan rehte hain.
Newton's laws — inertial frames — co-moving-frames edge case ( $v_{A B} = 0$ ) isliye hai ki inertial frames indistinguishable hote hain.
Rain-man umbrella problem — same subtract-the-observer logic; iske traps par bhi khud ko test karo.
Kinematics in 2D — projectile motion — "perpendicular components independent hote hain" idea reuse hota hai.