Average velocity vs instantaneous velocity — Hinglish notes

1.1.14 · Physics › Measurement, Vectors & Kinematics

1. First principles se setup karna

Hum motion ko ek position vector $r (t)$ se describe karte hain — object time $t$ par kahan hai.

HUM KYA chahte hain: ek aisi quantity jo "position change per unit time" measure kare.

$Δ t$ se divide kyun karte hain? Kyunki hum ek rate chahte hain — position har second kitna change hoti hai. Change ko liye gaye time se divide karna "per unit time" ka universal recipe hai.

2. Instantaneous velocity derive karna (the limit)

Average velocity se shuru karo aur $Δ t \to 0$ shrink karo:

$v_{in s t} = lim_{Δ t \to 0} \frac{Δ r}{Δ t}$

Ye limit derivative ki definition hai. Toh:

Geometric meaning (Dual Coding): Position–time graph par, average velocity do points ko join karne wali secant line (chord) ka slope hai; instantaneous velocity ek point par tangent line ka slope hai. Jaise-jaise do points saath slide karte hain, secant tangent ban jaati hai.

Figure — Average velocity vs instantaneous velocity

3. Worked examples

4. Common mistakes

Recall Feynman: ek 12-saal-ke-bache ko explain karo

Socho tum apne dost ke ghar gaye aur wapas aaye. Tumhari average velocity poochti hai, "Kya tum actually kahi gaye?" — aur kyunki tum ghar wapas aaye, jawab hai kahin nahi, toh ye zero hai! Lekin tumhari average speed tumhare har kadam count karti hai, toh ye ek bada number hai. Ab, tumhari instantaneous velocity ek speedometer par arrow dekhne jaisi hai — ye batati hai tum clock ke ek tiny tick par kitni fast aur kis direction mein ja rahe ho. Ise paane ke liye, tum measure karte ho ki tum ek teeny-tiny sliver of time mein kitna move karte ho aur divide karte ho. Sliver ko chhhota aur chhhota banao, aur tum us exact moment par speed paa lete ho.

5. The 80/20 core

Woh 20% jo 80% deta hai:

$v_{a v g} = Δ r /Δ t$ (secant slope).
$v = d r / d t$ (tangent slope, upar wale ki limit).
Velocity displacement use karti hai; speed distance use karti hai. Ye differ karte hain jab direction reverse hoti hai.

Connections

Position and displacement vectors
Distance vs displacement
Average speed vs instantaneous speed
Acceleration as derivative of velocity
Slope and tangent in calculus
Position-time graphs
Equations of motion under uniform acceleration

Flashcards

Vectors ke terms mein average velocity define karo.

v_{a v g} = \frac{Δ r}{Δ t} = \frac{r ( t _{2} ) - r ( t _{1} )}{t _{2} - t _{1}}

, displacement ke along ek vector.

Instantaneous velocity define karo.

Limit

v = lim_{Δ t \to 0} \frac{Δ r}{Δ t} = \frac{d r}{d t}

— position ka derivative;

x

–

t

graph par tangent ka slope.

Position–time graph par, average velocity kya represent karta hai?

Do endpoints ko join karne wali secant (chord) ka slope.

Position–time graph par, instantaneous velocity kya represent karta hai?

Us instant par tangent line ka slope.

Average velocity zero kyun ho sakti hai jabki average speed nahi?

Average velocity net displacement use karti hai (jo round trip mein zero ho sakta hai), jabki average speed total path distance use karti hai (agar tum move kiye toh kabhi zero nahi).

∣ v_{a v g} ∣

average speed ke barabar kab hota hai?

Sirf jab motion kabhi direction reverse na kare (path length = displacement magnitude).

x (t) = 3 t^{2} - 2 t

ke liye,

v (t)

do.

v (t) = 6 t - 2

m/s.

Uniform acceleration ke liye, interval ke kahan average velocity instantaneous velocity ke barabar hoti hai?

Interval ke midpoint time par.

Motion ke turning point par instantaneous velocity ki value kya hai?

Zero (flat tangent), chahe acceleration nonzero ho.

Velocity ki SI units?

metres per second, m/s.