Foundations — Average velocity vs instantaneous velocity

1.1.14 · D1 · Physics › Measurement, Vectors & Kinematics › Average velocity vs instantaneous velocity

Parent note padhne se pehle, tumhe usme aane wale notation ke chhote-chhote pieces mein fluent hona chahiye. Neeche, har ek symbol kamaya jaata hai: pehle simple words mein, phir ek picture, phir topic ko yeh kyun chahiye. Koi bhi cheez use nahi hoti jab tak build na ho jaaye.

1. "Position" kya hoti hai? — symbol $\vec{r}$

Picture. Apne page pe ek dot banao jiska naam origin hai — use $O$ kaho. Object kahin aur baitha hai. $O$ se seedha object tak ek arrow khiincho. Woh arrow hi $\vec{r}$ hai.

Topic ko yeh kyun chahiye. Velocity motion ke baare mein hai, aur motion matlab position change hoti hai. Jab tak position ko name nahi de sakte, position mein change ki baat nahi kar sakte. $\vec{r}$ woh name hai.

Poori construction ke liye dekho Position and displacement vectors.

2. Time ek label ki tarah — symbol $t$ aur $\vec{r}(t)$

Picture. SochO moving object ki ek film. Har frame pe ek time $t$ (seconds mein) stamp hai. $\vec{r}(t)$ ek machine hai: ek time daalo, aur us frame ka arrow $\vec{r}$ bahar aata hai.

Topic ko yeh kyun chahiye. Yeh poochne ke liye ki "yeh kitna fast move kar raha hai?" tum abhi kahan hai aur thodi der pehle kahan tha compare karte ho. Uske liye do alag times pe position pin karni padti hai — exactly wahi jo $\vec{r}(t_1)$ aur $\vec{r}(t_2)$ dete hain.

3. Greek letter $\Delta$ ("delta") — "change in"

Picture. Origin se do arrows: chhota wala $\vec{r}(t_1)$ hai (start), lamba wala $\vec{r}(t_2)$ hai (end). $\Delta \vec{r}$ woh naya arrow hai jo start-arrow ki tip se end-arrow ki tip tak khiincha jaata hai — jahan the wahan se jahan gaye wahan tak ka shortcut.

Topic ko yeh kyun chahiye. Yeh arrow $\Delta \vec{r}$ displacement kehlaata hai — position mein net change. Velocity ka poora idea hai "displacement divided by time," toh $\Delta$ is machinery ka dil hai. Displacement aur path length mein comparison dekho Distance vs displacement mein.

4. Ek rate — change ko laage hue time se divide karna

Picture — ek secant line. Ek graph banao jisme time neeche ho aur position side mein (ek position–time graph — dekho Position-time graphs). Do points $(t_1,x_1)$ aur $(t_2,x_2)$ mark karo. Inhe jodne wali straight line secant hai (ek chord). Uski steepness — har ek step across pe kitna rise karta hai — wahi average velocity hai.

Topic ko yeh kyun chahiye. "Average velocity = slope of the secant" poore parent note ke do headline facts mein se ek hai. Jab tak slope aur secant ka matlab nahi pata, yeh nahi dikh sakta.

5. Interval ko shrink karna — limit $\lim$ aur $\Delta t\to 0$

Picture. Start point fixed rakho aur secant ke end point ko uske paas aur paas slide karo. Chord pivot karta hai. Limit mein — jab do points touch karte hain — secant tangent ban jaati hai: woh straight line jo curve ko sirf us ek point pe just graze karti hai.

6. The derivative — symbol $\frac{d\vec{r}}{dt}$

Topic ko yeh kyun chahiye. "Instantaneous velocity = slope of the tangent = the derivative" doosra headline fact hai. Jab $\lim$, tangent, aur $d/dt$ apne ho jaate hain, parent note ka boxed formula bilkul plain English ho jaata hai.

Wahi limit idea, position ki jagah velocity pe apply hone se, deta hai Acceleration as derivative of velocity.

7. Do families alag rakhni hain: vector vs scalar

Picture. Velocity ek arrow hai; uski magnitude $|\vec{v}|$ sirf arrow ki length hai — woh length speed hai. Direction throw away ho jaata hai. Isi wajah se ek round trip ki velocity zero ho sakti hai (arrows cancel ho jaate hain) par speed nonzero (lengths add ho jaati hain). Compare karo Average speed vs instantaneous speed mein.

Ye position functions aate hain Equations of motion under uniform acceleration se.

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