Foundations — Reference frames — Galilean transformations

1.1.22 · D1 · Physics › Measurement, Vectors & Kinematics › Reference frames — Galilean transformations

Isse pehle ki tum parent note ki ek bhi line padho, tumhe us alphabet ki poori samajh honi chahiye jisme woh baat karta hai. Neeche, har ek symbol ko bilkul scratch se banaya gaya hai: seedhe words → ek picture → aur kyun is topic ko uski zaroorat hai. Upar se neeche padho; har rung apne neeche wale par khada hai.

1. Space mein ek point, aur uska arrow

Sab kuch koi cheez kahan hai is idea se shuru hota hai. "Kitni door" nahi — kahan, matlab ek agreed starting point se direction aur distance.

Ek bade graph paper ki sheet ko imagine karo. Koi ek crossing par pin ghusta hai aur kehta hai "yeh $O$ hai". Do rulers us pin se nikalty hain: ek east ($x$), ek north ($y$). Yeh ek coordinate system hai.

Upar di gayi figure dekho. Crossing par kala dot $O$ hai. Coral arrow $O$ se green object tak jaata hai — woh arrow hi position vector hai. Butter-yellow dashes uske do components dikhate hain: $x$ ke saath daayein chalo, phir $y$ ke saath upar, aur tum tip par pahunch jaate ho.

Arrow kyun, aur bas do numbers kyun nahi? Kyunki ek arrow ek nazar mein direction dikhata hai, aur — is topic ke liye bahut zaroori — arrows ko tip-to-tail add kiya ja sakta hai. Yehi ek property hamein allow karti hai "train kahan hai" aur "train ke andar ball kahan hai" ko combine karne ki. Tumhare dimaag mein Vectors — addition and components bilkul fresh hona chahiye.

Is doosri figure mein char rangin arrows charon corners par land karte hain. Har ek ki tip padho: top-right arrow hai $(+,+)$ = east-aur-north; top-left hai $(-,+)$ = west-aur-north; bottom-left $(-,-)$ = west-aur-south; bottom-right $(+,-)$ = east-aur-south. Tumhe donon signs ke saath comfortable hona chahiye pehle se vectors subtract karne se, kyunki $\vec{r}-\vec{V}t$ routinely negatives produce karega.

2. Samay ka ek pal, aur clock

Akela space kaafi nahi hai. Ek firecracker "pop" karta hai — uska kahan aur ek kab hota hai.

Ek flashbulb jalta hua imagine karo. Uski position ek arrow $\vec{r}$ hai; uska time clock par frozen ek number $t$ hai. Is topic ka poora game yeh hai: do observers ek hi flash par point karte hain aur alag $\vec{r}$ likhte hain lekin (Galilean world mein) same $t$.

3. Velocity — arrow kaise move karta hai

Velocity ko vector ki zaroorat kyun hai, sirf speed kyun nahi? Kyunki "5 m/s north" aur "5 m/s east" alag motions hain even though speed identical hai — aur parent ke Example 2 mein (river boat) donon directions ko arrows ki tarah add karna padega, jisse 5 m/s diagonal milta hai, "3 + 4 = 7" nahi.

Hum yahaan derivative use karte hain aur koi doosra tool nahi kyunki "position ka rate of change" exactly velocity ka matlab hai. Word aur symbol ek hi idea hain.

Is figure mein, lavender curve time ke saath position upar jaate dikhaata hai. Seedha coral line green dot par sirf chhuta hai — uski steepness (slope) hi us instant par velocity $\vec{u}$ hai. Steep coral line ⇒ tez motion. Yeh "velocity = slope" picture yaad rakho: hum abhi ek poore equation ko differentiate karne wale hain, aur differentiate karne ka matlab bas "donon sides ki slope lo" hai.

4. Acceleration — velocity kaise change hoti hai

Position par derivative do baar apply karo aur tumhe acceleration milti hai; woh stacked symbol $\frac{d^2}{d{t}^2}$ ka bas matlab hai "derivative ka derivative". Is topic ko $\vec{a}$ ki itni parwah kyun hai? Kyunki yeh woh ek quantity hai jis par do inertial observers agree karte hain — poore chapter ka anchor aur wajah ki Newton's laws of motion sabke liye ek jaisi dikhti hain.

5. Do frames $S$ aur $S^{\prime }$, aur relative velocity $\vec{V}$

Ab hum donon observers ko ek baar aur hamesha ke liye naam dete hain, taaki koi symbol bina parichay ke na aaye.

Is figure ko dhyan se padho — isme poora topic hai. Coral star ek flash (ek event) hai jise dono observers dekhte hain. Ground observer $S$ slate arrow $\vec{r}$ ko $O$ se star tak kheenchta hai. Train observer $S^{\prime }$ coral arrow $\vec{r}{}^{\prime }$ ko $O^{\prime }$ se usi star tak kheenchta hai. Neeche mint arrow drift $\vec{V}t$ hai — $O^{\prime }$ kitna slide hua hai. Teeno arrows ek tip-to-tail triangle banate hain.

6. Ek observer ke numbers ko dusre mein convert karna — derivation

Upar sab vocabulary tha. Yahaan payoff hai: dictionary jo $S$ ke numbers ko $S^{\prime }$ ke numbers mein convert karti hai.

Step 1 — triangle padho. §5 ki figure mein, $O$ se star tak pahunchne ke liye tum seedha ja sakte ho ($\vec{r}$), ya lamba rasta le sakte ho: pehle $O^{\prime }$ tak drift karo (woh hai $\vec{V}t$, mint arrow), phir $O^{\prime }$ se star tak (woh hai $\vec{r}{}^{\prime }$, coral arrow). Same destination, toh donon routes equal hain: $\vec{r}=\vec{V}t+\vec{r}{}^{\prime }.$ Yeh step kyun? Pure tip-to-tail vector addition — abhi tak koi physics nahi, bas triangle ki geometry jo tum dekh sakte ho.

Step 2 — primed position ke liye solve karo. $\vec{V}t$ ko donon sides se subtract karo taaki jo train observer measure karta hai woh ground observer ke measure ke terms mein mile:

Line $t^{\prime }=t$ woh hidden universal-time assumption hai jo ise Galilean banati hai. Yeh boxed pair hai "aha" moment: mujhe ground frame mein kisi bhi event ka $(\vec{r},t)$ do aur main tumhe train ka $(\vec{r}{}^{\prime },t^{\prime })$ de dunga.

Step 3 — velocity rule paane ke liye differentiate karo. $\vec{r}{}^{\prime }=\vec{r}-\vec{V}t$ ke donon sides ka slope-in-time ($\frac{d}{dt}$) lo. Kyunki $t^{\prime }=t$ hai donon clocks saath chalta hain, aur $\vec{V}$ constant hai toh $\vec{V}t$ ka slope bas $\vec{V}$ hai: $\frac{d\vec{r}{}^{\prime }}{dt}=\frac{d\vec{r}}{dt}-\vec{V}⟹\vec{u}{}^{\prime }=\vec{u}-\vec{V}.$ $\vec{V}$ ko doosri taraf move karo aur tumhare paas everyday form aa jaata hai:

Is figure mein velocities ke liye wahi triangle idea aata hai: coral arrow $\vec{u}{}^{\prime }$ (train mein velocity) ko tip-to-tail mint arrow $\vec{V}$ (train ki velocity) ke saath rakhne se slate arrow $\vec{u}$ (ground se velocity) milta hai. Position triangle ko differentiate karne se bas ek velocity triangle bana — isliye addition law bilkul wahi shape ka hai.

Step 4 — ek baar aur differentiate karo. Constant $\vec{V}$ ka slope-in-time zero hai, toh: $\vec{a}{}^{\prime }=\vec{a}.$ Dono observers same acceleration measure karte hain — woh invariance jis par poora chapter tika hua hai.

7. Prime marks aur hat

Prerequisite map

Ise neeche se upar padho: origin, sign convention aur vectors position arrow banate hain; ek clock add karo aur tumhe events milte hain; differentiate karo velocity paane ke liye, phir acceleration; constant frame velocity $\vec{V}$ dalo aur tum Galilean transformation bana sakte ho, jiska payoff yeh hai ki Newton's laws sabke liye unchanged rehte hain. Direct application ${\vec{u}}_{A/B}={\vec{u}}_A-{\vec{u}}_B$ ke liye Relative velocity dekho.

Equipment checklist

Khud test karo — daayein taraf cover karo aur reveal karne se pehle jawab do.