Reference frames — Galilean transformations

1.1.22 · Physics › Measurement, Vectors & Kinematics

WHAT is a reference frame?

Ek event kuch bhi hai jo ek definite jagah aur time par hota hai, jaise "ek patakha position $r$ par time $t$ par phatata hai". Alag-alag observers usi same event ko alag-alag numbers se label karte hain.

HOW the transformation is built (derive from scratch)

Do frames set up karo:

$S$ ("ground"/unprimed frame) origin $O$ ke saath.
$S^{'}$ ("train"/primed frame) origin $O^{'}$ ke saath, jo $S$ ke relative constant velocity $V$ se move kar raha hai.
Arrange karo ki origins $t = 0$ par coincide karein, aur clocks same padhein: $t^{'} = t$ .

Step 1 — $O^{'}$ ko $S$ se dekho. Kyunki $O^{'}$ $O$ se shuru hota hai aur $V$ se move karta hai, time $t$ ke baad yeh $V t$ par hoga.

Yeh step kyun? Yeh bas moving origin ke liye "distance = velocity × time" hai.

Step 2 — Vectors ko add karo. Kisi bhi event ke liye jo $S$ mein $r$ par hai aur $S^{'}$ mein $r^{'}$ par, $O$ se event ki position barabar hai (O ke relative $O^{'}$ ki position) + ( $O^{'}$ ke relative event ki position): $r = V t + r^{'}$

Yeh step kyun? Yeh vectors ke liye head-to-tail rule hai — pure geometry, abhi tak koi physics nahi.

Step 3 — Primed coordinates ke liye solve karo aur time assumption add karo.

$t^{'} = t$ wali line — absolute (universal) time — woh hidden assumption hai jo is transformation ko Galilean define karti hai.

Step 4 — Velocities paane ke liye differentiate karo. $r^{'} = r - V t$ ka $\frac{d}{d t}$ lo, yaad rakhte hue ki $t^{'} = t$ isliye $\frac{d}{d t ^{'}} = \frac{d}{d t}$ aur $V$ constant hai: $u^{'} = u - V$

Yeh step kyun? Velocity position ki time-derivative hai; position rule ko differentiate karna hi velocity rule hai.

Step 5 — Acceleration paane ke liye phir differentiate karo. $V$ constant hai, isliye $\frac{d V}{d t} = 0$ : $a^{'} = a$

Figure — Reference frames — Galilean transformations

Worked examples

Forecast-then-Verify

Recall Padhne se pehle predict karo

Do cars head-on approach kar rahi hain, dono $30 m/s$ se road ke relative. Forecast: car B ke frame mein car A ki speed kya hai? Verify: $u_{A / B} = u_{A} - u_{B}$ use karo. East ko positive lo: $u_{A} = + 30$ , $u_{B} = - 30$ . Phir $u_{A / B} = 30 - (- 30) = 60 m/s$ . Closing speed road-speeds ka sum hai kyunki woh opposite direction mein move kar rahi hain.

Common mistakes (steel-manned)

Recall Ek 12-saal ke bachche ko samjhao (Feynman)

Socho tum ek smoothly chalti train mein ho aur tum ek ball seedhi upar uchhalte ho. Tumhare liye woh seedhi upar jaati hai aur seedhi neeche aati hai. Lekin platform par tumhara dost dekhta hai ki ball upar-neeche jaate hue train ke saath aage move karti hai — unhe ek curve dikhta hai! Tum dono alag-alag paths aur alag-alag speeds dekhte ho. LEKIN yahan cool part hai: tum dono is baat par agree karte ho ki ball kitni fast speed up ya slow down hoti hai (uska acceleration). Yeh shared agreement hi wajah hai ki physics ke same rules train mein aur ground par kaam karte hain. "Galilean transformation" bas woh recipe hai jo tumhari numbers ko tumhare dost ki numbers mein turn karti hai: positions ko "train-speed × time" se shift karo, aur time same rakho.

Connections

Vectors — addition and components — velocity addition vector addition hai.
Relative velocity — direct application: $u_{A / B} = u_{A} - u_{B}$ .
Inertial and non-inertial frames — pseudo-forces tab appear hoti hain jab $V$ constant nahi hota.
Newton's laws of motion — Galilean transforms ke under invariant hain (Galilean relativity).
Special relativity — Lorentz transformation — Galilean ko replace karta hai jab $v \sim c$ ho, $t^{'} = t$ drop kar deta hai.

Flashcards

Reference frame kya hota hai?

Ek observer ka coordinate system (origin + axes) plus ek clock, jo events ko position aur time assign karne ke liye use hota hai.

Galilean position transformation batao.

r^{'} = r - V t

with

t^{'} = t

, jahan

V

S^{'}

S

ke w.r.t. constant relative velocity hai.

Woh hidden assumption kya hai jo Galilean ko Lorentz transforms se alag karti hai?

Absolute/universal time,

t^{'} = t

(clocks sabhi frames mein same tick karte hain).

Galilean velocity addition formula?

u = u^{'} + V

(ground frame mein object velocity = moving frame mein velocity + frame velocity).

Acceleration sabhi inertial frames mein same kyun hota hai?

Kyunki

V

constant hai,

d V / d t = 0

, isliye

u = u^{'} + V

ko differentiate karne par

a = a^{'}

milta hai.

Inertial frame kya hota hai?

Ek frame jo doosre inertial frame ke relative constant velocity se move kar raha ho; Newton ke laws pseudo-forces ke bina hold karte hain.

Do cars head-on dono 30 m/s par; ek ki doosre ke relative speed?

u_{A / B} = u_{A} - u_{B} = 30 - (- 30) = 60

m/s.

Accelerating frames mein pseudo-forces kyun appear hoti hain?

Kyunki

a^{'} = a - A

; extra

- A

ek aisi force ki tarah act karta hai jo kisi real interaction se nahi hai.

Ek boat 4 m/s north par 3 m/s east river mein — ground speed?

3^{2} + 4^{2} = 5

m/s,

\approx 5 3^{\circ}

north of east par.

Galilean (Newtonian) relativity kya hai?

Yeh principle ki mechanics ke laws (

F = m a

) sabhi inertial frames mein same form lete hain.