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1.1.22Measurement, Vectors & Kinematics

Reference frames — Galilean transformations

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🇮🇳 Yeh note Hinglish mein padho →

WHAT is a reference frame?

An event is anything that happens at a definite place and time, e.g. "a firecracker pops at position r⃗\vec r at time tt". Different observers label the same event with different numbers.

HOW the transformation is built (derive from scratch)

Set up two frames:

  • SS (the "ground"/unprimed frame) with origin OO.
  • S′S' (the "train"/primed frame) with origin O′O', moving at constant velocity V⃗\vec V relative to SS.
  • Arrange that the origins coincide at t=0t=0, and clocks read the same: t′=tt' = t.

Step 1 — Locate O′O' as seen from SS. Since O′O' starts at OO and moves at V⃗\vec V, after time tt it sits at V⃗t\vec V t.

Why this step? This is just "distance = velocity × time" for the moving origin.

Step 2 — Add the vectors. For any event at r⃗\vec r in SS and r⃗ ′\vec r\,' in S′S', the position of the event relative to OO equals (position of O′O' relative to OO) + (position of event relative to O′O'): r⃗=V⃗t+r⃗ ′\vec r = \vec V t + \vec r\,'

Why this step? It is the head-to-tail rule for vectors — pure geometry, no physics yet.

Step 3 — Solve for the primed coordinates and add the time assumption.

The line t′=tt'=t — absolute (universal) time — is the hidden assumption that defines this transformation as Galilean.

Step 4 — Differentiate to get velocities. Take ddt\dfrac{d}{dt} of r⃗ ′=r⃗−V⃗t\vec r\,' = \vec r - \vec V t, remembering t′=tt'=t so ddt′=ddt\frac{d}{dt'}=\frac{d}{dt} and V⃗\vec V is constant: u⃗ ′=u⃗−V⃗\vec u\,' = \vec u - \vec V

Why this step? Velocity is the time-derivative of position; differentiating the position rule is the velocity rule.

Step 5 — Differentiate again to get acceleration. V⃗\vec V is constant, so dV⃗dt=0\dfrac{d\vec V}{dt}=0: a⃗ ′=a⃗\vec a\,' = \vec a

Figure — Reference frames — Galilean transformations

Worked examples

Forecast-then-Verify

Recall Predict before reading

Two cars approach head-on, each at 30 m/s30\text{ m/s} relative to the road. Forecast: what is car A's speed in car B's frame? Verify: Use u⃗A/B=u⃗A−u⃗B\vec u_{A/B} = \vec u_A - \vec u_B. Take east positive: uA=+30u_A = +30, uB=−30u_B = -30. Then uA/B=30−(−30)=60 m/su_{A/B} = 30 - (-30) = 60\text{ m/s}. The closing speed is the sum of the road-speeds because they move oppositely.

Common mistakes (steel-manned)

Recall Explain to a 12-year-old (Feynman)

Imagine you're on a smooth-moving train and you toss a ball straight up. To you it goes straight up and comes straight down. But your friend on the platform sees the ball move forward with the train as it goes up and down — they see a curve! You both see different paths and different speeds. BUT here's the cool part: you both agree on how fast the ball speeds up or slows down (its acceleration). That shared agreement is why the same rules of physics work on the train and on the ground. The "Galilean transformation" is just the recipe to turn your numbers into your friend's numbers: shift positions by "train-speed × time", and keep time the same.

Connections

  • Vectors — addition and components — velocity addition is vector addition.
  • Relative velocity — direct application: u⃗A/B=u⃗A−u⃗B\vec u_{A/B} = \vec u_A - \vec u_B.
  • Inertial and non-inertial frames — pseudo-forces appear when V⃗\vec V isn't constant.
  • Newton's laws of motion — invariant under Galilean transforms (Galilean relativity).
  • Special relativity — Lorentz transformation — replaces Galilean when v∼cv \sim c, drops t′=tt'=t.

Flashcards

What is a reference frame?
An observer's coordinate system (origin + axes) plus a clock, used to assign position and time to events.
State the Galilean position transformation.
r⃗ ′=r⃗−V⃗t\vec r\,' = \vec r - \vec V t with t′=tt' = t, where V⃗\vec V is the constant relative velocity of S′S' w.r.t. SS.
What hidden assumption distinguishes Galilean from Lorentz transforms?
Absolute/universal time, t′=tt' = t (clocks tick the same in all frames).
Galilean velocity addition formula?
u⃗=u⃗ ′+V⃗\vec u = \vec u\,' + \vec V (object velocity in ground frame = velocity in moving frame + frame velocity).
Why is acceleration the same in all inertial frames?
Because V⃗\vec V is constant, dV⃗/dt=0d\vec V/dt = 0, so differentiating u⃗=u⃗′+V⃗\vec u=\vec u'+\vec V gives a⃗=a⃗′\vec a = \vec a'.
What is an inertial frame?
A frame moving at constant velocity relative to another inertial frame; Newton's laws hold without pseudo-forces.
Two cars head-on each at 30 m/s; speed of one relative to the other?
uA/B=uA−uB=30−(−30)=60u_{A/B}=u_A-u_B = 30-(-30)=60 m/s.
Why do pseudo-forces appear in accelerating frames?
Because a⃗′=a⃗−A⃗\vec a' = \vec a - \vec A; the extra −A⃗-\vec A acts like a force not from any real interaction.
A boat at 4 m/s north on a 3 m/s east river — ground speed?
32+42=5\sqrt{3^2+4^2}=5 m/s, at ≈53∘\approx 53^\circ north of east.
What is Galilean (Newtonian) relativity?
The principle that the laws of mechanics (F⃗=ma⃗\vec F=m\vec a) take the same form in all inertial frames.

Concept Map

assigns

two examples

two examples

moves at

head-to-tail geometry

enters

defines as Galilean

differentiate once

differentiate again

so observers agree on

corrected by

Reference frame: origin axes clock

Event: place plus time

Frame S ground

Frame S' train

Constant velocity V

Absolute time t'=t

Position rule r'=r-Vt

Velocity rule u=u'+V

Acceleration invariant a'=a

Same laws of mechanics

Einstein relativity

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, reference frame ka simple matlab hai ek observer jiske paas ruler aur ghadi hai. Train me baitha banda apne aap ko rest pe samajhta hai, platform wala usko move karta hua dekhta hai. Dono sahi hain — bas unke numbers alag hain. Galilean transformation wahi dictionary hai jo ek observer ke measurements ko doosre ke measurements me convert karta hai, ye maante hue ki time sabke liye same hai (t′=tt'=t).

Formula nikalna easy hai: agar S′S' frame constant velocity V⃗\vec V se chal raha hai, to uska origin tt time me V⃗t\vec V t aage chala jaata hai. Vector jodne se r⃗=V⃗t+r⃗ ′\vec r = \vec V t + \vec r\,'. Isko time ke saath differentiate karo to velocity rule milta hai: u⃗=u⃗′+V⃗\vec u = \vec u' + \vec V — yaani object ki ground speed = train ke andar ki speed + train ki apni speed. Phir se differentiate karo to a⃗′=a⃗\vec a' = \vec a, kyunki V⃗\vec V constant hai uska derivative zero. Matlab acceleration sab inertial frames me same rehta hai.

Yahi reason hai ki Newton ke laws (F=maF=ma) har inertial frame me bilkul same dikhte hain — isko Galilean relativity kehte hain. Yaad rakho: position me VtVt ka shift, velocity me VV ka shift, acceleration me kuch nahi. Ek bada mistake students karte hain — 2D me speeds ko seedha jod dete hain; nahi, ye vector equation hai, components me karo. Ye model tab tak perfect hai jab speed light se bahut kam ho (v≪cv \ll c); high speed pe Lorentz transformation aata hai jahan t′≠tt' \ne t ho jaata hai.

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Connections

Non-inertial reference frames — pseudo forcesPhysics · 1.2.13Special relativity — Michelson-Morley experimentPhysics · 2.3.25Relative motion — 1D and 2D; river-boat problemsPhysics · 1.1.216DOF equations — translational (Newton), rotational (Euler's equations)Physics · 3.4.5IMU — integrated accelerometer + gyroscopePhysics · 3.5.15GPS — pseudorange, trilateration, dilution of precisionPhysics · 3.5.18