The big idea: What you measure motion against is your reference frame: your own coordinate grid and clock.
On a train: Walk down the aisle of a moving train and you clock yourself at 3 km h⁻¹ relative to the train. Someone on the platform measures your speed against the ground and gets a bigger number.
Neither is wrong: Neither observer is 'wrong': all motion is relative, so 'how fast' only means anything compared to something else.
What is an inertial frame?: An inertial reference frame is one that moves at constant velocity — it does not accelerate (no speeding up, slowing down, or turning).
In an inertial frame an object with no resultant force stays still or keeps moving in a straight line at constant speed — Newton's first law holds.
Inertial (constant velocity)
- A train cruising at a steady 30 m s⁻¹ in a straight line
- A spaceship drifting with engines off
- The ground (good enough for most problems)
Non-inertial (accelerating)
- A train braking into a station
- A car going round a bend
- A spinning roundabout
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The Galilean transformation converts a position or velocity measured in one frame into the other. It is the everyday, low-speed rule for combining velocities.
The set-up: Suppose frame S' (e.g. a train) moves at constant velocity v relative to frame S (e.g. the ground).
- position measured in the moving frame (m)
- position measured in the ground frame (m)
- speed of the moving frame relative to the ground (m s⁻¹)
- time (the same in both frames, s)
- time of the event in the moving frame: the same as t (s)
- object's velocity measured in the moving frame (m s⁻¹)
- object's velocity measured in the ground frame (m s⁻¹)
Where u′ = u − v comes from: An object moves Δx in the ground frame in a time Δt. In the same time the train frame moves vΔt, so in the train frame the object moves Δx′ = Δx − vΔt, and the time is the same, Δt′ = Δt.
Divide: Δx′/Δt′ = Δx/Δt − v, which is u′ = u − v.
Mind the signs: Pick one positive direction and stick to it. If the object and the frame move the same way you subtract; if they move in opposite directions a sign flips and the speeds effectively add. Always sketch arrows first.
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A person walks at 1.5 m s⁻¹ toward the front of a train. The train moves at 12 m s⁻¹ relative to the ground. How fast does the person move relative to the ground?
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Car A travels east at 30 m s⁻¹ and car B travels east at 20 m s⁻¹. What is the velocity of car B as measured by the driver of car A? (Take east as positive.)
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No frame is special: Galileo's principle of relativity: the laws of mechanics are the same in every inertial frame.
Inside a smooth train: No mechanics experiment done inside a smoothly moving train can tell you the train is moving. Drop a ball and it falls straight down, just as on the platform.
Why Newton's laws agree: The two frames differ by a constant velocity v. Add the same constant to every velocity and every change in velocity stays the same, so both frames measure the same acceleration.
Same mass, same acceleration: F = ma gives the same force. That is why Newton's laws hold in every inertial frame, and why an accelerating frame is not inertial.
No absolute rest: There is no absolute rest frame: 'truly at rest' has no meaning, only 'at rest relative to ...'.
Where Galileo breaks down: Galilean velocity addition is an excellent approximation for everyday speeds. But measure the speed of light: it comes out the same — about 3 × 10⁸ m s⁻¹ — in every inertial frame, no matter how the source moves. Simple addition (u' = u − v) fails here. Fixing this is the job of special relativity (1.5.2).
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How this is tested — Galilean relativity is HL only (A.5):
Paper 1A
- A quick 'which is an inertial frame?' or 'is there an absolute rest frame?' (no).
- A one-step relative-velocity sum.
Paper 2
- Determine a relative velocity for objects moving along a line.
- State the principle of relativity and its limit.
The classic trap: Forgetting to sign the velocities. Objects moving in opposite directions have speeds that add, not subtract — a lost minus sign turns 55 m s⁻¹ into 5 m s⁻¹. Pick one positive direction and label every velocity first.
Three easy marks: (1) Choose a positive direction and label every velocity with a sign. (2) Same direction ⇒ subtract; opposite directions ⇒ the speeds add. (3) Remember the limit: Galilean addition works at low speed but not near the speed of light.
The words to know
| Word | What it means |
|---|---|
| Reference frame | A set of axes and a clock that positions and times are measured against |
| Inertial frame | A frame that is not accelerating: Newton's first law holds in it |
| Event | Something that happens at one place at one instant: (x, t) |
| Galilean transformation | x′ = x − vt and t′ = t, so u′ = u − v |
Two trains travel toward each other along the same straight track. One moves at 25 m s⁻¹ and the other at 30 m s⁻¹, both measured relative to the ground. Determine the speed at which the gap between them closes.
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