The big idea: All motion is relative — you can only say how fast something moves compared to something else. The thing you measure against is your reference frame (your own coordinate grid and clock).
Walk down the aisle of a moving train and you measure 3 km h⁻¹ relative to the train. Someone on the platform measures your speed relative to the ground — a different number. Neither is 'wrong'; they just use different frames.
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
Suppose frame S' (e.g. a train) moves at constant velocity v relative to frame S (e.g. the ground). 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.
- 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)
- object's velocity measured in the moving frame (m s⁻¹)
- object's velocity measured in the ground frame (m s⁻¹)
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.
Learn what examiners really want
See exactly what to write to score full marks. Our AI shows you model answers and the key phrases examiners look for.
Worked example — walking on a train
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?
Solution
- Ground velocity = the person's velocity in the train plus the train's velocity (rearranging u = u' + v). Both point the same way (forward), so the speeds add:
- Substitute the numbers:
- Work it out — keep the unit:
Final answer
u = 13.5 m s⁻¹ forward. (Walk toward the back instead and it would be 12 − 1.5 = 10.5 m s⁻¹.)
Worked example — one car seen from another
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.)
Solution
- Use the given velocity transformation, with the observer's car A as the moving frame (v = 30):
- Substitute (u = 20 for car B, v = 30 for car A):
- Work it out — keep the unit:
Final answer
u' = −10 m s⁻¹, i.e. 10 m s⁻¹ westward — to driver A, car B appears to drift slowly backwards.
No frame is special: Galileo's principle of relativity: the laws of mechanics are the same in every inertial frame. 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.
This means 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).
Never wonder what to study next
Get a personalized daily plan based on your exam date, progress, and weak areas. We'll tell you exactly what to review each day.
Where it shows up: Galilean relativity is HL only (A.5):
- Paper 1A — a quick 'which is an inertial frame?', 'is there an absolute rest frame?' (no), or a one-step relative-velocity sum. - Paper 2 — determine a relative velocity for objects moving along a line, or state the principle of relativity and its limit.
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.
IB-style question — two approaching trains
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 (their relative speed of approach).
Solution
- Use the given velocity transformation, taking one train's frame as moving. Let train 1 go +25 and train 2 go −30 (opposite directions):
- Substitute (u = −30 for train 2, v = +25 for train 1's frame):
- Take the magnitude — keep the unit:
Final answer
The trains approach at 55 m s⁻¹ — because they move oppositely, the speeds add.