At a glance
A train and a plane both travel East along one road, and three observers measure them: Alice on the ground, Bella on the train, and the pilot. The sim shows how the same motions are described by each observer, and how those descriptions are linked.
What you'll learn
- How an observer's own motion enters the velocity they read for any object.
- Which quantities depend on the observer, and which do not.
- How to find the size and sign of any reading from the ground velocities.
- Where this everyday rule stops working, and what replaces it.
Speeds as Alice, on the ground, measures them.
What's on screen
- Alice, Bella and the plane, each with a velocity arrow and a signed km/h label. The arrow's length is proportional to speed, and its direction is the sign.
- Frame badge (top right): whose frame is active and its speed W relative to the ground. The position ruler is in that frame.
- Velocities panel: each object's reading in the active frame, with the arithmetic underneath.
See itStart in Alice's ground frame. Watch for: three arrows above the road, and three signed readings in the Velocities panel.
The controls
Press ⌖ on a row to see where that control sits in the sim.
| Control | What it changes | Range | |
|---|---|---|---|
| Reference frame | Which observer's measurements the sim shows: Alice (W = 0), Bella (W = V) or the plane (W = u). Choosing a frame restarts the run from the centre. | 3 frames opens on Alice | |
| Train speed V | Bella's speed over the ground, East. | 0–400 km/h opens at 100 | |
| Plane speed u | The plane's speed over the ground, East. | 0–600 km/h opens at 200 | |
| Show plane | Adds or removes the plane, its slider, its frame button and its readout row. | on / off opens on | |
| Velocities | The reading of each object in the active frame, with the arithmetic that produced it. | panel | |
| ▶ Play / ⏸ Pause | Runs the motion or freezes it. | top bar | |
| Speed | Plays the animation faster or slower. The velocities do not change. | 0.25×–4× opens at 1× | |
| ↻ Reset | Restores V = 100, u = 200, Alice's frame and Show plane on. It also clears your Guided Inquiry answers. | top bar |
Fixed: all motion is along one line, East positive. V and u are always measured in Alice's frame. On screen, 1 hour passes in 2 seconds at 1× speed. The scene wraps when objects leave the edge.
Two frames, one clock
A frame is an observer's coordinate system. Bella's frame moves East at V relative to Alice's, with the two origins together at t = 0, and both frames use one shared clock. By time t Bella's origin has moved Vt, so each position she quotes is Alice's minus Vt.
The Galilean transformation. The course notes write the primed frame with a tilde; the sim uses primes.
Because time is shared, a rod has the same length in both frames: mark both ends at one instant and each position loses the same Vt.
The sim's defaults, V = 100 and u = 200 km/h, read in two frames. Each arrow on the right is the left arrow minus W.
Key idea. Galilean relativity assumes time is the same for every observer. The rest of this sheet follows from that one assumption.
See itSwitch to Bella's frame. Watch for: Bella's own reading falling to 0, and the frame badge on the canvas showing W = V.
Subtract the frame's speed
Divide the shifted separation by the unshifted time and a velocity rule comes out:
- Subtract, never add. Both fly East, so Bella is chasing the plane. Adding the speeds would give the closing speed of two objects heading toward each other.
- Every reading shifts by the same amount. The whole velocity axis slides by W, so the gaps between any two readings stay put.
- The sign is a direction. If V is larger than u, Bella reads the plane as moving West, although it still flies East for Alice.
The same three velocities read twice: W = 0 above, W = V = 100 below.
See itMake the train faster than the plane (V = 300, u = 200), in Bella's frame. Watch for: the plane's arrow pointing West, with a negative reading.
Any observer, the same rule
W is the active frame's ground velocity: 0 for Alice, V for Bella, u for the pilot. Every reading in the Velocities panel is the ground velocity minus W. The pilot's frame works exactly like Bella's.
V
u
These sliders move the sim too.
See itSwitch to the pilot's frame. Watch for: the plane's own reading falling to 0, and the frame badge on the canvas showing W = u.
What every observer agrees on
- Differences. Subtract any two readings in one frame and the shift W cancels. The plane's velocity relative to the train is u − V in all three frames.
- Accelerations. V is constant, so differentiating again gives a′ = a. Mass and force are the same in both frames too, so F = ma keeps its form in every frame that moves at constant velocity.
- No special frame. Among frames moving at constant velocity, the ground is one choice, not the true one. No mechanical experiment inside a train at constant velocity can tell that it moves.
See itVisit all three frames in turn and compare the plane with the train in each. Watch for: the same difference every time, though every reading changes.
Where plain subtraction fails
The sim always subtracts exactly. The relativistic law, shown here for comparison, divides the same difference by a factor that includes c = 299 792 458 m/s ≈ 1.08 × 10⁹ km/h:
- At road speeds the factor equals 1 to within 10⁻¹². Even at V = 400 and u = 600 km/h, uV/c² is about 2 × 10⁻¹³, so no road experiment can tell the two laws apart.
- For light the laws disagree. Subtraction predicts c − V. Put u = c in the relativistic law and every observer reads c, whatever V is.
The plane arrows are to scale. The dashed c − V is schematic: at V = 100 km/h the real shortfall would be about 1 part in 10⁷.
See itSet the fastest settings (V = 400, u = 600) in Bella's frame. Watch for: the plane read as exactly 600 − 400 = +200 km/h, with uV/c² of about 2 × 10⁻¹³.
Where it matters
- Navigation. An aircraft's velocity over the ground is its velocity through the air plus the air's velocity over the ground. This is the same bookkeeping, and it is how groundspeed is found from airspeed and wind.
- Overtaking. A car at 90 km/h passing a lorry at 80 km/h closes at 10 km/h. The time to pass depends on that relative speed, not on 90.
- Galileo's ship. Galileo argued that nothing seen or done below deck on a smoothly moving ship reveals its motion. Section 06 gives the modern reason: accelerations are the same in every constant-velocity frame.
Go deeper: when the correction matters
- The factor 1 − uV/c² matters only when u and V are a sizeable fraction of c. With u = 0.9c and V = 0.5c, subtraction gives 0.4c, but the relativistic law gives 0.4c / 0.55 ≈ 0.727c.
- For |u| and |V| below c, the relativistic result never reaches c. Lecture 7 shows what replaces absolute time.
Summary
In one line. An observer moving at W reads every ground velocity v as v − W. That rule is exact if time is absolute, and at road speeds it matches relativity to better than 1 part in 10¹².
Key points
- A velocity belongs to an object and an observer together. Change the observer and the reading changes.
- Readings shift by W; differences between readings, and accelerations, do not.
- The sign of a reading is a direction in the active frame, East positive.
- For light the rule fails: the relativistic law gives c in every frame.
Common mix-ups
- "Bella reads the plane at 300 km/h." Adding gives the closing speed of objects heading toward each other. Both fly East, so she reads u − V.
- "Alice is at rest, so Bella reads 0 for her." Alice is at rest only in her own frame. In Bella's frame the roadside moves West at V.
- "A negative reading means the plane turned around." The sign is a direction in the active frame. At V = 300 the plane still flies East for Alice.
Things to try on your own
- Set V = 250 and u = 200. Work out all nine readings (three objects, three frames), then check each one in the Velocities panel.
- Find a setting where the pilot reads Bella at exactly 0, then read the plane in Bella's frame.