Relativistic Velocity Addition

Source

Collection: Wiki SR simulations (v1) — SuperSTEM special-relativity engine.
Concept: Velocities combine as u⊕v = (u+v)/(1+uv) (c = 1), so two sub-light speeds never add up to c or beyond.
Serves: Intuition that the naive sum u+v overshoots the true result; the denominator (1+uv) keeps every combination strictly below the speed of light.
Inquiry: What does relativity say two 0.9c speeds combine to? What happens as both inputs approach c? What sets the gap between the Galilean and relativistic sums?
Misconception: That velocities simply add (0.9c + 0.9c = 1.8c), or that two large speeds can combine to exactly reach c.

Pedagogy principles applied

ID	Principle	How it shows up here
H1	Misconception is structural, not a text card	The "velocities just add to 1.8c" and "exactly reaches c" errors are wrong prediction choices; the rebuttal appears only if the student picks one.
D6/D7	Guided inquiry reveals in stages	Five sidebar cards run predict → read the number line → push both toward c → isolate the gap → resolve with the law.
A2	Fewest objects	Opens with two velocity sliders and their two sums; no metric cards or counts that no step reads.
C	Every number earns its place	u, v, the relativistic sum, the Galilean sum and their gap are each read, predicted, or compared by a step.
Phys	No false animation	Velocities are static sliders (SR frames are inertial); nothing auto-sweeps β over time and no marker rides a curve untethered from a slider.

Speed

Number linewhere does each sum land?

u v Galilean u+v Relativistic u⊕v forbidden (|value| > c)

The race against lighthow far each sum travels vs a light pulse

light (c) relativistic sum Galilean sum

Guided Inquiry

1 · Predict the combined speed

A ship flies at 0.9c; it fires a projectile forward at 0.9c relative to the ship. Galilean addition gives 0.9c + 0.9c = 1.8c. What does relativity say the ground observer measures?

2 · Read the number line

The sliders are set to u = v = 0.6c. On the number line, the blue and green arrows are your inputs; the red arrow is the Galilean sum and the cyan arrow is the relativistic sum. Notice the cyan arrow sits inside the ±c barriers while red can spill past.

3 · Push both toward c

Drag u and v each up to 0.99c. As both inputs approach c, the relativistic result …

4 · What controls the gap?

The gap between the Galilean and relativistic sums is (u+v)·uv / (1+uv). Set one slider to 0 and the other near c (e.g. u = 0, v = 0.9c). What happens to the gap?

5 · The velocity-addition law

Relativity replaces u + v with u⊕v = (u+v)/(1+uv). The denominator keeps the result below c for any sub-light inputs, and the correction is set by the product uv — it vanishes at everyday speeds (uv ≈ 0) and bites hardest near c. Turn on ∑ Formal for the algebra, or keep exploring with the sliders. Hit Finish to tuck this away.

Set the two velocities

▾

Velocity u +0.60 c

−0.99c+0.99c

Velocity v +0.60 c

−0.99c+0.99c

Results

▾

Relativistic u⊕v—

Galilean u+v—

Gap (u+v − u⊕v)—

Velocity-addition law

Gap: Galilean − relativistic

Sub-light stays sub-light