Speeds do not simply add.

Not modeled: accelerated motion; spin dynamics and Thomas precession of real bodies (only the kinematic rotation is shown); dispersion and the medium's own physics in Fizeau-type setups; gravity.

/**
 * 1 - U/c for collinear composition, never `1 - composeCollinear`.
 * At 0.99 ⊕ 0.99 this is 1/19801.
 */
export function composedSpeedShortfall(beta1: number, beta2: number): KinematicResult<number> {
  const a = refuseBeta(beta1);
  if (a) return a;
  const b = refuseBeta(beta2);
  if (b) return b;
  return ok(((1 - beta1) * (1 - beta2)) / (1 + beta1 * beta2));
}

/** Cancellation-free (u ⊕ w) - u. */
export function compositionIncrement(
  u: number,
  w: number,
  c = speedOfLightMetresPerSecond(),
): KinematicResult<number> {
  if (![u, w, c].every(finite) || c <= 0) {
    return outsideDomain("nonfinite-input", "Increment needs finite speeds and a positive c.");
  }
  if (Math.abs(u) >= c || Math.abs(w) >= c) {
    return outsideDomain("superluminal-observer", "Increment inputs must be subluminal.");
  }
  return ok((w * (1 - (u * u) / (c * c))) / (1 + (u * w) / (c * c)));
}

/** Paper 3 §5 printed forms. V is c. alpha in radians. */
export function composePrinted(
  v: number,
  w: number,
  alpha: number,
  c = speedOfLightMetresPerSecond(),
): KinematicResult<number> {
  if (![v, w, alpha, c].every(finite) || c <= 0) {
    return outsideDomain(
      "nonfinite-input",
      "Printed composition needs finite speeds and a positive c.",
    );
  }
  if (Math.abs(v) >= c || Math.abs(w) >= c) {
    return outsideDomain("superluminal-observer", "Printed composition inputs must be subluminal.");
  }
  const cosA = Math.cos(alpha);
  const sinA = Math.sin(alpha);
  const den = 1 + (v * w * cosA) / (c * c);
  const inner = v * v + w * w + 2 * v * w * cosA - ((v * w * sinA) / c) ** 2;
  if (inner < 0 || den === 0) {
    return outsideDomain(
      "invalid-parameter",
      "Printed composition radicand or denominator is invalid.",
    );
  }
  return ok(Math.sqrt(inner) / den);
}

Speeds do not simply add. Two speeds below light's combine to a speed below light's, and light keeps its speed whatever other speed is added to it.

§5 of the relativity paper asks how a point moving with velocity w in the moving system k moves as seen from K. Putting its motion through the transformation of §3 gives, for w along the motion, U = (v + w)/(1 + vw/V²), and for w at an angle α to it, U = √((v² + w² + 2vw cos α) − (vw sin α/V)²)/(1 + vw cos α/V²). The parallelogram law of velocities therefore holds only to a first approximation. With both speeds at 0.6c along one line the lab gives 0.882c, where the Galilean sum would be 1.2c; at right angles it gives 0.768c instead of 0.849c. §5 draws two consequences: two speeds below V always compose to a speed below V, and composing V with any smaller speed gives V again, which the lab shows when the second speed is set to c. It adds that the collinear formula also follows from applying two transformations one after the other, and that such parallel transformations form a group. Two boosts that are not parallel do something §5 does not discuss: the result is a boost combined with a rotation of the axes, 12.68° for 0.6c along x followed by 0.6c at right angles. The lab also shows, as a later check, Fizeau's flowing water: light in water of index 1.333 moving at 7.06 m/s gains 3.09 m/s, the drag u(1 − 1/n²) that the same formula gives to first order.

Take v = 0.6c and w = 0.6c along the same line. The numerator is v + w = 1.2c. The denominator is 1 + vw/c² = 1 + 0.36 = 1.36. So U = 1.2c/1.36 = 0.882c. The denominator is always more than 1 when both speeds point the same way, so the result is always less than the plain sum. To see that it stays below c, write v = c − κ and w = c − λ with κ and λ positive, as §5 does; then U = c(2c − κ − λ)/(2c − κ − λ + κλ/c), and the extra κλ/c in the denominator makes the fraction less than 1. Now let w = c: U = (v + c)/(1 + v/c) = c(v + c)/(c + v) = c, whatever v is. At right angles, α = 90°, so cos α = 0 and sin α = 1, and U = √(0.36 + 0.36 − 0.36²) c = √(0.72 − 0.1296) c = √0.5904 c = 0.768c. In components the lab gives 0.6c along x and 0.48c across; the sideways part is w√(1 − v²/c²) = 0.6 × 0.8 = 0.48, because the moving clock that times the point's sideways motion runs slow. The inverse check takes the result back: removing the frame's 0.6c returns the original 0.6c. The rapidity view turns the collinear rule into addition. Each speed has a rapidity atanh(v/c), here atanh(0.6) = ln 2 = 0.693, and the composed speed's rapidity is their sum, 1.386, with tanh(1.386) = 0.882. For the two boosts at right angles, multiplying their 4 × 4 transformation matrices gives a matrix that is not a pure boost; written as a boost followed by a rotation, the rotation is 12.68° and the combined γ is 1.25 × 1.25 = 1.5625. Last, Fizeau: to first order in the water's speed u, light in the water moves at c/n + u(1 − 1/n²). For n = 1.333, 1 − 1/n² = 0.437, so water flowing at 7.06 m/s adds 7.06 × 0.437 = 3.09 m/s.