A carrier ship moves relative to the laboratory and launches a probe at a velocity measured in the carrier's frame. The Galilean answer adds the two speeds; the relativistic answer is v/c = (β + u′/c)/(1 + βu′/c). The simulator draws both so you can watch the classical prediction exceed c while the correct result never does.
• A moving carrier ship, the laboratory probe trajectory, a classical comparison lane and a rapidity composition display. • Controls for carrier velocity (-0.8c to +0.8c), probe velocity in the carrier frame (-0.95c to +0.95c), and toggles for the classical lane and explanatory markers. • Readouts: relativistic lab velocity over c, classical sum over c, carrier rapidity, probe relative rapidity, total rapidity and the margin 1 − |v/c|. • Experiments: two 0.8c velocities give about 0.9756c relativistically versus the invalid classical 1.6c, and opposite velocities leave the probe at rest in the lab.
Define rapidity η = atanh(β). For collinear boosts η_total = η_carrier + η_probe, and v/c = tanh(η_total). Because tanh never reaches 1, no pair of subluminal inputs can combine to reach or exceed c. The rapidity panel in the lab shows this addition directly.
Motion is collinear and inertial, with subluminal input velocities. The lanes share an illustrative position scale and the launch age repeats every 4 animation seconds. Negative values mean the opposite direction. The classical lane is a counterexample and not a valid relativistic motion.
No. The relativistic formula always returns a value below c for inputs below c, and in rapidity terms tanh of a finite sum is always less than 1.
Rapidity. The simulator displays carrier, probe and total rapidity, and converts the total back to velocity with tanh.
The probe is at rest in the laboratory frame. The Opposite velocities experiment sets this up: rapidities cancel and the lab velocity is zero.
When both velocities are much smaller than c, the denominator 1 + βu′/c is almost 1, so the relativistic result approaches the simple sum. The gap only grows noticeably at high speeds.