How this calculator works
Velocities along one axis combine using the Lorentz transformation rather than ordinary addition. The moving frame travels at v relative to the lab, and the projectile travels at w measured in that moving frame. Positive signs point along the chosen axis; negative signs point against it. Inertial frames require |v| < c. Projectile velocities may equal plus or minus c for light. Two observations of light in the same direction cannot determine the frame speed uniquely. Transverse velocity components require a different transformation.
Formula
u = (v+w)/(1+vw/c²); w = (u-v)/(1-uv/c²); v = (u-w)/(1-uw/c²).
Worked example
A ship at 0.6c launching a projectile at 0.6c in its own frame gives lab speed (1.2/1.36)c, about 0.882352941c. Launching at -0.6c instead gives zero lab velocity.
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Frequently asked questions
Can a negative result be valid?
Yes. It means motion opposite the positive axis defined for the two frames.
Does this keep light at c?
Yes. A light ray with w = c is measured at u = c in every admissible inertial frame.
Can I combine sideways motion?
This calculator covers collinear velocities. A transverse component also depends on the Lorentz factor.