Relativistic Velocity Addition Law from Machine Gun Analogy

Many derivations of the relativistic addition law of parallel velocities without use of the Lorentz transformations (LT) are known.1–5 Some of them are based on thought experiments that require knowledge of the time dilation and the length contraction effects.1,4,5 Other derivations involve the Doppler effect in the optic domain considered from three inertial reference frames in relative motion.6 A few derivations simply involve only the principle of constancy of the light velocity.2 Such derivations are interesting for the teaching of special relativity theory since the relativistic addition of velocities leads directly to the LT.7 The derivation we propose is based on a machine gun-target analogy8 of the acoustic Doppler effect, considered from the rest frame of the machine gun and from the rest frame of the target.