Relativistic Analogue Hidden in Projectile Motion

We present a four-vector analogue U^μ ≡ (U⁰, U) = (|V′|, V) that can be directly constructed from the velocity of a projectile. Here, V and V′ are the velocities of a projectile at heights h + H and h, respectively. In the non-relativistic regime, the mathematical structure of this four-vector analogue has an exact correspondence to the relativistic counterpart four-velocity u^μ = γ(1, β)c of a massive particle. Based on this observation, we illustrate the design of an introductory laboratory experiment to investigate the Lorentz invariance and its covariant nature by measuring the velocity of a projectile. The experiment may help students to acquire a concrete perspective of Lorentz covariance through their own measurements and analyses of a free-fall motion.