Relativistic Analogue Hidden in Projectile Motion
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Relativistic Analogue Hidden in Projectile Motion Jae Hoon Lim, Dong-Won Jung, U-Rae Kim , Sungwoong Cho and Jungil Lee
∗
KPOPE Collaboration, Department of Physics, Korea University, Seoul 02841, Korea (Received 4 August 2020; accepted 31 August 2020) We present a four-vector analogue Uμ ≡ ( 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 uμ = 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. Keywords: Special relativity, Lorentz transformation, Four-velocity, Free falling DOI: 10.3938/jkps.77.624
I. INTRODUCTION Relativity [1] education [2–19] on an undergraduate level confronts various obstacles [20–24]. Instructors as well as students are strongly conformed with the fundamental postulates of Euclidean geometry that are intuitively acceptable based on their own experiences with empirical evidence. These experiences are fortified by the Cartesian approach to analytic geometry involving right triangles, circles, and spheres. While both instructors and students are well aware of Euclidean geometry, they are rarely acquainted with the structure of the Minkowski spacetime. When teaching Einstein’s special theory of relativity, one usually follows the deductive approach based only on the postulates of relativity. The reason is that no easy-to-experience examples, except for extreme situations that can be realized only in an advanced laboratory, are available. Although some efforts have been made to introduce such experiments to the undergraduate laboratory [25–31], those experiments are not so popular. Thus, both instructors and students are liable to rely on abstract approaches without a direct demonstration. The situation is worsened by their own stereotyped concepts involving the Euclidean space. In this paper, we introduce a four-vector analogue Uμ = (|V |, V ) that can be extracted from the wellknown formula for the conservation of total mechanical energy in a projectile motion under a constant gravitational field: V 2 − V 2 = 2gH,
(1)
∗ Director of the Korea Pragmatist Organization for Physics Education (KPOP E), E-mail: [email protected]
pISSN:0374-4884/eISSN:1976-8524
where V and V are the initial and the final velocities, respectively, of a projectile whose height changes by −H and g is the gravitational acceleration. The employment of this popular equation as the starting point has advantages over the conventional approaches in the aspect that this example is already familiar to bo
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