Projectile Motion in Special Theory of Relativity: Re-Investigation and New Dynamical Properties in Vacuum

The projectile motion (PP) in a vacuum is re-examined in this paper, taking into account the relativistic mass in special relativity (SR). In the literature, the mass of the projectile was considered as a constant during motion. However, the mass of a projectile varies with velocity according to Einstein’s famous equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>=</mo><mfrac><msub><mi>m</mi><mn>0</mn></msub><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>v</mi><mn>2</mn></msup><mo>/</mo><msup><mi>c</mi><mn>2</mn></msup></mrow></msqrt></mfrac></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mn>0</mn></msub></semantics></math></inline-formula> is the rest mass of the projectile and <i>c</i> is the speed of light. The governing system consists of two-coupled nonlinear ordinary differential equations (NODEs) with prescribed initial conditions. An analytical approach is suggested to treat the current model. Explicit formulas are determined for the main characteristics of the relativistic projectile (RP) such as time of flight, time of maximum height, range, maximum height, and the trajectory. The relativistic results reduce to the corresponding ones of the non-relativistic projectile (NRP) in Newtonian mechanics, when the initial velocity is not comparable to <i>c</i>. It is revealed that the mass of the RP varies during the motion and an analytic formula for the instantaneous mass in terms of time is derived. Also, it is declared that the angle of maximum range of the RP depends on the launching velocity, i.e., unlike the NRP in which the angle of maximum range is always <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi><mo>/</mo><mn>4</mn></mrow></semantics></math></inline-formula>. In addition, this angle lies in a certain interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mi>π</mi><mo>/</mo><mn>4</mn><mo>,</mo><mi>π</mi><mo>/</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula> for any given initial velocity (<<i>c</i>). The obtained results are discussed and interpreted. Comparisons with a similar problem in the literature are performed and the differences in results are explained.