Vector Field of the Velocity in Projectile Motion: Mathematical Modeling and Empirical Validation

This work presents a detailed study on projectile motion when two marbles meet at their maximum height. I observed that the classic formulas in textbooks, as well as those found on web pages, do not mention the exceptional case of the moment when two marbles coincide in the air at their maximum height. Through an experimental analysis documented by high-precision videography and the use of integral and differential calculus tools, I was able to observe this finding, through the use of GraviTrax for kinematic experiments which, when combined with the rigor of physics and mathematics, has enormous educational value. After a persistent effort and careful preparation using complex formulas, I reinforced my hypothesis after observing the videos. The frames per second at the moment of collision are evidence that the velocity of each marble increases as a force is created at that point, attracting both marbles. As an independent researcher, I present this ansatz to the scientific community, demonstrating that the proposed formula accurately predicts the behavior of the marbles at that moment. Consequently, the validity of this study depends on monitoring the marbles' temperature at that moment. As demonstrated in the Methods section, the absence of temperature variation confirms the thermal stability necessary to validate the model. NOTE: The definitive, expanded Second Edition—featuring the exclusive 3-part Supplement: Analytical Extension in Kinematics Engineering and high-resolution color graphics—is available in print format on Amazon.