Baran Complex Kinematics: A Unified Complex-Number Formulation of Constant-Acceleration 2D Motion

This paper presents a compact complex-number formulation of two-dimensional motion under constant acceleration. The central result is a general differential equation z̈ = a·e^(iβ), where a is the magnitude of acceleration and β is its direction angle in the complex plane. Integrating twice with initial velocity v₀·e^(iα) yields the position formula z(t) = v₀·t·e^(iα) + (1/2)·a·t²·e^(iβ), where α and β are independent polar angles encoding the directions of velocity and acceleration respectively.The special case β = −90° gives e^(iβ) = −i, recovering the projectile motion formula z(t) = v₀·t·e^(iα) − (i/2)·g·t². All standard projectile cases — free fall, vertical throw, horizontal throw, and oblique projectile — follow from this single formula by choosing α and v₀.To the author's knowledge, this formulation has not previously appeared in the literature in this compact form. Classical treatments invariably use two separate scalar equations for horizontal and vertical components. The present approach encodes both magnitude and direction of each kinematic vector in a single complex exponential, providing a more natural and unified description of planar motion. All ideas, intuitions, and discoveries are due to Abdullah Baran - Van Çatak. This document was typeset in LaTeX by Claude (Anthropic) acting as amanuensis.