Semianalytical Approximation of the Fluid Drag in Projectile Motion Using Newton's Equation of Motion

A body moving in a fluid experiences some of the fluid-induced forces, which may derail the body from its fluid-free trajectory and consequently transform its maximum height reached, time of flight, and range. These forces include upthrust, drag, and lift. Other than upthrust, an accurate description of drag and lift remains uncertain. Within the quadratic velocity model, the drag acting opposite to the direction of motion and the lift acting upwards and perpendicular to the drag are usually empirically expressed using the same formula, with the difference arising only in the direction of the force and in the coefficients of drag and lift, respectively. Starting from the equations of motion, we derive the quadratic velocity model for the drag and lift forces analytically. Besides the Magnus force, which is the upward lifting force, we show that there is a corresponding push force acting in the forward direction, while the drag acts in the backward direction. The derived model can explain the origin of a spin-induced aerodynamic lift when a body is catapulted along a horizontal runway with some tilt angle. In our work, we adopt a general drag formula, integrating both linear and quadratic drag, while neglecting the effects of lift, and then solving the resulting nonlinear coupled second-order ordinary differential equations using a mean field approximation and an infinite power series method. Our study suggests the existence of a nice analytical formula for the trajectory of the projectile incorporating the effects of both linear and quadratic drag forces simultaneously as independent processes with well-defined scaling functions. The effect of the conservative part of the drag equation can be determined perturbatively, while the dissipative part is exactly calculated, within the suggested mixed model framework. Our findings demonstrate that at higher velocities, quadratic drag effects dominate, while the linear drag effects can be treated as a simple perturbation to the quadratic drag.