The high-precise parametrical equation for the trajectory of a point mass projectile in medium with quadratic drag under head-, tail- or side wind conditions

Precise trajectory equation is deduced by using dual-projective variables for a heavy projectile motion in medium with quadratic in speed longitudinal wind. By integration by parts there were received the power type formulas for low angle trajectories with initial slopes $\Theta_0<15^\circ$. They use the following key parameters of motion, namely $b_0=\operatorname{tg}\Theta_0$, with $\Theta_0$ as an angle of throwing, $R_a$ as the top curvature radius and $\beta_0$ as dimensionless speed square in the highest point of the trajectory. These formulas for the coordinates and time $x(b)$, $y(b)$, $z(b)$ and $t(b)$ with $b=\operatorname{tg}\Theta$ being the current slope of the trajectory display strongly the effect of self-improving of accuracy due to diminishing of $\beta_0$ with the $b_0$ growing. Their precision when compared to exact integral formulas occurs to consist of 0.1–0.3 %% and this takes place in wide range of wind speeds up to 40 mps and with starting drag forces of 1.15 mg value. Due to their simplicity and quasi-algebraic type the formulas may be easily implemented in ballistic calculator, especially for the guns shooting as they moving at high speeds and in moving targets.