Section

Mechanics

Title

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

Author(s)

Chistyakov V.V.^{abc}

Affiliations

Mikhailovskaya Artillery Military Academy^{a},
Mozhaisky Military Space Academy^{b},
Saint Petersburg National Research University of Information Technologies, Mechanics and Optics^{c}

Abstract

Precise trajectory equation is deduced by using dualprojective 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 $b_0 = \rm{tg}\,\Theta_0$, 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 $b = \rm{tg}\, \Theta$ being the current slope of the trajectory display strongly the effect of selfimproving 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 0.10.3 %% and this takes place in wide range of wind speeds up to $40\, mps$ and with starting drag forces of $1.15$ $\rm{m\,g}$ value. Due to their simplicity and quasialgebraic 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.

Keywords

quadratic air drag, head/tail/side wind, projectile, dualprojective coordinates,
ballistic, direct fire angle, moving gun, trajectory, parametrical equation

UDC

531.55

MSC

34B15, 34C15, 70E15, 70K75

DOI

10.20537/vm150410

Received

14 September 2015

Language

Russian

Citation

Chistyakov V.V. The highprecise parametrical equation for the trajectory of a point mass projectile in medium with quadratic drag under head, tail or side wind conditions, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2015, vol. 25, issue 4, pp. 534543.

References

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 Routh E.J. A treatise on dynamics of a particle with numerous examples, Cambridge: Pergamon, 1898, 603 p.
 Yabushita K., Yamashita M., Tsuboi K. An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method, Journal of Physics A: Mathematical and Theoretical, 2007, vol. 40, no. 29, pp. 403412.
 Chistyakov V.V. On one resolvent method for integrating the low angle trajectories of a heavy point projectile motion under quadratic air resistance, Komp’yuternye Issledovaniya i Modelirovanie, 2011, vol. 3, no. 3, pp. 265277 (in Russian).
 Chistyakov V.V. On one numerical method of integrating the dynamical equations of projectile planar flight affected by wind, Bulletin of Peoples' Friendship University of Russia. Series Mathematics. Information Sciences. Physics, 2014, no. 3, pp. 138149.

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