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Section
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Mechanics
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Title
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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
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Author(-s)
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Chistyakov V.V.abc
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Affiliations
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Mikhailovskaya Artillery Military Academya,
Mozhaisky Military Space Academyb,
Saint Petersburg National Research University of Information Technologies, Mechanics and Opticsc
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Abstract
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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 $b_0 = \rm{tg}\,\Theta_0$, 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 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 $1.15$ $\rm{m\,g}$ 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.
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Keywords
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quadratic air drag, head/tail/side wind, projectile, dual-projective coordinates,
ballistic, direct fire angle, moving gun, trajectory, parametrical equation
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UDC
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531.55
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MSC
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34B15, 34C15, 70E15, 70K75
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DOI
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10.20537/vm150410
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Received
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14 September 2015
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Language
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Russian
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Citation
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Chistyakov V.V. 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, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2015, vol. 25, issue 4, pp. 534-543.
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References
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- Euler L. Recherches sur la veritable courbe que decrivent les corps jettes dans l'air ou dans un autre fluide quelconque, Memoires de 'academie des sciences de Berlin, 1755, vol. 9, pp. 321-352.
- 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. 403-412.
- 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. 265-277 (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. 138-149.
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Full text
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