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## Archive of Issues

Russia Yaroslavl
Year
2012
Issue
1
Pages
120-132
 Section Mechanics Title On integrating the projectile motion equations of a heavy point in medium with height decreasing density Author(-s) Chistyakov V.V.a Affiliations Yaroslavl State Agricultural Academya Abstract The resolvent method based on Legendre transformation was applied to integrate ballistic equations of a heavy point mass in inhomogeneous medium with the drag force being power-law with respect to speed, at that the coefficient of the drag force decreases linearly with height $y$. General expressions were obtained for resolvent function $a''_{bb}(b)$ with $a(b)$ being an intercept and $b={\rm tg}\,\theta$, where $\theta$ is inclination angle. In the second order by gradient $c/m^{-1}$ of perturbative approach, the universal formulas for $\delta a''_{bb}(b)$-, $\delta x(b)$-, $\delta y(b)$-additions were derived. The case of Releigh resistance was considered particularly in frames of the method above and inhomogeneity influence on the motion was investigated. The falling of gravity $g(y)$ is taken into consideration too. Keywords Legendre transformation, resolvent function, power law air drag, linear density inhomogenity UDC 531.55, 514.85 MSC 70E15, 34A26, 34A34 DOI 10.20537/vm120110 Received 12 December 2011 Language Russian Citation Chistyakov V.V. On integrating the projectile motion equations of a heavy point in medium with height decreasing density, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, issue 1, pp. 120-132. References http://www.snipercountry.com/ballistics/tables/ Paul Weinacht, Gene R. Cooper, James F. Newill. Analytical Prediction of Trajectories for High-Velocity Direct-Fire Munitions, Army Research Laboratory Report. http://www.arl.army.mil/arlreports/2005/ARL-TR-3567.pdf Thomas R.N. Some Comments on the Form of Drag Coefficient at Supersonic Velocity: report no. 542, U.S. Army Ballistic Research Laboratory: Aberdeen Proving Ground, MD, April, 1945. Robert F. Lieske. Determination of aerodynamic drag and exterior ballistic trajectory simulation for the 155MM, DPICM, M864 base-burn projectile, Memorandum report BRL-MR-3768, Aberdeen, Maryland, June, 1989. N. de Mestre. The Mathematics of Projectiles in Sport, Cambridge University Press, New York, 1990, 175 p. Edward J.R. A Treatise on Dynamics of a Particle with Numerous Examples, Cambridge Univ. Press, Cambridge, 1898, 418 p. Chudinov P.S. Numerical-analytical algorithmfor constructing the envelope of the projectile trajectories in the air, Vest. Perm. Univ. Mat. Mekh., 2009, no. 7 (33), pp. 90–94. Chistyakov V.V. On one resolvent method for integrating the low angle trajectories of a heavy point projectile motion under quadratic air resistance, Komp. Issled. Model., 2011, vol. 3, no. 3, pp. 265–277. Full text