|
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
|
|
+7 (3412) 91 60 92
Russian