The Envelope of a Family of Elliptic Orbits and Ballistic Trajectories
Keywords:
safety domain, Keplerian orbits, foci of the ellipse, elliptical mirror, ballistic trajectories
Abstract
The paper presents a simple geometric derivation for the “safety domain” boundary for the family of Keplerian orbits of satellites or ballistic missiles, moving in a central field of gravity, and for the family of parabolic trajectories of projectiles in a uniform gravitational field launched from a common starting point at equal velocities. Examples of practical application of the envelope of a family of orbits for solving ballistics problems are discussed. These examples are illustrated by computer simulations.
References
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[2] A. P. French, "The Envelopes of Some Families of Fixed-Energy Trajectories," Am. J. Physics, vol. 61, no. 9, pp. 805-811, 1993; doi:10.1119/1.17429
[3] A. G. Alenitsyn, E. I. Butikov, and A. S. Kondrat'ev, Kratkii fiziko-matematicheskii spravochnik [Quick physical-mathematical reference book], Moscow: Nauka, 1997 (in Russian).
[4] M. Baće, S. Ilijić, Z. Naranci欬, and L. Bistricić, “The envelope of projectile trajectories ,” Eur. J. Phys., vol. 23, no. 6, pp. 637–642, 2002; doi:10.1088/0143-0807/23/6/305
[5] E. I. Butikov, “Comment on ‘The envelope of projectile trajectories',” Eur. J. Phys., vol. 24, no. 4, pp. L5–L9, 2003; doi: 10.1088/0143-0807/24/4/101
[6] E. I. Butikov, A. A. Bykov, A. S. Kondrat'ev, Fizika dlya postupayushchikh v vuzy [Physics for entering university], Мoscow: Nauka, 1991 (in Russian).
[7] E. I. Butikov, A. S. Kondrat'ev, Fizika dlya uglublennogo izucheniya [Physics-depth study] Moscow: Fizmatlit, 2000 (in Russian).
[8] E. I. Butikov, “Planets and Satellites. Educational software package”, in Eugene Butikov personal page, Oct. 2015. [Online]. Available: http://butikov.faculty.ifmo.ru (in Russian).
[9] L. D. Landau and E. M. Lifshits, Mekhanika [Mechanics], Moscow: Nauka, 1965 (in Russian).
[10] Ch. Kittel', U. Nait, and M. Ruderman, Mekhanika [Mechanics], Moscow: Nauka, 1971 (in Russian).
[11] E. I. Butikov, Motions of Celestial Bodies, IOP Publishing Ltd., 2014; doi: 10.1088/978-0-750-31100-7
[2] A. P. French, "The Envelopes of Some Families of Fixed-Energy Trajectories," Am. J. Physics, vol. 61, no. 9, pp. 805-811, 1993; doi:10.1119/1.17429
[3] A. G. Alenitsyn, E. I. Butikov, and A. S. Kondrat'ev, Kratkii fiziko-matematicheskii spravochnik [Quick physical-mathematical reference book], Moscow: Nauka, 1997 (in Russian).
[4] M. Baće, S. Ilijić, Z. Naranci欬, and L. Bistricić, “The envelope of projectile trajectories ,” Eur. J. Phys., vol. 23, no. 6, pp. 637–642, 2002; doi:10.1088/0143-0807/23/6/305
[5] E. I. Butikov, “Comment on ‘The envelope of projectile trajectories',” Eur. J. Phys., vol. 24, no. 4, pp. L5–L9, 2003; doi: 10.1088/0143-0807/24/4/101
[6] E. I. Butikov, A. A. Bykov, A. S. Kondrat'ev, Fizika dlya postupayushchikh v vuzy [Physics for entering university], Мoscow: Nauka, 1991 (in Russian).
[7] E. I. Butikov, A. S. Kondrat'ev, Fizika dlya uglublennogo izucheniya [Physics-depth study] Moscow: Fizmatlit, 2000 (in Russian).
[8] E. I. Butikov, “Planets and Satellites. Educational software package”, in Eugene Butikov personal page, Oct. 2015. [Online]. Available: http://butikov.faculty.ifmo.ru (in Russian).
[9] L. D. Landau and E. M. Lifshits, Mekhanika [Mechanics], Moscow: Nauka, 1965 (in Russian).
[10] Ch. Kittel', U. Nait, and M. Ruderman, Mekhanika [Mechanics], Moscow: Nauka, 1971 (in Russian).
[11] E. I. Butikov, Motions of Celestial Bodies, IOP Publishing Ltd., 2014; doi: 10.1088/978-0-750-31100-7
Published
2017-06-03
How to Cite
Бутиков, Е. И. (2017). The Envelope of a Family of Elliptic Orbits and Ballistic Trajectories. Computer Tools in Education, (3), 50-56. Retrieved from http://cte.eltech.ru/ojs/index.php/kio/article/view/1401
Issue
Section
Computers in the teaching process
This work is licensed under a Creative Commons Attribution 4.0 International License.
