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The Exact Solution of the Falling Body Problem in Three-Dimensions: Comparative Study

1
Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk 71491, Saudi Arabia
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Department of Mathematics, Faculty of Sciences and Humanities in Al-Kharj, Prince Sattam bin, Abdulaziz University, Al-Kharj 11942, Saudi Arabia
3
Department of Basic Engineering Science, Faculty of Engineering, Menofia University, Shebin El-Kom 32511, Egypt
*
Authors to whom correspondence should be addressed.
Mathematics 2020, 8(10), 1726; https://doi.org/10.3390/math8101726
Received: 17 August 2020 / Revised: 16 September 2020 / Accepted: 24 September 2020 / Published: 8 October 2020
(This article belongs to the Section Engineering Mathematics)
Very recently, the system of differential equations governing the three-dimensional falling body problem (TDFBP) has been approximately solved. The previously obtained approximate solution was based on the fact that the Earth’s rotation (ER) is quite slow and hence all high order terms of ω in addition to the magnitude ω2R were neglected, where ω is the angular velocity and R is the radius of Earth. However, it is shown in this paper that the ignorance of such magnitudes leads, in many cases, to significant errors in the estimated falling time and other physical quantities. The current results are based on obtaining the exact solutions of the full TDFBP-system and performing several comparisons with the approximate ones in the relevant literature. The obtained results are of great interest and importance, especially for other planets in the Solar System or exterior planets, in which ω and/or ω2R are of considerable amounts and hence cannot be ignored. Therefore, the present analysis is valid in analyzing the TDFBP near to the surface of any spherical celestial body. View Full-Text
Keywords: falling body problem; angular velocity; projectile motion; three dimensions; Earth’s rotation; Laplace transform falling body problem; angular velocity; projectile motion; three dimensions; Earth’s rotation; Laplace transform
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MDPI and ACS Style

Ebaid, A.; Alharbi, W.; Aljoufi, M.D.; El-Zahar, E.R. The Exact Solution of the Falling Body Problem in Three-Dimensions: Comparative Study. Mathematics 2020, 8, 1726. https://doi.org/10.3390/math8101726

AMA Style

Ebaid A, Alharbi W, Aljoufi MD, El-Zahar ER. The Exact Solution of the Falling Body Problem in Three-Dimensions: Comparative Study. Mathematics. 2020; 8(10):1726. https://doi.org/10.3390/math8101726

Chicago/Turabian Style

Ebaid, Abdelhalim, Weam Alharbi, Mona D. Aljoufi, and Essam R. El-Zahar 2020. "The Exact Solution of the Falling Body Problem in Three-Dimensions: Comparative Study" Mathematics 8, no. 10: 1726. https://doi.org/10.3390/math8101726

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