Projectile Motion in Special Theory of Relativity: Re-Investigation and New Dynamical Properties in Vacuum

Ebrahem A. Algehyne (), Abdelhalim Ebaid (), Essam R. El-Zahar, Musaad S. Aldhabani, Mounirah Areshi and Hind K. Al-Jeaid
Additional contact information
Ebrahem A. Algehyne: Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia
Abdelhalim Ebaid: Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia
Essam R. El-Zahar: Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, P.O. Box 83, Al-Kharj 11942, Saudi Arabia
Musaad S. Aldhabani: Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia
Mounirah Areshi: Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia
Hind K. Al-Jeaid: Department of Mathematical Sciences, Faculty of Applied Sciences, Umm Al-Qura University, Makkah 21955, Saudi Arabia

Mathematics, 2023, vol. 11, issue 18, 1-21

Abstract: The projectile motion (PP) in a vacuum is re-examined in this paper, taking into account the relativistic mass in special relativity (SR). In the literature, the mass of the projectile was considered as a constant during motion. However, the mass of a projectile varies with velocity according to Einstein’s famous equation m = m 0 1 − v 2 / c 2 , where m 0 is the rest mass of the projectile and c is the speed of light. The governing system consists of two-coupled nonlinear ordinary differential equations (NODEs) with prescribed initial conditions. An analytical approach is suggested to treat the current model. Explicit formulas are determined for the main characteristics of the relativistic projectile (RP) such as time of flight, time of maximum height, range, maximum height, and the trajectory. The relativistic results reduce to the corresponding ones of the non-relativistic projectile (NRP) in Newtonian mechanics, when the initial velocity is not comparable to c . It is revealed that the mass of the RP varies during the motion and an analytic formula for the instantaneous mass in terms of time is derived. Also, it is declared that the angle of maximum range of the RP depends on the launching velocity, i.e., unlike the NRP in which the angle of maximum range is always π / 4 . In addition, this angle lies in a certain interval [ π / 4 , π / 6 ) for any given initial velocity (< c ). The obtained results are discussed and interpreted. Comparisons with a similar problem in the literature are performed and the differences in results are explained.

Keywords: projectile motion; nonlinear ordinary differential equation; analytic solution (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/11/18/3890/pdf (application/pdf)
https://www.mdpi.com/2227-7390/11/18/3890/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:11:y:2023:i:18:p:3890-:d:1238520

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().