Skip to main content
Log in

Analysis of the two-dimensional fractional projectile motion in view of the experimental data

  • Original paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript


This paper addresses the modeling of projectile motion using fractional models vis-a-vis experimental data. Recently, it was shown that an auxiliary parameter (\(\sigma \)) needs to be included in the fractional modeling to preserve the dimensionality of the physical quantities. In previous studies, \(\sigma \) was subjected to several restrictions without considering clear and meaningful reasons. Such problems are overcome here and a method for estimating \(\sigma \) using the experimental data is introduced. A new solution for the two-dimensional projectile motion using the Caputo’s fractional derivative is obtained. An explicit formula for the trajectory of the projectile in vacuum is first derived. Then, the projectile parametric equations in a resistant medium are expressed in terms of the Mittag–Leffler function. The transcendental equations for the time of flight and the time of maximum height are solved numerically. The model agrees with the classical one as the fractional order tends to 1. In view of the superior results, the current numerical modeling approach is validated for this real-world application.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others


  1. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific Publishing Company, Singapore (2000)

    Book  MATH  Google Scholar 

  2. He, J.H.: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Meth. Appl. Mech. Eng. 167, 57–68 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Agrawal, O.P.: A new Lagrangian and a new Lagrange equation of motion for fractionally damped systems. J. Appl. Math. 68, 339–340 (2001)

    MathSciNet  MATH  Google Scholar 

  4. Kiryakova, V.: Generalized Fractional Calculus and Applications. Pitman Research Notes in Mathematics. Longman, Harlow (1994)

    MATH  Google Scholar 

  5. Luchko, Yu F., Srivastava, H.M.: The exact solution of certain differential equations of fractional order by using operational calculus. Comput. Math. Appl. 29(8), 73–85 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Sebaa, N., Fellah, Z.E.A., Lauriks, W., Depollier, C.: Application of fractional calculus to ultrasonic wave propagation in human cancellous bone. Signal Process. 86, 2668–2677 (2006)

    Article  MATH  Google Scholar 

  7. Machado, J.Tenreiro: A fractional approach to the Fermi–Pasta–Ulam problem. Eur. Phys. J. Spec. Top. 222(8), 1795–1803 (2013).

    Article  Google Scholar 

  8. Wang, S., Xu, M., Li, X.: Green’s function of time fractional diffusion equation and its applications in fractional quantum mechanics. Nonlinear Anal. Real World Appl. 10, 1081–1086 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Tarasov, V.E.: Fractional Heisenberg equation. Phys. Lett. A 372, 2984–2988 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ding, Y., Yea, H.: A fractional-order differential equation model of HIV infection of CD4\(^{+}\)T-cells. Math. Comput. Model. 50, 386–392 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Achar, B.N.Narahari, Hanneken, J.W., Enck, T., Clarke, T.: Dynamics of the fractional oscillator. Phys. A 297, 361–367 (2001)

  12. Song, L., Xu, S., Yang, J.: Dynamical models of happiness with fractional order. Commun. Nonlinear Sci. Numer. Simul. 15, 616–628 (2010)

  13. Goldstein, H.: Classical Mechanics. Addison-Wesley, California (1980)

    MATH  Google Scholar 

  14. Bedford, A., Fowler, W.: Engineering Mechanics. Addison-Wesley, California (1996)

    Google Scholar 

  15. Hayen, J.C.: Projectile motion in a resistant medium. Part I: exact solution and properties. Int. J. Nonlinear Mech. 38(3), 357–369 (2003)

  16. Hayen, J.C.: Projectile motion in a resistant medium. Part II: approximate solution and estimates. Int. J. Non-linear Mech. 38, 371–380 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Weinacht, P., Cooper, G.R., Newill, J.F.: Analytical prediction of trajectories for high-velocity direct-fire munitions. Technical report ARL-TR-3567, US Army Research Laboratory (2005)

  18. Yabushita, K., Yamashita, M., Tsuboi, K.: An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method. J. Phys. A Math. Theor. 40, 8403–8416 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Benacka, J.: Solution to projectile motion with quadratic drag and graphing the trajectory in spreadsheets. Int. J. Math. Educ. Sci. Technol. 41(3), 373–378 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Benacka, J.: On high-altitude projectile motion. Can. J. Phys. 89(10), 1003–1008 (2011)

    Article  Google Scholar 

  21. Ebaid, A.: Analysis of projectile motion in view of the fractional calculus. Appl. Math. Model. 35, 1231–1239 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ahmad, B., Batarfi, H., Nieto, Juan J., Otero-Zarraquinos, Oscar, Shammakh, Wafa: Projectile motion via Riemann–Liouville calculus. Adv. Differ. Equ. (2015).

    MathSciNet  MATH  Google Scholar 

  23. Rosales, J.J., Guia, M., Gomez, F., Aguilar, F., Martinez, J.: Two-dimensional fractional projectile motion in a resisting medium. Cent. Eur. J. Phys. 12, 517–520 (2014)

  24. Garcia, J.Juan Rosales, Calderon, M.Guia, Ortiz, Juan Martinez, Baleanu, Dumitru: Motion of a particle in a resisting medium using fractional calculus approach. Proc. Romanian Acad. Ser. A 14(1), 42–47 (2013)

    MathSciNet  Google Scholar 

  25. Gómez-Aguilar, J.F., Rosales-García, J.J., Bernal-Alvarado, J.J.: Fractional mechanical oscillators. Revista Mexicana de Física 58, 348–352 (2012)

    Google Scholar 

  26. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  27. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  28. Kittel, C., Knight, W., Ruderman, M., Helmholz, K., Moyer, B.: Berkeley Physics Course Mechanics, vol. 1. McGraw Hill, New York (1973)

    Google Scholar 

Download references


The authors extend their appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research Group No (RG-1439-003).

Author information

Authors and Affiliations


Corresponding author

Correspondence to A. F. Aljohani.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ebaid, A., El-Zahar, E.R., Aljohani, A.F. et al. Analysis of the two-dimensional fractional projectile motion in view of the experimental data . Nonlinear Dyn 97, 1711–1720 (2019).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: