In the last three decades, much work has been performed on fractional calculus (FC). FC is the natural generalization of ordinary derivatives and integrals. It deals with operators having a non-integer (arbitrary) order. Today, there are excellent books and reviews on fractional calculus and its applications in science and engineering [1 2 3 4 5 6 7]. FC has been employed in numerous practical and theoretical problems, but, for the sake of brevity, we mention a few of them as follows: the design of optimal control systems ; the analysis of anomalous relaxation and diffusion processes , and the regularized long-wave equation ; the study of a bead sliding on a wire ; and the investigation of chaotic systems , heat conduction , nerve impulse transmission , chemical kinetic systems , and oscillating circuits . However, this generalization of the ordinary calculus is still inconsistent in physical and geometric interpretation, because several definitions of fractional derivatives have been proposed. These definitions include the Riemann-Liouville, Grunwald-Letnikov, Weyl, Caputo, Marchaud and Riesz fractional derivatives [1 2 3 4 5]. These fractional derivatives seldom satisfy the well-known formulae, e.g., the product, quotient, and chain rules. As a result, researchers have been trying to construct new definitions of fractional derivatives and integrals. As an illustrative example, helpful work on the theory of derivatives and integrals is done in .
Let f : [0, ∞) → ℝ be a function. Then, the γ-th order conformable derivative of f is defined by  as
where γ ∈ (0,1]. If f is γ-differentiable in some (0,a), a > 0, and exists, then, it is defined as
and the conformable integral of a function, , starting from a ≥ 0, is defined as,
The integral is the usual Riemann improper integral, and γ ∈ (0,1]. The most important properties of this conformable derivative are given in the following theorem.
Let γ ∈ (0,1], and f and g be γ-differentiable at any point t > 0. Then,
Tγ(af + bg) = aTγ(f) + bTγ(g) for all a,b ∈ ℝ,
Tγ (tp) = ptp-γ for all p ∈ ℝ,
Tγ(λ) = 0 for all constant function f(t) = λ,
Tγ (fg) = fTγ (g) + gTγ(f),
, if f is differentiable.
This conformable derivative has attracted the interest of researchers in recent years since it seems to satisfy all the requirements of the standard derivative , . Hence, there is a large amount of work done in this area at the present time [20 21 22 23 24].
We would like to point out that the term “fractional” is not the one used in the notation of fractional calculus . However, the properties of this conformable derivative make it suitable for investigating real systems and to get new insights due to the presence of the fractional parameter 0 < γ ≤ 1.
In this paper, we use the conformable derivative to discuss two-dimensional projectile motion. First, we consider free two-dimensional projectile motion. Subsequently, new formulae for the trajectory, height, range, flight time, optimal angle, and maximum range are obtained. We show that these formulae depend on the conformable derivative order 0 < γ ≤ 1, and in the particular case of γ = 1, they become ordinary. Finally, we consider two-dimensional projectile motion in a resistive medium.
This paper is organized as follows. In Section 2, the main results of ordinary free projectile motion are reviewed. In Section 3, new analytical solutions for this motion are obtained. In Section 4, we consider motion in resistive media. Finally, the conclusions are given in Section 5.
2 Ordinary free projectile motion
We study the motion of a projectile moving in a free two-dimensional space. The projectile is treated as a particle of mass (m), under a uniform gravitational force (g), and disregarding any other resisting or external force. Assuming the particle starts from rest (i.e., x0 = y0 = 0 m), with an initial velocity of modulus v0 and angle θ with respect to the horizontal axis x, then, the classical equations of motion for this system, in the x, y plane, are,
with the initial conditions,
The solutions, given in parametric form, are
and the corresponding velocities are
By eliminating the time t in the expressions (8), we obtain the trajectory equation, given by the parabola
Moreover, three interesting features for studying two-dimensional projectile motion are described as follows. Range (R) is defined as the horizontal distance traveled by the particle, from start to end. It can be obtained with (10) assuming the landing condition, y(x = R) = 0 m,
Flight time (Tflight) is the amount of time that the projectile spends in the air, i.e., from its shooting to its landing . It is determined as
3 Free projectile motion in view of conformable derivative
Usually, authors replace integer derivative operators with fractional ones on a purely mathematical basis [26 27 28]. However, this practice is not completely correct, from the physical and engineering point of view, and a dimensional correction in the new equation is necessary. Having this in mind, in  has been proposed a systematic way to construct fractional differential equations; this has been successfully applied in [30 31 32 33]. This procedure considers the introduction of parameters such as σt and σx with an appropriate dimensionality. In other words,
where γ is an arbitrary parameter which represents the order of the derivative 0 < γ ≤ 1. σt is a parameter representing the fractional time components in the system with dimensionality in time unit [t]; likewise, the parameter σx has dimension of length unit [l], and represents the spatial fractional components . These are the only conditions imposed on parameters σt and σx.
and arrive at 
where we have omitted the subscript t in σ. Substituting into (9), we obtain the corresponding conformable differential equations
We can integrate them directly
We also determine the components of the conformable velocity of the system by differentiating with respect to time as
Unlike expressions (8) and (9), the coordinates and velocities in (19)-(22) depend on the fractional parameter 0 < γ ≤ 1. In the particular case γ = 1, those are reduced to (8) and (9). We want to emphasize that the solutions given by (19) and (20) are qualitatively different from the solutions previously obtained in ,  and . Therefore, in the projectile motion under the conformable derivative sense, we can achieve new formulae for the trajectory, maximum height, range, flight time, maximum range, and optimal angle, as will be shown:
Maximum height (H): This is obtained by using the condition vy(t;γ) = 0 m/s, and employing (22), to find
since tmax indicates the time when the particle has reached its maximum height; then, substituting this time in (20), we obtain the maximum height H, as
We have the ordinary maximum height (12), when γ = 1.
Range (R): The maximum displacement is calculated from the trajectory equation, and it occurs when y (x = R) = 0 m. That is,
Note again, if γ = 1, we have (11).
Flight time (Tflight): This is defined as the time value T at which the projectile hits the ground, y(t = T) = 0 m, as
In the case γ = 1, we have (13).
Maximum range (Rmax): This feature is considerably interesting for both practical and theoretical studies. It can be determined by finding the optimal projection angle θop, and evaluating it in Rmax = R(θop). The necessary condition to maximize the range, R → Rmax is given by . Thus, from (26) we have
By solving this equation with respect to θop, we obtain
In the case γ = 1, the optimal angle is θop = π/4 (as expected). Hence, using trigonometric formulae,
and, by replacing this expression in (26), we obtain the maximum projectile range Rmax as
From this expression, in the case γ = 1, we get the classical maximum range .
These new formulae have the same dimensionality as in the ordinary case, because σ has dimension of seconds. Furthermore, for demonstrative purposes, Figure 1 shows the trajectory (Figure 1a), the maximum height (Figure 1b), and the range (Figure 1c) of a particle in projectile motion with different values of γ.
4 Resistive projectile motion in view of conformable derivative
The motion equations of a projectile going through an isotropic medium in two-dimensional space with a resistive force proportional to its velocity, have the form :
where k is a positive constant and its dimensionality is the inverse of seconds [k] = s-1. The initial conditions are given by
namely, the projectile starts from rest, with an initial velocity of module v0 and an angle θ with respect the x-axis. The solutions of these equations (33) and (34) and satisfying the initial conditions (35) and (36) are given in 
These expressions are ordinary linear, homogeneous and non-homogeneous differential equations, hence, they can be solved easily using the ordinary methods. The analytical solutions that satisfy initial conditions (35) and (36) are:
Therefore, we only have a vertical conformable trajectory .
As an illustrative example, Figure 2 presents the projectile motion for a defined medium with k = 0.1 s-1, in terms of x(t;γ), y(t;γ) and t. The main tri-dimensional view of this behavior is displayed with its planes.
In addition, the vertical displacement for different values of γ, k, velocities and angles are shown in the Figure 3. It is observed that the trajectory varies with γ (cf. Figure 1), but there are negligible changes for different values of k.
In this work, we have considered free and resistive two-dimensional projectile motion in the conformable derivative sense. New formulae for the trajectory (23), height (25), range (26), flight time (27), optimal angle (29), and maximum range (32) were obtained. Also, calculations were performed in the case of a resistive medium (46) and (47). The obtained formulae are qualitatively different from the solutions previously obtained in , ,  and . Moreover, we found that the new formulae depend on the conformable derivative order 0 < γ ≤ 1, and they became ordinary in the particular case γ = 1. We noticed that the obtained trajectories never exceeded the ordinary ones where γ = 1, unlike the results obtained in other studies .
The authors acknowledge the fruitful discussion with I. Lyanzuridi and D. Baleanu. This work was supported by CONACyT and DICIS-University of Guanajuato.
Oldham K.B., Spanier J., The Fractional Calculus. Theory and Applications of Differentiation and Integration of Arbitrary Order, 1974, Academic Press, New York Google Scholar
Podlubny I., Fractional Differential Equations, 1999, Academic Press, New York Google Scholar
Kilbas A., Srivastava H., Trujillo J., Theory and Applications of Fractional Differential Equations, 2006, Math. Studies, North-Holland, New York Google Scholar
Hilfer R., Applications of Fractional Calculus in Physics, 2000, World Scientific, Singapore Google Scholar
Baleanu D., Guvenc Z.B., Machado J.A.T., New Trends in Nanotechnology and Fractional Calculus Applications, 2010, Springer New York. Google Scholar
Uchaikin V., Fractional Derivatives for Physicists and Engineers, 2013, Springer H.D., London, New York. Google Scholar
Baleanu D., Jajarmi A., Hajipour M., A New Formulation of the Fractional Optimal Control Problems Involving Mittag-Leffler Nonsingular Kernel, J. Optim. Theory Appl., 2017, 175(3), 718-737. CrossrefWeb of ScienceGoogle Scholar
Kumar D., Singh J., Baleanu D., Sushila, Analysis of regularized long-wave equation associated with a new fractional operator with Mittag-Leffler type kernel,Phys. A Stat. Mech. Appl., 2018, 492, 155-167. CrossrefGoogle Scholar
Hajipour M., Jajarmi A., Baleanu D., An Efficient Non-standard Finite Difference Scheme for a Class of Fractional Chaotic Systems, J. Comput. Nonlinear Dyn., 2018, 13(2), 021013, 1-9. Google Scholar
Zhao D., Singh J., Kumar D., Rathore S., Yang X., An efficient computational technique for local fractional heat conduction equations in fractal media, J. Nonlinear Sci. Appl., 2017, 10(4), 1478-1486. CrossrefWeb of ScienceGoogle Scholar
Kumar D., Singh J., Baleanu D., A new numerical algorithm for fractional Fitzhugh-Nagumo equation arising in transmission of nerve impulses, Nonlinear Dyn., 2018, 91(1), 307-317. Web of ScienceCrossrefGoogle Scholar
Singh J., Kumar D., Baleanu, D., On the analysis of chemical kinetics system pertaining to a fractional derivative with Mittag-Leffler type kernel, Chaos, 2017, 27(10), 103113. CrossrefWeb of ScienceGoogle Scholar
Kumar D., Agarwal R.P., Singh J., A modified numerical scheme and convergence analysis for fractional model of Lienard’s equation, J. Comput. Appl. Math.,2018, 339, 405-413. CrossrefWeb of ScienceGoogle Scholar
Katugampola U.N., A new fractional derivative with classical properties, 2014, arXiv:1410.6535v1. Google Scholar
Abu Hammad I, Khalil R., Fractional Fourier series with applications, Am. J. Comput. Applied Math., 2014, 4(6), 187-191. Google Scholar
Ahmad B., Batarfi H., Nieto J.J., Otero-Zarraquinos O., Shammakh W., Projectile motion via Riemann-Liouville calculus, Adv. Diff. Equations, 2015, 63(1), 1-14. Google Scholar
Rosales J.J., Gomez J.J., Guía M., Tkach V.I., Fractional Electromagnetic Waves, 2011, LFNM2011 International Conference on Laser and Fiber-Optical Networks Modeling 4-8 September, Kharkov, Ukraine. Google Scholar
Rosales J.J., Guía M., Gomez J.F., Tkach V.I., Fractional Electromagnetic Wave,Disc. Nonl. Compl.,2012, 1(4), 325-335. Google Scholar
Gomez-Aguilar J.F., Rosales-Garcia J.J., Bernal-Alvarado J.J., Cordova-Fraga T., Guzman-Cabrera R., Fractional Mechanical Oscillator, Rev. Mex. Fis., 2012, 58, 348-352. Google Scholar
Rosales García J.J., Guía M., Martínez J., Baleanu D., Motion of a particle in a resistive medium using fractional calculus approach, Proceed. Rom. Acad. Ser. A, 2013, 14, 42-47. Google Scholar
Rosales J.J., Guía M., Gomez F.,Aguilar F., Martínez J., Twodimensional fractional projectile motion in a resisting medium, Cent. Eur. J. Phys., 2014, 12(7), 517-520. Google Scholar
Thornton S.T., Marion J.B., Classical dynamics of particles and systems, 2004, Thomson Brooks/Cole Google Scholar
About the article
Published Online: 2018-10-16
Citation Information: Open Physics, Volume 16, Issue 1, Pages 581–587, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2018-0076.
© 2018 A. Ortega Contreras et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0