The Falling Body Problem in Quantum Calculus

The quantum calculus, q-calculus, is a relatively new branch in which the derivative of a real function can be calculated without limits. In this paper, the falling body problem in a resisting medium is revisited in view of the q-calculus to the first time. The q-differential equations describing the vertical velocity and distance of the body are obtained. Accordingly, exact expressions for the vertical velocity and the vertical distance are provided. The solutions are expressed in terms of the small q-exponential function which is an elementary function in the q-calculus. The dimensionality of the obtained formulae of the velocity and the distance are also analyzed. In addition, the present exact solutions reduce to the corresponding solutions in classical Newtonian mechanics when the quantum parameter q tends to one.


INTRODUCTION
Basically, the regular calculus uses limits in calculating the derivatives of real functions. However, the calculus without limits is nowadays known as quantum calculus or q-calculus. Historically, in the eighteenth century, Euler obtained the basic formulae in q-calculus. However, Jackson [1] may have been the first to introduce the notion of the definite q-derivative and q-integral. Currently, there is a significant interest in implementing the q-calculus due to its applications in several areas, such as mathematics, number theory, and combinatorics [2]. Ernst [3,4] pointed out that the majority of scientists who use q-calculus are physicists. Baxter [5] introduced the exact solutions of several models in Statistical Mechanics. Bettaibi and Mezlini [6] solved some q-heat and q-wave equations. Many interesting results in such area of research were also introduced by several authors in the literature [7][8][9][10][11][12].
In this paper, we aim to extend the applications of the q-calculus to study the falling body problem in a resisting medium. This problem and also the full projectile motion have been investigated by several authors [13][14][15][16][17] using various definitions in fractional calculus. However, the present paper may be the first to analyze the falling body problem in view of the q-calculus.
The basic formulae in q-calculus will be used to analyze the motion of a falling body in a resisting medium. Moreover, it will be shown that the exact solutions for the vertical velocity and distance reduce to the classical ones as q → 1. The paper is organized as follows. Section 2 presents the main aspects of the q-calculus. Sections 3 discusses the application of the q-calculus on the falling body problem. Section 4 includes an additional analysis. Finally, section 5 outlines the conclusions.

THE MAIN ASPECTS OF THE q-CALCULUS
Let q ∈ R and n ∈ N, then [n] q is defined as (first chapter in [18]) and as q → 1, we have lim q→1 [n] q = n.
The q-factorial [n] q ! of a positive integer n is given by The definition of q-differential is such that if f is differentiable at t, and we have at t = 0 that According to (4) we have The small q-analog of the exponential function e t denoted by e q (t) (also called the small q-exponential function) is given as The definite Jackson q-integral is defined by and according to (4) and (9), we have The indefinite Jackson q-integral of the small q-exponential function e q (αt) is given as [18] e q (αt) where c is a real constant. The correctness of dimensionality of the physical quantities is actually guaranteed by the definition (4).

THE FALLING BODY PROBLEM
Consider the falling of an object of mass m in the Earth gravitational field through the air from a height h with initial velocity v 0 . The classical equation of motion for the particle is given by [15,16] where k is a positive constant and its dimensionality is the inverse of seconds, i.e., [k] = s −1 . The initial conditions are given as where z(t) is the vertical distance of the particle at arbitrary time t and dz(t) dt = v(t). The equation of motion (12) in view of the quantum calculus becomes In order to solve Equation (14), we assume the solution in the series form: v(t) = ∞ n=0 a n t n , and therefore [n] q a n t n−1 , [n] q a n t n−1 , where [0] q = 0, Substituting (15) and (16) into (14), yields ∞ n=0 [n + 1] q a n+1 t n = −g − k ∞ n=0 a n t n , [n + 1] q a n+1 t n = −g − ka 0 − k ∞ n=1 a n t n , (18) which gives From (19), we have .
The instantaneous velocity is obtained as v(t) = a 0 + ∞ n=1 a n t n , which can be written as In terms of the small exponential function e q (−kt), we have Applying the first initial condition in (13) on (24), we obtain a 0 = v 0 and therefore v(t) becomes which can be simplified as The vertical distance z(t) in quantum calculus is governed by, where v(t) = D q z(t). Integrating (27), it then follows; and hence, or i.e., where [1] q = 1. The exact solutions (26) and (31) should be reduced to the corresponding solutions in classical Newtonian mechanics when q → 1. In addition, if the acceleration due to gravity is measured in ms −2 , then the vertical velocity in (26) must has dimension ms −1 and the vertical distance in (31) must has dimension m. These issues are addressed in the following section.

ANALYSIS AND APPLICATIONS
First of all, we investigate the solutions (26) and (31) when q → 1. In this case, the small exponential function e q (−kt) reduces to the standard exponential function e −kt in classical calculus. Hence, which is the analytic expression for velocity in the case of the classical Newtonian mechanics (see Equation 16 in reference [15]). Besides, the vertical distance in (31) reduces to which is also the analytic expression for the vertical distance in the classical Newtonian mechanics (see Equation 17 in reference [15]). In addition, in the case of no air resistance, i.e., the parameter k vanishes, we obtain from (32) that v(t)| k→0 = lim