The Falling Body Problem in Quantum Calculus

Abstract

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.