Abstract
•The vertical motion in fractional generalized Newtonian mechanics is studied.•The nonlinear FDE describing the motion has been analytically solved.•The escape velocity in fractional calculus is discussed to the first time.
Based on the Riemann-Liouville (R-L) fractional derivative and the generalized Newtonian law of gravitation, the nonlinear fractional differential equation describing the vertical motion of a particle is solved. Such solution is investigated to obtain the escape velocity (EV) following the fractional Newtonian mechanics. It is well known that the EV from the Earth’s gravitational field is about 11.18 km/s within the paradigm of the classical Newtonian mechanics using integer derivatives, but its value has not been yet determined in the scope of fractional calculus. Therefore, we can pose the question: Is the classical value of the EV identical when analyzed under the light of the fractional mechanics? The paper answers this question for the first time. It is found that the fractional escape velocity (FEV) depends on the non-integer order α and a parameter σ with dimension of seconds. The general relation between σ and α is established. The results reveal that the values of the FEV approaches the classical one when α → 1 and σ ≈ 5 × 103 seconds.