We consider the acceleration of a single sine-Gordon (SG) soliton kink wave by an external time-dependent force $\ensuremath{\chi}(t)$, first without any dissipation, and then in the presence of a weak damping effect. We use the method of Fogel, Trullinger, Bishop, and Krumhansl [FTBK, Phys. Rev. B 45, 1578 (1977)] which consists in perturbing the SG equation about its kink solution and solving the resulting linear inhomogeneous equation for the perturbation function by expanding it in the complete set of eigenfunctions of the Schr\"odinger operator with potential $1\ensuremath{-}2{sech}^{2}x$. Our results concerning the accelerated soliton dynamics strongly disagree with the FTBK conclusion that the soliton should undergo an acceleration proportional to $\ensuremath{\chi}$ (this is the so-called Newtonian dynamical behavior of SG soliton, which is also predicted by all existing perturbation theories dealing with the perturbed SG equation). On the contrary, we find that this Newtonian acceleration is exactly balanced by a reaction effect of the continuous phonon spectrum excited by the external force $\ensuremath{\chi}$, upon the moving kink, so that there is no soliton acceleration at all within the frame of this linear perturbation theory, i.e., for small time values. Actually, we show by the simple example of a static external force that the acceleration of an initially static kink is a higher-order effect (proportional to $\ensuremath{\chi}{t}^{2}$, where $t$ is the time, instead of being constant and proportional to $\ensuremath{\chi}$). We emphasize that this last result has already been checked by numerical experiments and show, both by theory and by numerical simulations, that it does not qualitatively change when a small damping effect is taken into account.