The hyperbolic motion of an idealized globular particle with the Planck length radius is described by means of special relativity in order to derive an upper quantum limit of acceleration. Furthermore, it is shown that the existence of a relative acceleration can be excluded by principle. Finally, by assuming validity of Mach's principle, it is deduced that the acceleration of the rate of increase of the radius of the (closed) Universe in the Planck era, is equal, in its absolute value, to one-half of the maximum quantum limit of acceleration; the result is then shortly discussed.