We have obtained 782 real spectra and used them as inputs for 6700 automatic cross-correlation runs to the aim of investigating the radial velocity accuracy that GAIA could potentially achieve as function of spectral resolution and signal-to-noise ratio. We have explored the dispersions 0.25, 0.5, 1 and 2 Ang/pix (bracketing the 0.75 Ang/pix currently baselined for the 8490--8740 Ang GAIA range centered on the near-infrared CaII triplet) over S/N ranging from 10 to 110. We have carefully maintained the condition FWHM (PSF) = 2 pixels during the acquisition of the 782 input spectra, and therefore the resolutions that we have explored are 0.5, 1, 2 and 4 Ang corresponding to resolving powers R=17200, 8600, 4300 and 2150. We have investigated late-F to early-M stars (constituting the vast majority of GAIA targets), slowly rotating (V_{rot} sin i = 4 km/sec, as for field stars at these spectral types), of solar metallicity ( = -0.07) and not binary. The results are accurately described by the simple law: lg sigma = 0.6(lg S/N)^2 - 2.4(lg S/N) + 1.75(lg D) + 3, where sigma is the cross-correlation standard error (in km/sec) and D is the spectral dispersion (in Ang/pix). The spectral dispersion has turned out to be the dominant factor governing the accuracy of radial velocities, with S/N being less important and the spectral mis-match being a weak player. These results are relevant not only within the GAIA context but also to ground-based observers because the absence of telluric absorptions and proximity to the wavelengths of peak emission make the explored 8490--8740 Ang interval an interesting option for studies of cool stars with conventional telescopes.

12 pages, 2 figures, 5 tables, in press in Baltic Astronomy