Final answer:
The best method to measure the speed of a star moving away from the Sun is through Doppler spectroscopy, which detects shifts in a star's spectrum due to the Doppler effect, revealing its radial velocity.
Step-by-step explanation:
The best way to measure the average speed with which a star is moving away from the Sun is by using Doppler spectroscopy. This method involves analyzing the star's spectrum to detect shifts in the wavelength of spectral lines due to the Doppler effect. If a star is moving away from us, the spectral lines will be shifted toward the red end of the spectrum, known as redshift, which indicates its radial velocity. The amount of shift is directly related to the speed at which the star is moving. Doppler spectroscopy does not depend on the distance of the star from the observer and is therefore a reliable method for detecting motion at significant distances as long as a bright enough spectrum and a large telescope are available.