Observing the Light Intensity of M31
You know that the apparent brightness of an object changes as its distance changes, so you should be able to see how much the M31 moves by observing its change in brightness over a given time period. Finding its velocity is then just a matter of dividing this distance by the time interval. A snap. You can use a light curve of M31 to discern how the galaxy's brightness changes over time.
The equation that describes how apparent brightness changes with distance is given by the relationship:
In words, the intensity light varies as the inverse square of the separation from the light source (r).