Katherine Freeze continues her description of the hunt for dark matter in The Cosmic Cocktail describing how the rotational speeds of galaxies, or at least the hypothesized ones, provide more evidence for the phenomena.
In short, some very smart people applied a 300 year old equation to data that Mr. Newton himself couldn't have imagined. The equation was Newton' gravitational force equation which says F = (GMm)/r^2 which then would equal centripetal force or F = (mv^2)/r giving you the speed of a body to be v = ((GM)/r)^(1/2).
They took this simple equation and applied it to the speeds of nearby galaxies. But what they found was quite intriguing. It is simple to understand based on the equation stated above that as r increases the velocity should decrease. But the data they found was contrary to this understanding for the speed of multiple galaxies. They noticed that there was a small bump in the beginning of the graph which was expected since much of the mass in a galaxy is not in the center. They expected the velocity to drop off after that, however it did not and was still quite large very far away from the center of the galaxy.
How could they explain this increased speed? Well M would have to be much bigger, and since they had already calculated the visible masses of these galaxies, they knew that the mass must be invisible, such as dark matter is inferred to be.
The speed of the galaxy would require more mass to make it go faster, dark matter explains why even as r increases v does not drop off. The graph below shows this idea as the observed velocity is much lower than the observed one, and a line showing dark matter is also included to account for this.
Something surely is out there propelling these galaxies, and it sure is interesting that an equation learned by high school physics students was so integral in providing one answer for it!