The gravitational two-body problem is a popular concept in the study of planetary bodies. In essence, it models the paths taken by two massive objects orbiting around each other. Earlier today, I was thinking about our Earth's orbit around the sun, and how while it is easy to think that the Earth doesn't move the sun, it does. So while our solar system surely doesn't have only two bodies, I decided to assume it did (and with a perfectly circular orbit), and calculate just what the orbital radius of the sun is.
Beginning this, I actually had no idea the gravitational two-body conundrum existed, so I tried to solve for part of it myself. Knowing the force of gravity on both objects (Gm1m2/r^2), and the centripetal force necessary to maintain said orbits (mv^2/r - note, however, that the r in both equations is not the same: the r for gravity is the sum of the radius of both bodies around the barycenter, or center of the orbit). Using this knowledge (and I won't bore you with the steps), I arrived at Ea/w^2 = Sb/v^2, with E=mass of earth, a=distance from Earth to barycenter, w=velocity of the sun, S=mass of sun, b=distance from sun to barycenter, and v=velocity of the Earth. However, I felt this required too many known values, and could be simplified.
From there I looked into the laws of momentum. Knowing that the force of gravity was the same on both bodies, and that it acted (obviously) over the same time period, I deduced that the momentum imparted to both was equal. However, momentum is also mass times velocity. Using that knowledge, with my previously derived equations, I could further simply and eliminate my velocities, netting me, eventually, the distance of each mass to the barycenter (equivalent to the mass of the object times the radius between objects, divided by the sum of the masses).
While you may not think this is too exciting, I found that deriving this, and then finding out I was actually right was interesting. All of these equations that were used were simple, but when applied together in the correct situation, they have the ability to solve more complex problems.