Image explains it pretty well, I'm wondering what the best way would be to create something like this. Another example would be a fire-hose that's not being held at one end and ejecting water at a super high velocity - imagine how the hose would be flopping around like crazy, connected at one end and free to move on the other. Is something like possible using dynamics or some other method? I'm interested in being able to create these sorts of complicated animations without needing to key frame any bones/avoid traditional animation/would like a realistic solution for the problem

Initial velocity?

Both of your examples could be done with Vellum Hair. For the string/ball, pin the top point and apply a POP Axis force to spin the string. You can then copy a sphere to the end point.

For the hose, again, pin one point and then apply a POP Force and adjust the noise settings.