As you are deriving for homework, the mobility tensor of a body of revolution with fore-and-aft symmetry (e.g., it looks the same from both ends, like a cylinder or a football) can be determined from just two experiments. We use these to determine how rapidly a rod moves off to the side when it falls through a viscous fluid.
This problem has been studied by many investigators over the years, a classic reference being the work of G. K. Batchelor that may be found here. Interestingly, substantially different behavior may be seen if the body of revolution settles near a lateral boundary: as we shall see, the object either impacts the wall and "flips" before drifting away, or "swoops" and drifts away without impacting, depending on its initial orientation. The details of this effect are described in the paper by Russel and co-workers found here.