I agree with Mr. Bridge: the design of your experiment (length of the free fall distance, the timing equipment, estimating or measuring the inherent measurement errors) will be non-trivial and should be done carefully before starting the experiment.
I suggest doing some basic research on the aerodynamic drag of spheres - a Wikipedia search, for example, will quickly reveal the range of values for spheres' drag coefficients. It's however true that the drag coefficient will depend on the speed and diameter of the sphere (the "Reynolds number" effects at work here).
If possible, reconsider whether or not a ball bearing is the best test article to use. A ball bearing is small but relatively heavy for its size (moderate to high "ballistic coefficient" - another data item to research). What this means is that the ball bearing will respond quicker to gravity ("fall fairly quickly") and not be slowed down much by aerodynamic drag; contrast this to a whiffle ball or pingpong ball, both which have much lower
"ballistic coefficient" values. A slower falling test article will ease your
measurement problems, I should add.
If you are intent on using a ball bearing, you can make a pretty good
estimate of its drop speed by applying Newton's laws, F= ma and integrating to get velocity. This first calculation will neglect aerodynamic drag but should give you an idea of the drop speeds you can expect. A more refined estimate would be to rerun the calculations using the range of drag coefficient values as mentioned above.
A final suggestion: the drop-measurement problems would be made much easier if a "test medium" was chosen which still allows descents of spheres, but has higher viscosity which would slow down the test object.
Water or lightweight vegetable oil in a tall glass column, for example, could be a practical test setup that still allows controlled testing of your hypotheses.
Whew! There's quite a bit of background work I've described here but pre-test background research, and some pre-test experiment planning, will give you good insight into the physics of the problem. Good luck!
Senior Project Leader
The Aerospace Corporation
El Segundo CA