I am working on a project which tests the rate of speed for a variety of rated bearings. I am lacking any real scientific equations and was hoping you could give me some ideas and also how I should write the scientific equation for speed vs. time. Thanks!

More info is really required to answer this question in terms of the design of your experiment but I will make some suggestions. It seems to me that the easiest experiment to test the effectiveness of bearings would be to roll the cart containing the bearings down a ramp. You will need either a ticker/spark timer setup or a photogate setup to get the time it takes to roll down the ramp. If this is a very rough estimate you could use a stopwatch and a very long ramp on a very small slope but I wouldn't recommend it.

By rate of speed I assume you mean acceleration. You will need to measure the distance it travels down the ramp and the time. You will also need to keep all other variables (the cart, the tires, the ramp, etc.) constant. Start each experiment from rest (stopped) and then use the following equation to solve for acceleration:

Acceleration = (2 x distance) ÷ (time squared)

Note: This is a simple non-calculus based solution that doesn't account for changes in frictional effects as speed changes.