If you are looking to collect data at a reasonably fast rate and test it against the basic laws of physics, then I'd suggest that you study the libration of the moon.
With the equipment you have, you can easily collect photos of the moon throughout its phase cycle and compared the relative nodding of the moon back and forth. Using basic software, you can then measure the relative spatial displacement of a fixed point relative to the viewable face, analyzing the data and then formulate en equation to predict the amount of libration at any given time.
Designing and executing this experiment would satisfy all the requirements listed... and I believe that you could collect data over three months and have a reasonably amount to create a predictive model.
OIC!
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