Looking for research ideas? Here are a few low-cost suggestions:
Explore the structure of the electromagnetic modes in your microwave using a layer of cheese slices. Where are the nodes and anti-nodes? Can you make a more sensitive testing arrangement using, say, a uniform grid of syrup drops?
Suppose you have an unfair die…say, a cube that is "half-empty". What is the likelihood that it will land heavy side up? Renowned cosmologist, Sir Hermann Bondi, explored the case of a cylindrical die in 1993 in an attempt to answer the question "How likely is it that a nickel will land on its edge?", in a 1993 article in The European Journal of Physics, but the experimental aspects of his predictions are largely untested, as are most details about other shapes.
What is the meaning of the SPF rating on sunscreens? Does a rating of 30 block twice as much light as a 15?…at what frequency? Does it matter how thick it is applied? Does the angle of incidence matter?
Estimate how high and far you can spray water using your thumb in a water hose, before doing the experiment. In your model, what is the mathematical form of the envelope of points the water can reach?
If you know the gas prices along your route, where should you stop to fuel up so as to minimize your costs? It is not always wise to stop to get the cheapest gas. (See "The Rental Car Problem", November 1996, The College Mathematics Journal; there are many alternate unsolved versions of this problem.)