Close examination of the floor of my bathtub has revealed a long, thin crack in the bathtub itself. The bathtub is one of the common plastic-feeling variety (though I am not sure if it is actually a plastic.) What is the best way to patch this crack?
Most bathtubs that aren't cast iron or steel are acrylic plastic.
If the crack is right through the tub and water is leaking though, you should replace the tub. If it's an emergency and you need a short-term repair, I'd use a two-part epoxy. Choose a thin epoxy (2:1 ratio) from the selection at the home center and use a putty knife to try to force it into the crack. Let it cure for 24 hours. This will be at best a short term solution because if tub cracked it must be because it flexed in that spot - it will flex again and the brittle epoxy repair will crack again quickly.
If the crack is a surface scratch, it can be repaired with a repair kit such as offered by http://www.surface-repair.com/. The problem is color matching: your tub will have changed color over time and it will be difficult or impossible to get a good match.
Another approach is using fine sandpaper. You can get very fine wet-and-dry sandpaper at an auto parts store like Napa. Get 320, 600, 1200 and 2000 grit. Apply some elbow grease to the scratch with the 320. Try to keep the scratches localized and in one direction. Use the 320 until the scratch is gone. Then move to the 600 and keep it up until the scratches from the 320 are gone. By the time you're done 1200 you may be satisfied, but if not you should be able to get a fairly glossy finish with the 2000.
If your crack is at the bottom of the tub and it is acrylic or fiberglass then most likely the crack will affect the structure of the tub. In order to do a lasting repair, you need to repair the structure. You can take a two pronged approach:
Step 1) Drill relief holes at either sides of the crack. Then drill several small holes around the areas of the tub that flex the most when you step on them. Buy some expanding foam for "Gaps and Cracks". Spray it into all of the holes as far as it will go. The point is to provide some support under the floor. Wait a day and then sand down or physically remove any foam that expanded above the floor of the tub.
Step 2) Purchase a repair kit that includes fiberglass mat(s) and resin. The fiberglass will provide a new structure and the resin will bond it completely with the old surface. Do not expect color matching at this step. You will most likely need to do another project if you care about color matching. However, now your repair will be structurally sound and will very likely last a long time.