I am trying to set up a fountain that will run off grid. I want to use a 100w solar power system to charge a deep cycle 12v battery. I am looking at a 120v 4.8 amp water pump. It will be ran 100 ft from the solar panel system. My question is how long will the pump run, how will it take the 100w solar panel to charge the battery, and how low should I let the battery get before I charge it. I plan on setting a 24h timer to run the pump in intervals of x hrs then letting it charge x hrs and repeating. Hoping to run 3-4 hours in the morning then again in the evening, (something like 5-9 then 2-6). Thanks for any help
First, as has been pointed out in comment, if you really are using a 120 VAC 4.8 A pump, you are simply out of luck. 120 V x 4.8 A is 600 watts. I do hope you see the problem with charging from a 100 W panel.
But let's be optimistic and figure you misspoke, and the proper number is 12 volts at 4.8 A. After all, 600 watts is just under 1 HP, and a 1 HP water pump will provide something like 20 gal/min at 50 psi, which would be a pretty impressive fountain. If you really do want such a fountain, you're going to need a much bigger panel array.
So, 12 volts at 4.8 A is about 60 watts. Can you do this? Sure. Sort of. You need to keep in mind that, particularly near full charge, the charge efficiency of a battery, particularly a lead-acid, starts to drop off, and 60% is not an outrageously low number. Charge efficiency, of course, is simply the ratio of the stored charge out divided by the charging charge in for a given state of charge. Using this number, you can sort of figure that (on a bright sunny day) you'll get about 1 hour of fountain per hour of charge.
Another thing you need to keep in mind is that, although the sun might shine all day, in early morning and late afternoon it won't be very bright, and unless you have a sophisticated tracking panel array, most of the time the panel will not be directly facing the sun and will receive even less energy. The general rule of thumb is that you can count on the equivalent of 6 hours charging per day.
So, as a rough number, assuming perfect weather, you'll be able to get 6 hours of operation per day. In a desert.
As for the battery, you might consider a bit of basic research. Like, you know, Wikipedia? How about this? The general rule is to not discharge below about 50%.