/\ you could figure out with mathematical equations what the required volume of liquid water you would need to duplicate the volume of the other barrel of water when frozen.
SO on to my point
the barrels are NOT in freefall so the equations w=mg, F=ma do not apply. there are many different factors to consider when something is rolling down a hill.
we are going to assume that the paths they take are identical and the surface along those paths is also identical. we are also going to assume that the barrel containing the ice has the same temperature on the outside as the barrel with the water in it(yes not possible but this is a hypothetical question and there has to be constants established and certain things must be assumed). The coefficient of kinetic friction between the hill and the barrels are also identical. the initial velocitys of the barrels are the same. the ice in the one barrel is bonded to the side of the barrel and there is no air in this barrel, and lastly there is absolutely zero air in the barrel filled with water, there is water in it and nothing but.
so with all of that said.
the barrel full of ice is one solid mass, the ice stays bonded and does not move against the sides of the barrel. the barrel filled with water is not one mass. the water does move within the barrel the molecules of the water are in motion or as somebody above me said, they have inertia. as the barrel full of water is sent down the hill the water molecules will not rotate with the barrel because the barrel cannot 'grab' the entire mass of water and spin it at the same rate as itself. because of this the water stands still sort to speak and there is friction between the water and the inside of the barrel. to demonstrate the fact that the water will not rotate at the same rate as the barrel fill a cup with water and put an ice cube in it. with the ice cube stationary rotate the glass with signifigant speed and watch the ice cube move not at all or very little. either way it will move much less than the amount you've moved the glass. the ice cube doesnt move because its floating in the water and the water did not move within the glass very much. this is exactly what is hapenning inside the barrel full of water. there is always friction between two moving masses, no matter how small, THERE IS FRICTION BETWEEN THE WATER AND THE BARREL.
now somebody above me said that the friction within the barrel does not matter, only the friction of the surface its rolling down. this is false. to demonstrate(use your imagination on this one): picture a suspended cylinder spinning in one spot rubbing against a surface at a constant velocity. if you were to stick your finger inside that cylnder and touch and of the walls of it it would most definitly slow down and may even stop. FRICTION ANYWHERE ON THE BARREL MATTERS.
SO with that established we can say that the water within the barrel does in fact rotate slower than the barrel itself and this does crate friction and friction on the inside does affect the barrel. so the ice creating no frction against the barrel will roll faster than the barrel with the water creating frction inside of it and the ice barrel wins. you are all owned, thank you im going to sleep now.
'I thought i was going to dead.'
-Charles Gagnier