Polyurethane foams are made using a prepolymer that is the reaction product of an organic polyisocyanate, a hydroxy-functional acrylate or methacrylate, and a polyol. This prepolymer is reacted at specified volume ratios and isocyanate indices with a polyol component that includes at least one polyol. The foam is made in the presence of a blowing agent and catalyst. Preferred blowing agents are water and chemical blowing agents that generate carbon dioxide in the reaction. Preferred catalysts have primary or secondary amine groups that allow them to react into the resulting polymer structure, thus decreasing the level of volatile components.