People have strong intuitions about the influence objects exert upon one another when they collide. Because people's judgments appear to deviate from Newtonian mechanics, psychologists have suggested that people depend on a variety of task-specific heuristics. This leaves open the question of how these heuristics could be chosen, and how to integrate them into a unified model that can explain human judgments across a wide range of physical reasoning tasks. We propose an alternative framework, in which people's judgments are based on optimal statistical inference over a Newtonian physical model that incorporates sensory noise and intrinsic uncertainty about the physical properties of the objects being viewed. This "noisy Newton" framework can be applied to a multitude of judgments, with people's answers determined by the uncertainty they have for physical variables and the constraints of Newtonian mechanics. We investigate a range of effects in mass judgments that have been taken as strong evidence for heuristic use and show that they are well explained by the interplay between Newtonian constraints and sensory uncertainty. We also consider an extended model that handles causality judgments, and obtain good quantitative agreement with human judgments across tasks that involve different judgment types with a single consistent set of parameters. (Contains 14 figures and 6 footnotes.)