Newtonian mechanics is not just a physical theory; this is a general paradigm applicable in very different area of experience. Indeed, it does not incorporate any specifically physical ideas; all it does is to link the overall dynamics of a system to some informally introduced "forces". To consider a physical force, we need a special model deriving it from the nature of particles, fields etc. In exactly the same manner, we can describe the interactions of any other entities, which may result in a science very far from physics, but still using the same mechanical paradigm (up to reutilizing its mathematical formalism).
For example, a biological system of any level (from a single molecule to the biosphere in general) can be characterized with a few fundamental parameters so that one combination of values should be treated as essentially different from another. This means that all such states are representable with the points of an appropriate configuration space, the way a material body is said to occupy a particular spatial position in physics. Now, once urged from the outside, the organism (in the most general sense) reacts with a certain sequence of acts that, assuming both a definite pace (the analog of velocity) and a kind of inertia (the analog of mass). For that, we need the notion of a mechanical momentum. While the organism keeps on with the same metabolic process (including outer behavior), this does need any special attention, it just happens. As soon as something relevant occurs in the organism's environment, the state of motion will change, which is effectuated through a number of physiological effects that may, in higher animals, be treated as an emotional response (under a permanent pressure becoming an emotional mood, a feeling). The paradigmatic value of mechanical acceleration (as the inner response to an external force) thus comes out.
Now, consider the usual weather variations as described by current temperature, atmospheric pressure, humidity etc. An organism experiences them as weather provided the characteristic times of the change are comparable with the duration of metabolic cycles. Slower variations are associated with the alterations of the seasons, or climatic shifts; faster processes may be just impossible to describe with the statistical parameters of that kind.
The mechanical paradigm predicts that it is the acceleration values (the second derivative of the state function) that produce significant inner responses: the living organism does not pay much to steady grow or decrease, but will immediately report on any abrupt changes. Weather dependent people (especially the aged ones) can easily check the validity of this conclusion, and it seems to adequately arrange the facts. However, there is yet another aspect of the same that is generally overlooked, as the traditional whether services only report some averaged trends, while real atmospheric processes may be much more complicated. Indeed, observe that, while the information about daily variations of the atmospheric pressure may be very useful for risk control, there are much faster pressure variations occurring within a few hours, or even within minutes (for instance, as an effect of the variations of solar irradiation, as the sun hides behind the clouds and comes back in occasional clearances). With the mechanical model, we can expect that such lower-scale variations may be much more dangerous for a weather dependent person since they result in stronger forces (represented in the model as the product of acceleration and mass) just tearing the organism into pieces (as people often feel it). We can also predict that weather dependence is more pronounced in physiologically stiff people (characterized with greater mechanical mass), with less responsive organism, which may be either a genetic predisposition or a trace of disease.
The same holds for any other mode of motion. Thus, people adapt themselves to the gradual landscape changes that are bound to develop during millennia; however, the same change happening within a year (or a day) would be a terrible catastrophe. Psychologically, minor nuisances happening every day are much more disturbing than a steady moral climate (however negative). Gradual improvement of scientific theories or technological schemes is nothing compared to drastic reorganization on some entirely new principles. No economic and social crises can shatter the layman's innate fear of a revolution.
Paradigms are abstractions, at the highest possible level. However, it is this generality that makes them extremely practical. One does not need an "exact" mathematical formulation to get many qualitative predictions. In fact, the same paradigm accepts different kinds of formalization; moreover, new mathematics is always born from an already existing paradigm in the course of its adaptation to a particular application area. Weather dependence may be pictured using other paradigms; still, the mechanical outlook, too, is bound to find its objective niche.