We seem to make a big deal in intro mechanics that x- and y- motions are “independent”. There certainly are in a limited number of cases (constant forces, linear drag forces, and linear restoring forces, for example) in which the force dynamics described among cartesian coordinates are separate. But this condition isn’t true in most situations (gravitational forces, electric forces, magnetic forces, non-linear drag, non-linear restoring forces). Still, in many situations, we can still find a coordinate system in which the forces are separable. For example, all the non-linear restoring forces (gravity, electric, rubberbands) are separable in (r,θ) coordinates, but the magnetic and non-linear drag forces I don’t think are separable, at least not generally so.
I could be wrong, but I think all (or most) force (fields) that can be described by the gradient of a scalar function will have at least one coordinate choice for which the force dynamics separate. This requires some further inspection, but it has to do with curl being zero. Linear drag forces are, of course, non-conservative; but they “nicely” separate due to a cancellation of factors that would otherwise couple them. This nice cancellation doesn’t occur with v² dependent drag forces, so non-linear drag forces are not separable.
I’m not sure, butI think when we teach this notion of “independent motions”, we are confusing independence with orthogonality, or perhaps separability with orthogonality. X and y cartesian coordinates are certainly orthogonal, but motions described using x and y coordinates are only independent (or separable) under circumstances in which they aren’t coupled.
Anybody know the history of teaching “horizontal and vertical forces” are independent? My suspicion is that it is an over-generalization from projectile motion.