PHY.4:5.1 - Restrict an unknown resistive force

Seek the instantaneous mean resistive force on a body moving through a homogeneous isotropic medium at a fixed material state, in a classical three-dimensional regime. Assume the body and preparation introduce no preferred direction, relevant memory is negligible, and relative velocity w is the only varying input. These assumptions define the proposed comparison; their adequacy in a particular experiment remains a physical question.

Rotating the entire relevant situation rotates w and the force together. Because there is no further directional input, F(Qw)=QF(w) for every spatial rotation Q. For nonzero w, rotations about w rule out a perpendicular force component. Equal-length velocities are related by rotation, so write:

F(w)=-a(|w|^2)*w for w different from zero.

At w=0, rotational symmetry gives F(0)=0. The coefficient a is an unknown scalar function at the fixed material conditions. Its dimension is mass divided by time. Memoryless passive resistance gives:

F(w) dot w=-a(|w|^2)*|w|^2 <= 0,

so a(s)>=0 for s>0.

The result determines a direction and sign while leaving the speed dependence unresolved. Both a(s)=a0 and a(s)=b0*sqrt(s) with suitable nonnegative dimensional constants satisfy these restrictions. They predict different force magnitudes when speed changes. The comparison alone therefore provides neither a linear nor a quadratic drag law.

For a direction-only question, use the result directly. To predict stopping time, supply further information about a over the speeds involved, or obtain a sufficient bound over the admissible family. MMP.11 keeps that unresolved relation visible instead of inserting a familiar drag coefficient.