PHY.Preface:4 - Worked connection - From an unknown resistance to a useful comparison

Consider a body moving through a medium with positive speed v. The work needs to know how far it travels while slowing from 1 m/s to 0.5 m/s. Use a stipulated effective inertia of 1 kg and no other force along the motion. The proposed account treats resistance as an instantaneous function of relative speed at fixed material conditions.

Constrain before selecting a law. PHY.4 recovers the medium, relative motion and the conditions kept fixed. In an isotropic comparison without another relevant direction, resistance opposes motion under the passivity premise, while its magnitude can retain an unknown speed dependence. A force magnitude of 1 N at 1 m/s alone leaves, among other possibilities, linear resistance D=b v and quadratic resistance D=c v². Take the corresponding candidates b=1 N·s/m and c=1 N·s²/m². These are two proposed accounts, not all laws admitted by the initial restriction.

Construct evolution and obtain its consequence. PHY.6 combines the resistance with the momentum balance and position change: m dv/dt=-D(v), dx/dt=v. MMP.10 retains these relations and their preparation. Eliminating time over the stated positive-speed interval gives dx/dv=-m v/D(v). The computation under C.29.2 can now use the two supplied integrals:

distance_L = (m/b) (v_0-v_1) = 0.5 m,

distance_Q = (m/c) ln(v_0/v_1) = ln(2) m, approximately 0.693 m.

If the available travel distance is 0.6 m, the two accounts give different answers. For an available distance above 0.7 m, both would meet this particular requirement under their premises; resolving their difference would then need another reason. Neither comparison licenses extrapolation to zero speed or a new physical regime.

Expose the difference that matters. PHY.10 asks for a preparation where the candidates diverge. At a maintained speed of 0.5 m/s they predict force magnitudes 0.5 N and 0.25 N. An available calibrated force arrangement with error bounded by 0.02 N separates their predicted indication ranges. An indication of 0.25 N is compatible with the quadratic account and incompatible with the linear one under these conditions. If no suitable readout is available, PHY.9 supplies its construction question; the distance comparison remains conditional meanwhile.

The steady comparison can inform coasting only if the retained instantaneous-response and material premises cover both preparations. PHY.5 examines a consequential wake, relaxation time or other omitted state if that transfer is doubtful. A memory effect can require a different evolution rather than a new value of the old coefficient.

Change a premise and return locally. Suppose the medium’s material condition changes between the initial and final force comparisons. The coefficient is no longer established as the same. Recover or constrain that dependence before treating the new indication as a test of the two original candidates. Their conditional integrals remain correct; their applicability to the changed run is what reopened.

The resulting work can be divided. A physical contributor supplies the interaction and preparation account, a mathematical contributor derives the permitted consequences, and a computational contributor obtains them in a useful form. B.5.MPC keeps those contributions connected to the distance question; B.5.MPC.R locates the affected return after the material change.