A Universal Scaling Law for Constant-Force Motion in Linearly Resistive Media

Abstract

What is the travel time for an object moving under constant force through a resistive medium? Here we answer this fundamental question with a surprising simplicity: all such motions, regardless of the specific parameters, collapse onto a single universal curve. We derive a universal dimensionless parameter β  that governs the time delay of an object moving under constant applied force F₀
in a linearly resistive medium with drag coefficient b. The normalized travel time τ  is the ideal (drag-free) time, satisfies the implicit equation 2τ /β − 2(1 − exp(−βτ) )/β² = 1. This relation, which we term the Aqueeq universal scaling law, shows that all possible motions collapse onto a single universal curve. We provide rigorous asymptotic expansions with explicit error bounds, establish the physical interpretation of β as the inverse square root of the ratio of stopping distance to target distance, and validate the universal collapse with numerical simulations spanning five orders of magnitude in parameter space. Applications to engineering design, experimental diagnostics, and energy analysis are presented, along with generalizations to nonzero initial velocity. The analysis is strictly valid follow Reynolds-number flows (Re ≪ 1) where the linear drag law applies; limitations are discussed in detail