Time Scale for Velocity to Track a Force

TL;DR

This paper derives a formula for the time required for a force to significantly change an object's velocity direction, considering viscous effects and generalizing to other vector quantities with constant derivatives.

Contribution

It provides a new analytical expression for the velocity turn-around time under constant and viscous forces, extending to general vector quantities.

Findings

Derived formula for velocity change time $ au$

Viscous forces reduce $ au$ logarithmically

Generalization to any vector with constant first derivative

Abstract

In this paper we derive and discuss the time it takes for a force to turn a velocity. More precisely, we derive the formula for the time $\tau$ it takes a constant force that makes an angle $\alpha$ with the initial velocity $\vec{v}(0)$ to have $\vec{v}(\tau)$ get within an angle $\theta<\alpha$ of the force. We then show how the addition of a viscous force decreases $\tau$ logarithmically. The result can be generalized to any vector quantity whose first time derivative is a constant.