The falling raindrop, revisited

TL;DR

This paper revisits the physics of a raindrop falling through mist, generalizing the accretion process with a power-law model and solving the resulting equations through a novel variable substitution.

Contribution

It introduces a generalized power-law model for raindrop accretion and provides a simplified solution method by eliminating time in favor of mass.

Findings

Derived analytical solutions for the raindrop's growth under power-law accretion.

Showed the effectiveness of variable substitution in solving coupled differential equations.

Extended classical raindrop models to more general accretion scenarios.

Abstract

I reconsider the problem of a raindrop falling through mist, collecting mass, and generalize it to allow an arbitrary power-law form for the accretion rate. I show that the coupled differential equations can be solved by the simple trick of temporarily eliminating time (t) in favor of the raindrop's mass (m) as the independent variable