Consider the following model:
A raindrop falls from rest with a small initial mass, As it falls, it accumulates mass by absorbing the smaller water drops in its path, which we approximate by the uniform density
Throughout the fall, it maintains the same characteristic shape defined by its width, which increases as it picks up mass at the rate (in time , it sweeps out a vertical cylinder of height ).
How will its velocity depend on time? Surprisingly, this complicated problem has a simple solution: after a short time, the raindrop will accelerate at the constant rate (convince yourself of this!).
What is the value of
Details and Assumptions:
Inspired by a similar problem by Tapas Mazumdar