Small Raindrop

Consider the following model:

A raindrop falls from rest with a small initial mass, $$m_0.$$ As it falls, it accumulates mass by absorbing the smaller water drops in its path, which we approximate by the uniform density $$\rho_\textrm{atm}.$$

Throughout the fall, it maintains the same characteristic shape defined by its width, $$2r,$$ which increases as it picks up mass at the rate $$\dot{m} = \rho_\textrm{atm}\pi r^2 v$$ (in time $$dt$$, it sweeps out a vertical cylinder of height $$vdt$$).

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 $$a=\frac gK$$ (convince yourself of this!).

What is the value of $$K?$$


Details and Assumptions:

• Neglect the effects of air resistance, wind speed, the density of air, or any other atmospheric factor except those stated in the problem.

Inspired by a similar problem by Tapas Mazumdar

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