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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