---
tags : Mechanics,Kinematics
title : 2020.4.13(9)
---
# Kinetics of a raindrop in a heavy force field
### The equation of motion of a small object such as a raindrop forced influence by viscorcity of air and gravity.

$\huge m\frac{dv}{dt} = mg - bv$

$\huge \frac{dv}{dt} = -\frac{b}{m}(v-\beta)$

$\large (\because \beta \equiv \frac{mg}{b})$

$\beta$ means the velocity that reach finarly caused by balancing viscorce resistance and gravity.
In other words, it is "**the last speed**"

($\because$ if we set $u \equiv v-\beta$, that shows decay because of $\frac{du}{dt}=- \frac{b}{m}$. )

The power of resistance against a ball (This disemmeter is a) which fall slowly in liquid is , for Stokes's rule ,

$\huge -bv = -6\pi a \eta v$

($\eta$ is a ratio of viscorce)

Therefore ,

$\huge \beta = \frac{mg}{6\pi a \eta} = \frac{2a^2\rho g}{9\eta}$

$(\large \because m = \frac{4}{3} \pi a^3\rho)$

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### These fact shows that the final velocity proportionate squared to the disemmeter a.
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