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When a raindrop falls, it increase in size and so its mass at time $ t $ is a function of $ t, $ namely, $ m(t). $ The rate of growth of the mass is $ km(t) $ for some positive constant $ k. $ When we apply Newton's Law of Motion to the raindrop, we get $ (mv)' = gm, $ where $ v $ is the velocity of the raindrop (directed downward) and $ g $ is the acceleration due to gravity. The terminal velocity of the raindrop is lim $ _{t \to \infty} v(t). $ Find an expression for the terminal velocity in terms of $ g $ and $ k. $

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$$\lim _{t \rightarrow \infty} v(t)=g / k$$

Calculus 2 / BC

Chapter 9

Differential Equations

Section 3

Separable Equations

Missouri State University

University of Nottingham

Idaho State University

Boston College

Lectures

13:37

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. An ordinary differential equation (ODE) is a differential equation containing one or more derivatives of a function and their rates of change with respect to the function itself; it can be used to model a wide variety of phenomena. Differential equations can be used to describe many phenomena in physics, including sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, quantum mechanics, and general relativity.

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A raindrop falls through …

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The mass $m$ of a raindrop…

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