# Chapter 1 - Mathematical Modeling and Engineering Problem Solving - Problems - Page 21: 1.1

$v(t)=v(0)e^{-\frac cmt}+\frac{mg}c(1-e^{-\frac cmt})$

#### Work Step by Step

Equation 1.9: $\frac{dv}{dt}=g-\frac cmv$ Multiply by $e^{\frac cmt}$ $v'e^{\frac cmt}+\frac cmve^{\frac cm t}=ge^{\frac cmt}$ Product rule $(ve^{\frac cm t})'=ge^{\frac cmt}$ $v(t)e^{\frac cmt}-v(0)=g\int\limits_0^te^{\frac cm\bar{t}}d\bar{t}$ Note: bar over t is to differentiate the integral variable from the time at an instant. $v(t)=v(0)e^{-\frac cmt}+\frac{mg}c(1-e^{-\frac cmt})$

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