Using differential equations to graph velocity over time of a falling object subject to wind resistance Wind resistance -- upwards acceleration, typically varies either linearly or quadratically by the current velocity.
There is a constant downward acceleration due to gravity.
How can we model the velocity over time of a falling object, subject only to wind resistance and downwards gravity?
I don't have much experience with differential equations, but I do know that this answer necessarily involves it, so could you possibly explain every step?
Thank you.
 A: $$\begin{align*}
\sum F &= ma\\
\frac{dv}{dt} &= a\\
&= \frac{\sum F}{m}\\
\sum F &= mg - kv\\
\frac{dv}{dt} &= g - \frac{k}{m} v\end{align*}$$
This is a differential equation with a solution of
$$\begin{align*}
v &= A + B \space exp\left(\frac{-k}{m} t\right)\\
\frac{dv}{dt} &= - B \cdot \frac{k}{m} \cdot \exp\left(\frac{-k}{m} t\right)
\end{align*}$$
Match terms and initial conditions ($v = 0$ at time $t = 0$) and you get 
$$\begin{align*}
&g - \frac{k}{m}A - \frac{k}{m} B\cdot \exp\left(\frac{-k}{m} t\right)\\
\implies&-B  \frac{k}{m} \cdot \exp\left(\frac{-k}{m} t\right)\\
 \implies& A = g \cdot  \frac{m}{k} \space \text{and} \space  B = -g \cdot \frac{m}{k}\\ 
\implies& v = \frac{mg}{k} \cdot  \left(1 - \exp\left(\frac{-k}{m} t\right)\right)\end{align*}$$
That's a linear differential equation ($\frac{dv}{dt}$ is a linear function of $v$); the $\sum F = -kv^2$ is a nonlinear differential equation (can't remember off the top of my head how to deal with that one; it may not have a closed form solution). It's a bit difficult to summarize the techniques in general, but any good book on differential equations would cover them.
