Calculate the free fall distance of an object with air resistance I would like to know how I could calculate the distance an object would fall with air resistance, assuming it's starting velocity is 0.
 A: As mentioned in the comments, the resistive force is proportional to the velocity and let the constant of proportionality be $b$.
SO, $ma=mg-bv$
$a=g-\frac{bv}{m}$
Let $b/m=k$
Then $a=g-kv$
$dv/dt=g-kv$
This is a linear differential equation. Solving this for v in terms of t, we get.
$ve^{kt}=\frac{ge^{kt}}{k} + c$
Applying the condition $v=0$ at $t=0$, we get $c=-g/k$
Thus, $$v=\frac{g(1-e^{-kt})}{k}$$
Write $v=ds/dt$
Take $ dt$ to the other side.
$$ds=\frac{g(1-e^{-kt})}{k}dt$$
Now, I leave this integration for you to solve. Solve this as indefinite integration and obtain the value of integration constant by putting $s=0$ for $ t=0$.
One question came to my mind while solving. Will the relation obtained between $s$ and $t$ follow the concept of terminal velocity?
Edit:
This relation is only approximate because in practical situations, $b$ does not remain constant for the whole journey. The value of $b$ can be approximated by conducting an experiment and using the relation between $s$ and $t$ . Suppose you release an object from a height $h$ (let) and it hits the ground in time $t$. Then, the value of $b$ can be easily determined.
