Terminal Velocity of Body under Fall Retarded Proportional to Velocity

Theorem
Let $B$ be a body falling in a gravitational field.

Let $B$ be falling through a medium which exerts a resisting force $k \mathbf v$ upon $B$ which is proportional to the velocity of $B$ relative to the medium.

Then the velocity of $B$ reaches a limiting value:
 * $v = \dfrac {g m} k$

This value is called the terminal velocity of $B$.

Proof
Let $B$ start from rest.

The differential equation governing the motion of $B$ is given by:


 * $m \dfrac {\mathrm d^2 \mathbf s} {\mathrm d t^2} = m \mathbf g - k \dfrac {\mathrm d \mathbf s} {\mathrm d t}$

Dividing through by $m$ and setting $c = \dfrac k m$ gives:


 * $\dfrac {\mathrm d^2 \mathbf s} {\mathrm d t^2} = \mathbf g - c \dfrac {\mathrm d \mathbf s} {\mathrm d t}$

By definition of velocity:


 * $\dfrac {\mathrm d \mathbf v} {\mathrm d t} = \mathbf g - c \mathbf v$

and so:

When $t = 0$ we have that $\mathbf v = 0$ and so:
 * $\mathbf c_2 = \mathbf g$

Hence by taking magnitudes:
 * $v = \dfrac g c \left({1 - e^{-c t} }\right)$

Since $c > 0$ it follows that $v \to \dfrac g c$ as $t \to \infty$.

Thus in the limit:


 * $v = \dfrac g c = \dfrac {g m} k$