Terminal Velocity of Body under Fall Retarded Proportional to Square of Velocity
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Theorem
Let $B$ be a body falling in a gravitational field.
Let $B$ be falling through a medium which exerts a resisting force of magnitude $k v^2$ upon $B$ which is proportional to the square of the velocity of $B$ relative to the medium.
Then the terminal velocity of $B$ is given by:
- $v = \sqrt {\dfrac {g m} k}$
Proof
Let $B$ start from rest.
The differential equation governing the motion of $B$ is given by:
- $m \dfrac {\d^2 \mathbf s} {\d t^2} = m \mathbf g - k \paren {\dfrac {\d \mathbf s} {\d t} }^2$
Dividing through by $m$ and setting $c = \dfrac k m$ gives:
- $\dfrac {\d^2 \mathbf s} {\d t^2} = \mathbf g - c \paren {\dfrac {\d \mathbf s} {\d t} }^2$
By definition of velocity:
- $\dfrac {\d \mathbf v} {\d t} = \mathbf g - c \mathbf v^2$
for some constant $c$.
and so taking magnitudes of the vector quantities:
\(\ds \int \dfrac {\d v} {g - c v^2}\) | \(=\) | \(\ds \int \rd t\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac 1 {2 c} \sqrt {\frac c g} \ln \paren {\frac {\sqrt {\frac c g} + v} {\sqrt {\frac c g} - v} }\) | \(=\) | \(\ds t + c_1\) | Primitive of $\dfrac 1 {a^2 - x^2}$: Logarithm Form |
We have $v = 0$ when $t = 0$ which leads to $\ln 1 = 0 + c_1$ and thus $c_1 = 0$.
Hence:
\(\ds \ln \paren {\frac {\sqrt {\frac c g} + v} {\sqrt {\frac c g} - v} }\) | \(=\) | \(\ds 2 t \sqrt {c g}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\sqrt {\frac c g} + v} {\sqrt {\frac c g} - v}\) | \(=\) | \(\ds e^{2 t \sqrt {c g} }\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sqrt {\frac c g} + v\) | \(=\) | \(\ds \paren {\sqrt {\frac c g} - v} e^{2 t \sqrt {c g} }\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds v \paren {e^{2 t \sqrt {c g} } + 1}\) | \(=\) | \(\ds v \paren {e^{2 t \sqrt {c g} } - 1} \sqrt {\frac g c}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds \sqrt {\frac g c} \paren {\frac {e^{2 t \sqrt {c g} } - 1} {e^{2 t \sqrt {c g} } + 1} }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {\frac g c} \paren {\frac {1 - e^{-2 t \sqrt {c g} } } {1 + e^{-2 t \sqrt {c g} } } }\) |
Since $c > 0$ it follows that $v \to \sqrt {\dfrac g c}$ as $t \to \infty$.
Thus in the limit:
- $v = \sqrt {\dfrac g c} = \sqrt {\dfrac {g m} k}$
$\blacksquare$
Sources
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 5$: Falling Bodies and Other Rate Problems: Problem $1$