Terminal velocity

{title2=$v_T=Mg/b,\quad\tau=M/b$}

Consider a <point mass> of fixed <mass> $M>0$ released from rest in a stationary medium. Choose downward as positive, assume constant <gravitational acceleration> $g>0$, neglect <buoyancy>, and impose <linear drag> $F_d=-bv$ with constant <linear friction coefficient> $b>0$. Here $v$ is the signed downward <velocity> component; it equals <speed> during this descent. <Newton's second law> becomes
$$
M\frac{dv}{dt}=Mg-bv,\qquad v(0)=0,\qquad v_T=\frac{Mg}{b},\quad\tau=\frac{M}{b}.
$$
The <terminal velocity> balances the two <forces>, and the <time constant> $\tau$ controls how quickly that balance is approached:
$$
v(t)=v_T(1-e^{-t/\tau}),\qquad a(t)=\frac{dv}{dt}=ge^{-t/\tau},\qquad t\ge0.
$$
For the illustrative parameters $M=1\,\mathrm{kg}$, $b=2\,\mathrm{kg/s}$ and $g=9.8\,\mathrm{m/s^2}$, the <time constant> is $0.5\,\mathrm s$ and the <terminal velocity> is $4.9\,\mathrm{m/s}$ downward. At $t=1\,\mathrm s=2\tau$, the downward <velocity> is $4.9(1-e^{-2})\,\mathrm{m/s}\simeq4.23686\,\mathrm{m/s}$ and the downward <acceleration> is $9.8e^{-2}\,\mathrm{m/s^2}\simeq1.32629\,\mathrm{m/s^2}$. This is a toy <linear drag> model, not a claim that every $1\,\mathrm{kg}$ falling object has this resistance law.

\Image[/linear-drag-terminal-velocity.png]
{title=Exponential approach to terminal velocity and decay of acceleration in an illustrative linear drag model}

For a proof outline, subtract the equilibrium value: $u=v-v_T$ obeys $du/dt=-u/\tau$. The <exponential decay> solution and the initial condition give $u=-v_Te^{-t/\tau}$; one <derivative> gives the <acceleration>. Every <time constant> multiplies the remaining gap $v_T-v$ by $e^{-1}$. The exact limit is reached only as $t\to\infty$. This makes the model useful for estimating a settling duration without mistaking vanishing <acceleration> for vanishing <velocity>.

Newton already analyzed vertical motion under uniform <gravity> and resistance proportional to <velocity> in Book II, Section I, Proposition III of the \i[Principia]. His geometric construction includes a greatest downward <velocity>, illustrating that a limiting rate can emerge from balancing a driving <force> and resistance. https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1729)/Book_2/Section_1[Newton, Principia, Book II, Proposition III; Motte's 1729 translation]{ref}


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 Incoming links: Terminal velocity under linear drag