content(warnings): Add W3-W5, N2-N3 — drag, work, and rolling warnings + notes

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2026-05-04 23:31:32 -05:00
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@@ -10,6 +10,10 @@ The negative sign means that the drag force always points opposite the velocity.
If an object falls vertically through the fluid and eventually moves with constant velocity, that steady velocity is called the \emph{terminal velocity}. At terminal velocity, the net force is zero, so the acceleration is zero.}
\wc{Terminal velocity is not zero speed}{Terminal velocity is a \emph{nonzero constant speed}: the object is still moving, but its acceleration is zero because drag balances the driving force. The object approaches $v_T$ asymptotically as $t\to\infty$; it never stops.}
\wc{Drag force is not kinetic friction}{Drag $F_D=bv$ or $F_D=\tfrac12 C\rho A v^2$ is velocity-dependent and acts in fluids. Kinetic friction $f_k=\mu_k N$ is approximately velocity-independent and acts at solid-solid contacts. They are physically distinct interactions.}
\thm{Vertical-fall ODE and terminal-speed result}{Choose the vertical axis positive downward. Let $v(t)$ denote the downward velocity of an object of mass $m$ at time $t$, let $g$ denote the gravitational field strength, and let $b>0$ denote the linear-drag coefficient. Then the vertical equation of motion is
\[
m\frac{dv}{dt}=mg-bv.