Terminal velocity on reentry?

Question

I've seen many references to "terminal velocity" on reentry, e.g., prior to a Falcon 9 landing burn.

And I'm confused by this.

Because terminal velocity comes from drag which depends closely on air density and drag coefficient.

And air density increases a lot as the rocket descends. And the drag coefficient changes a lot—and very nonlinearly—as the rocket goes from supersonic to subsonic.

(A Falcon 9 is traveling at hundreds of m/s when the final landing burn begins---right in that transonic region where the drag coefficient is highly nonlinear).

So the force balance on the rocket (before ignition) is:

$$ mg - \frac{1}{2} C_\texttt{D}(v) \rho(h) A v^2 = 0, $$

or:

$$ v = \sqrt{\left(\frac{2mg}{C_\texttt{D}(v) \rho(h) A}\right)}, $$

where velocity is not a constant, not even approximately, but a variable changing wildly (and nonlinearly) during descent.

Am I not seeing this right?

A parachuter normally jumps from no more than a mile or two above ground at no more than 150 mph, so changes in air density and drag coefficient are small and you can approximate them as constant---meaning it's OK to talk of terminal velocity there.

But a transonic rocket dropping rapidly and wildly in speed and altitude? At best, you can speak of a variable terminal speed, it seems? And if it's variable, then what meaning does "terminal" have anymore?

Answer 1

You're correct that terminal velocity varies.

For a given vehicle in a given attitude relative to the airstream (i.e. a particular angle of attack), though, it varies only with the air density and therefore primarily by altitude. The $C_D$ varies with airspeed, but it doesn't change very much over the slow subsonic regime of low-altitude terminal velocity, and it changes continuously, so terminal velocity does converge rather than somehow oscillating wildly.

So the meaning, and the exact value, of terminal velocity is always changing, and dependent on context.

For a falling object that's going to hit the ground, the "terminal" part of the term is definitive: it's the terminal, last, final velocity the object has before it hits the ground.

When speaking of a Falcon 9's first stage, "terminal velocity before the landing burn" means exactly that -- the natural falling velocity reached by the stage before the engines fire. It's the terminal (final, last) velocity of the free-falling portion of the flight.