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:

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


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

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?

  • 2
    $\begingroup$ Can you clarify what exactly you're asking a bit? I don't see an issue with a variable terminal velocity and how it relates to transonic flight. I mean, the speed of sound is dependent on a bunch of variables too, so it's not a fixed number either. $\endgroup$
    – Dragongeek
    Apr 13 at 20:14
  • $\begingroup$ "Terminal velocity" implies a hard limit on fast you can go. Jump out of a plane and you'll soon hit a max speed and keep going at that max speed. A variable terminal velocity no longer behaves as a limit, so it becomes no different from any other velocity calculated from a force balance. $\endgroup$
    – user39728
    Apr 13 at 20:17
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    $\begingroup$ Well, you've got your definition mixed up. Terminal velocity is "the constant speed that a freely falling object eventually reaches when the resistance of the medium through which it is falling prevents further acceleration", not some sort of speed limit. Also, it's not like this can't vary. For example, a skydiver jumping from space would have an extremely high terminal velocity high up because there is basically no atmosphere, and as she approaches the surface, her terminal velocity decreases. If the skydiver had a thruster to point upwards, they could simply exceed terminal velocity. $\endgroup$
    – Dragongeek
    Apr 13 at 20:21
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    $\begingroup$ @user39728 Terminal velocity is the speed at which the force of gravity equals the force of drag. It's not a speed limit - as a spacecraft re-enters the atmosphere, it invariably finds itself going faster than terminal velocity for the local atmospheric conditions, which is why aerobraking is a viable means to slow down. You must decelerate to terminal velocity, as opposed to a skydiver, who accelerates to terminal velocity. $\endgroup$ Apr 13 at 20:21
  • $\begingroup$ Sure. I'm not disputing the definition of terminal velocity. I know what it is. My point is that people suggest the rocket eventually settles at a constant terminal velocity. But there is no constant terminal velocity, and the rocket will continue to accelerate/decelerate as it drops through the atmosphere before the final landing burn. $\endgroup$
    – user39728
    Apr 13 at 20:25

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.

  • $\begingroup$ Thanks! That's what I thought. I just wish we had a better term for that velocity other than "terminal"---which clearly a lot of people casually interpret as a constant (and which I would too if I didn't know better). $\endgroup$
    – user39728
    Apr 13 at 21:30
  • $\begingroup$ But doesn't the drag coefficient change the fastest as you go from supersonic to subsonic? E.g., article.sciencepublishinggroup.com/journal/134/1341026/… $\endgroup$
    – user39728
    Apr 13 at 21:33
  • $\begingroup$ And air density changes by some 50% in the first 3 km above ground too: engineeringtoolbox.com/docs/documents/195/… $\endgroup$
    – user39728
    Apr 13 at 21:35
  • $\begingroup$ So both drag coefficient and air density change very much in the final few km before touch down... That terminal velocity is not even remotely close to constant. It is changing very, very rapidly as the rocket descends. $\endgroup$
    – user39728
    Apr 13 at 21:36
  • $\begingroup$ Ah, I meant to say subsonic, not transonic. Terminal velocity for an empty aluminum can is pretty low. $\endgroup$ Apr 13 at 21:37

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