OK, fine, the airliner is flying west, so west is the positive direction of our frame of reference. The velocity of the westbound airliner is +500 knots. The man is walking walking in the east direction , so that's -1 knot. a positive 500 knots, plus a negative 1 knot equals positive 499 knots, the man is still moving west at 499 knots. it doesn't become negative because of the direction he is facing.

"groundspeed" by definition, is relative to the ground. His velocity in his frame of reference is zero, because it's impossible for him to move relative to himself.

He is moving backwards. If I reverse my car at 30mph, what is my car's groundspeed? Clue: the speedo won't be reading 30mph.

Oh Christ. You're anchoring the +/- to the geographical direction. I was leaving them anchored to the body. Yes I can belabor the point and rework everything to keep to a geographical frame, but the whole purpose of the analogy was to relate it to what we all intuitively understand from flying bug smashers at 100 knots in 10 knot winds (which is where a positive sign on groundspeed always means that ground is moving opposite the direction the nose is pointing). Look at scenario A!

That's his speed relative to the airframe. When I move backwards relative to the ground, my groundspeed has a minus in front of it.

Well, yeah, what else are you going to measure ground speed relative to? Again, groundspeed is inherently relative to (in the frame of reference of) the ground.

You can't make any meaningful analysis when the axes of your reference system move whenever a person looks in a different direction.

Sure you can. If your last sentence is true, then it is not a meaningful analysis that in Scenario A the Cessna went from 90 to 110 groundspeed when turning East to West. Did you look at scenario A? Do you think that's meaningful or meaningless?

Hell we do the same thing about airspeed too, and necessarily so. The positive sign is always applied when air is coming from the direction of the nose, and we take that axis with us wherever we point the nose. If we didn't, then an airplane should fly equally well in a tailslide as in normal flight, and I can tell you from personal experience that it certainly doesn't.

From a physics standpoint, it is completely meaningless. You are making the identical mistake that Ian was making. In your Scenario A, From the frame of reference of the ground, the airplane went from a velocity +110 knots to a velocity of -90 for a total change in velocity of 200 knots.

Look, it's pretty clear that you simply are unable to grasp the concept of Velocity, how it is different than "speed", why it is imperative that velocity is used in physics and not speed, and why a constant, unchanging frame of reference is necessary to make any kind of a meaningful analysis. I apologize for having wasted your time trying to explain these things and I will stop now.

Say there's no wind (it's one of those perfect mornings) and you're flying around. You fly East for a while at 100 knots airspeed, and turn around and fly West. By that same premise, your airspeed is now -100 knots too, for a total change of 200 knots. Hmmmmm?

(if there's no wind then the airmass frame and the ground frame are the same, so you should just be able to do a find-and-replace of "ground" with "air" to your post and everything still hold true.)

No, it's perfectly valid, for certain contexts, to rotate the axes as needed.

The difference between these two examples is that in one the windspeed is less than the airspeed and in the other the windspeed is greater than the airspeed. In both cases airspeed is positive. In the first case the groundspeed reverses direction and in the second case the groundspeed is in the same direction. That means that in the first case you are entitled to take the difference between the groundspeeds but in the second case you have to take the sum. In other words along with common sense there is a minus sign missing.
I don't really mind where you stick it.

I should add that this is because calculation of groundspeed is a triangle of velocities problem envolving the cosine of the wind-angle. Less than 90 degrees is positive Greater than 90 degrees is negative

(1+500)+(1-500)
=501 +(-499)
=2

Quit being a martyr. You know it's perfectly reasonable, given the standard set of assumptions, to say that your groundspeed is 90 knots one way, then 110 the other way. In fact, before this thread, you would not have even blinked at saying exactly that. You know it.

It's not even a mistake, let alone the same mistake. The whole point is to explain to people like Ian, that what he saw was different to what the plane actually experienced.

Now, a ball can't travel backwards, but people and planes can so + and - is a way of expressing that relative movement.

No-it doesn't matter which way you are facing, it matters which way you have nominated as positive and negative in the reference frame you are using.

Acceleration is absolute- if his velocity changed by 1000kts in a fraction of a second, he'd be crushed by the Gs.

We are looking at is groundspeed not his velocity. His groundspeed can be + or -.

The magnitude of your ground VELOCITY is 30 MPH

What does a negative ground speed look like?

So if two people are standing in the isle facing each other, one is going positive and one negative? Does one therefore not get to the destination?

What is the ground speed of someone facing sideways?

Land a parachute with a negative groundspeed and have someone video you. Whether groundspeed it + or - does matter sometimes.