The idea that GS can’t fall into a negative value is really quite stupid. No need to report GS as mainly positive as its regarded as positive unless stated otherwise.

Absolute value.

Speed is distance over time. Velocity is distance over time with a direction component, a vector. The ground speed of this slow flying Cessna is 10 knots; it may not be making any progress towards its intended target, but its speed over the ground is 10 knots. Its velocity vector is 10 knots in the direction of the wind. There's no 'negative' speed, there's speed in the direction opposite to the intended travel. If you want to plot this on an X -Y axis then you could certainly make the direction it ends up traveling as -X, the speed (magnitude) is still 10 knots, the direction is negative.

How long is a bit of string?

Aka the speed is negative 10

GS isn’t calculated as the absolute value of airspeed - headwind. If you think it is then you should edit that wiki page accordingly.

You could measure that pretty easily. Start with the the the airplane stabilized in a 100 knot cruise, dive the airplane until the speed is above 110 knots, level out, start timing when the airspeed decreases through 110 knots, and stop timing when the airplane stabilizes at 100 knots again. That's how long it takes.

As I have said, I'm not a pilot, I'm a winemaker with an interest in aviation. I think we are confusing aviation terms with physics terms. I am thinking in physics terms; speed is a quantity, a magnitude, and combined with the direction of travel generates a vector. The length of the vector is the speed, the way the vector points is the direction. The vector sum of the wind speed and the slowly retreating Cessna is the resultant that defines the aircraft vector. In this case the speed is 10 knots and the direction is along the negative X axis. The speed is not negative, the resultant of the vectors is -10X after one hour. The plane has traveled 10 nautical miles in a direction opposite to where the pilot wanted to go.

When you are discussing ground speed I assume you are thinking in aviation terms, but this whole thread is not about aviation terms, it's about somehow magically gaining energy when turning windward in a steady wind environment. This seems to me to indicate that we need to treat this as a physics problem, not an interpretation of 'ground speed' in the aviation sense.

So call the Cessna's ground speed -10 knots/hr if you wish, and I'll call it 10 knots/hr at angle Θ, but I think we mean the same thing.

So, if you have a groundspeed of negative thirty knots for an hour is your displacement negative thirty miles from your origin?

Well, the math is that drag is now more than thrust, so it will cause acceleration in the "tailwards" direction.

This would be the same as flying at 100 kts, then quickly reducing throttle to the thrust you know will sustain 90 knots.

From experience I would say a matter of 5-10ish seconds.

ET TU Wiki-

https://en.wikipedia.org/wiki/Speed

in everyday use and in kinematics, the speed of an object is the magnitude of its velocity (the rate of change of its position); it is thus a scalar quantity.

A position can't have a negative rate of change.

Yes!! When defined as positive one way, negative the other. You know, like that nonsensical useless system called longitude.

Thus, you have added direction to magnitude and are talking velocity not speed.

The distance between two paces can't be negative thirty miles.

OK, and?

text filler

OK,

Scenario B2 (really fast running person inside airliner. He runs at 501 knots and can turn around on a dime.)
final groundspeed minus initial groundspeed
(airspeed + tailwind) - (airspeed - headwind)
or, if you like:
(walk + tailwind) - (walk - headwind)
(501+500)-(501-500)
1001-(1)
1000

Scenario B3 (really really fast running person inside airliner. He runs at 1000 knots and can turn around on a dime.)
final groundspeed minus initial groundspeed
(airspeed + tailwind) - (airspeed - headwind)
or, if you like:
(walk + tailwind) - (walk - headwind)
(1000+500)-(1000-500)
1500-(500)
1000

Holy schnike, it turns out that if you substitute any number for the airspeed, the groundspeed difference is 1000, which is twice the wind speed when going from upwind to downwind.

Unless that number is less than the wind speed, then it becomes a special case for some reason that wraps people around the axle over scalars and quantum chromodynamics and whatever else.

We were talking about ground speed, not ground velocity.

Get real. You know the two are collquially interchangable, and you know exactly what I meant. The point isn't just flying by you, you're actively dodging it. Look one post up.

No, speed and velocity are completely different things and are useful; in different circumstances.- and if your whole argument has been that they are the same, you have been sadly mistaken.

Your one post up is just silly- turning your back does not change your velocity.

Do you think there's an error in the math? If so, what is it? First you weren't happy that there were negative groundspeeds, so now that they're all positive, then what?
Do you apply that same criticism to scenario A? If not, why not? What's the difference?

Scenario A (Cessna)
final groundspeed minus initial groundspeed
(airspeed + tailwind) - (airspeed - headwind)
(100+10)-(100-10)
110-90
20

I don't dispute the math- I dispute you are making any pertinent point.

To make sure you haven't lost the plot -- what do you think my (attempted) point is?