[RedDevil]

Originally Posted by RoadRaceJosh

You previously wrote, and I read, all of that. You're skipping over the energy needed at Step 3. What you're describing is a propeller driven car moving slightly above the speed of its tail wind. And this requires a power input.

Good point.
I was already planning to write something about force versus power, this is a nice lead-in.
The power input, incidentally, is the wind blowing against the backturning propeller, putting a force on the wheels that makes them power the gears to turn the propeller; the energy is the distance the car moves over the ground while experiencing that force.
But that will become clear later; for now you only have to understand that the wind puts a force on the propeller.

The force versus power story is what enables both the spool to roll faster than the rope and the car to move faster than the wind.

There are only 2 contact points in the spool (actually 3, but let's take the wheels as 1): the point where the rope meets the axle and said wheels.
Naturally, when you pull the force on the rope tries to turn the spool backwards while the force on the wheels tries to turn it forwards.

But power is force times distance.
The spool has a smaller diameter than the wheels, so naturally the wheels cover a larger distance than the rope.
While the force on the rope and wheels is equal, it takes less power to roll up the rope on the spool than the wheels provide because the wheels cover more distance than the length of rope that got rolled up on the spool.

Likewise, with the car the force on the propeller and the force on the wheels are equal - but the distance they cover is not!
The propeller is moving the air backwards at a slower speed than the wheels move forward, so the wheels transfer more power to the gears than is needed to spin the propeller!

Let's quantify that to make it clear.
Suppose we set the car up with a 1/2 gear ratio so that the wheels drive the propeller to blow half as fast as the car moves.
Say the wind blows at 10 meter per second (using the metric system to keep conversions simple) pushing against the propeller with 100 Newton at standstill.
Then there's a force of 100 Newton at the wheels as well to drive the gears, to spin the propeller against that pressure of 100 Newton.

When the car covers the first meter the wheels have provided 100 Newton times 1 meter = 100 Joules of energy to the gears. But the fan needed only 50 Joules to push the air half a meter back against 100 Newton!
So there's 50 Joules of excess energy to overcome any friction.
We're supposing the friction is low, so the car accelerates.

Now assume we've reached wind speed.
The aerodynamic pressure is squarely related to speed; so with the propeller spinning backwards at half speed the force on the propeller is just 100N * 0.5 * 0.5 = 25N.
The force on the wheels is therefore also 25 N times 10 metes (as we're moving at 10 meters per second) = 250 Watt.
The power needed to spin the fan backwards is 25 N times 10 meters * 0.5 = 125 Watt, leaving 125 Watt to overcome the friction of the gears and propeller blades (as they are turning). But as the blades turn relatively slow, that is pretty minimal yet.

The fun ends when you near double the wind speed; it is impossible to go beyond the gear ratio. So to go faster you need a more aggressive setting, but then the initial acceleration is worse.

Let's do the same with a 4/5 ratio. For every meter the propeller blows the air back 80 centimeter.
At a standstill the force on the wheels and the resistance against them through the gears are at a 4/5 ratio too, so there's just 20 Newton left to set it all in motion. We need low friction here.

After the first meter the wheels produced 100 Joules and the fan needed 80 Joules to turn against the wind: only 20 Joules are available to accelerate and overcome friction.

At wind speed the force on the propeller is 100 N * 0.8 * 0.8 = 64 Newton.
The wheels provide 640 Watt to the gears, the gears use 512 Watt to spin the propeller leaving 138 Watt to overcome friction and accelerate. That is already more power than the 1/2 ratio provides at this speed!

At twice the speed the force on the propeller is 36 Newton, the power on the wheels is 20 * 36 = 720 Watt, the propeller needs 720 * 0.8 = 576 Watt, leaving 144 Watt for acceleration and friction.

At three times the speed the force is 16 Newton, wheel power is 480 Watt, the propeller uses 384 watt and we're left with 96 Watt to overcome friction. Which by now is surely bigger than that, as the propeller would be blowing the air back 2.4 times faster than the wind blows... only mitigated by the car moving forward 3 times as fast as the wind, or 6 times as fast as the wind speed relative to the propeller blades (0.4 times 10 m/s = 4 meter per second).
That's in fact faster than the car in the video could reach; its record is 2.8 times wind speed.

It is clear though why the car has variable gear ratios. You need a fairly low ratio to get going but a high ratio to maintain speed, and a gear ratio beyond 1 can be useful for braking and reversing.