toad wrote:

Do you accept that it is possible for the cart to move directly upwind?

It is essential that we assume that to be the case so you can explain
the paradox exposed by the windmill on powerboat example.

Is that yes?

If in a headwind the windmill pushes back harder than it is pushed
then it must do that no matter how that headwind comes about. Which
leaves us with a power boat with a windmill on it's foredeck getting a
net gain in energy from wind that it is creating.


OK. Let's assume wind at 1m/s (I like metric - but that's about 2 1/4
MPH)) and a nice simple force on the windmill of 1000n. (~10Kg, or 20lbs
- ish) Neglecting losses, you get a kilowatt out of the mill - 1000n at
1 m/s.

Suppose you use that kilowatt to pull the anchor cable. Again,
neglecting losses, it ought to be able to pull the winch in at 1m/s,
because the force is the drag on the mill.

However, you do have losses. Say the whole system loses half the power
(in drag on the boat, turbulence around the blades, losses in the
generator, etc) You can then wind in the winch at only 1/2 m/s.

Now it doesn't matter how much you lose, you will still have some power,
and you will still be able to wind the winch in.

Back to the 50% loss case. You're winding the winch at 0.5 m/s, and the
wind is 1 m/s. Hey look, the apparent wind has gone up - you get more
power. Yup, as the force has gone up to 2.25 the original value (it's
proportional to the square of the wind speed) and the speed has gone up
- so the available power is now the cube of 1.5, a tad over 3. You can
wind the winch in faster, the wind goes up, you're away. EXCEPT the
drag went up too - by 2.25 times - and as the speed goes up, the power
goes up too. You'll find it more than cancels out - which was the point
of the spreadsheet I posted the other day.

It's a bit different with a prop in the water, not a winch, because a
prop. will require some power just to hold still - but not fundamentally
different.

Now the windmill on the deck of the power boat - lets stick with the
1000n force & 1m/s. (OK - Canal boat?). The windmill is going to get
its kilowatt out of the air. But the power boat has to push it through
the air, against that 1000n drag. Because of losses, you'll need *more*
than a kilowatt of engine power to overcome the extra drag of the mill,
but you'll only get a kilowatt, at best, out of the mill. Less than you
put in.

Remember that a stand-alone windmill cannot produce a forward force. It
can only generate power that something else can make into a forward force.

Are we there yet?

Andy