One of the common themes in cart discussions is that different people require wildly different sorts of explanations or analogies for it to "click". Once it does click it's not uncommon for people to feel irritated that "it would have been so much easier if you just said X in the first place". Of course the explainer has no way to know in advance which explanation is going to click for any particular person.
I absolutely agree with you that a force/energy calculation is the gold-standard best possible proof that the cart works. Unfortunately most people can't read force/energy calculations. In most cases throwing a wall of numbers their face is... unhelpful.
Also note that the explanation which did work for you demonstrated that the cart can work, but in many ways (and for many people), it's a completely worthless non-explanation of how it works. For many people it still leaves a very . Since you are a calculations-person, check out the almost magical "how" of wind-energy-collection going on in the following calculation:
The wind is going left at 10 m/s, cart is going 20m/s in the same direction. From the point of view of the cart, the air is going 10 m/s to the right. The kinetic energy of 1 kg of air at 10 m/s is 50 joules. (ke=1/2*mv^2). The cart spends energy to run the prop, accelerating the air faster to the right. The air leaves the prop at 12 m/s rightwards In the frame of the cart). The kinetic energy of the air has increased to 72 joules. Simple enough... the cart spends energy running the prop accelerating the air, and naturally energy is transferred from the cart into the air.
But now look at it from the ground frame. The air is a wind going 10 m/s to the left. 1 kg of air has a kinetic energy of 50 joules. The cart prop is pushing that air rightwards, the resulting in 8 m/s in the ground frame. The kinetic energy of this air, in the ground frame, is now 32 joules. 50 joules - 32 joules = 18 joules of energy vanishing from the air. Where does the energy go? The only place it could go is into the cart, and a careful analysis shows the cart's kinetic energy increasing by the power applied to the prop PLUS the energy lost from the air. The power applied to the prop is then subtracted back out by the drag at the wheels. The net result is the wind energy being teleported into the cart. From the ground frame the air energy went *into* the cart (increasing the cart's kinetic energy). This clearly shows the energy being extracted from the wind. But remember what the car's point of view was.... the cart was spending energy running the prop to pump energy OUT of the cart and INTO the air. The cart saw energy flowing in the opposite direction! Doh!
I have no idea where you got the idea that airplanes need millions of horsepower.
Allow me to recap how we got there, from my point of view. I was trying to get across the idea that the prop can generate greater thrust than the drag at the wheels. Pulling energy from the wheels to drive the prop looks like a perpetual motion design, and your natural gut reaction was to apply conservation laws. Trying to use using a generator to power a motor never gives you more out than you put in. So you mentally had me in the "crackpot zone", or at least in the "potential crackpot" zone. You were actively looking to shoot down everything I said. When I tried to explain that it was "harder" for a prop to push against a faster wind, using an example that it was "hard" for a prop against a 100 MPH wind, you answered that it was trivially easy for a prop to push against 100 MPH air. You presented a graph of a half-million horsepower prop thrusting in 100 MPH air.
I rather naturally took your example and pointed out how it would take twice as much power to get the same thrust in air going twice the speed.
I was trying to get across that the ease/difficulty of generating thrust at the prop was completely relative to the air speed. You were countering/challenging that with a completely non-relative disproportionate example calling it "easy" to push against 100 MPH air.... and the gross million horsepower number was deliberately highlighting how the objection had wandered off in a flawed/silly direction.
The reduction of efficiency of propellers at low airspeed is a general effect, and I believe it is mostly due to recirculation of the airflow at the tips.
Yeah, although the interpretation "efficiency" be context-dependent, depending on what we are trying to obtain. Prop power efficiency breaks down at low speed, but to get the cart to work what we actually desire at the prop force, greater force than the drag at the wheels. Props are more efficient at converting power into force at low (or zero) airspeed. Perhaps it would be better wording to say the prop is "more effective" at producing force under those conditions.
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