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Nick_Tarleton comments on The Correct Contrarian Cluster - Less Wrong
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Here's where I am now:
Sailboats can move with a downwind component faster than the wind.
The first (windsock) video shows no evidence that the cart moves downwind faster than the wind.
The string video is more convincing, but I'm not convinced that this particular cart works as advertised. The rational offered for how it works is that when it moves at a velocity v, this causes the propeller to turn at a rate that thrusts air backwards with a velocity greater than v. Hmm... okay, maybe. The propeller blades moving perpendicular to the wind are a lot like the sails of a boat moving perpendicular to the wind.
This ignores mass. Thrust is (mass flow rate) * v_air, so it can get enough thrust by moving enough air at a velocity less than v_car. As for energy, power = 1/2 * F * v for air or car, so again, you can get enough thrust if v_air < v_car.
There's no perpetual motion, because as the originally linked solution says, eventually the apparent headwind becomes too strong. (The above assumes apparent wind is zero.)