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DDWFTTW

Is it possible to build a machine which goes directly downwind, faster than the wind, powered only by the wind? Sounds crazy, but it works!

If someone had asked me if I thought such a device/machine were possible, I think my intuition would have been that it isn’t. I mean, come on! If it’s going faster than the wind, then it sees an apparent headwind, from which it somehow has to extract energy so that it can move in the opposite direction, all without violating the conservation of momentum. But I’d (hopefully) also mutter something about the physics of sailing not being immediately straightforward as an admonition abut too hastily jumping to conclusions. Sailboats sail faster than the wind by making their own wind, after all. =)

Since the video pretty convincingly shows that it is possible, as physicists our job is to explain how/why it works. This is actually the most fun situation — when experiment contradicts intuition — since then we learn something! Although I don’t have a solid explanation, here’s (I think) a fruitful way to think about how it works.

An important clue that something strange is going on is best shown in this video:

Watch carefully as the cart starts. Which way does the propeller turn? Answer: Opposite to the way it “should”, i.e. opposite to the way the wind is pushing it. In this case, it turns clockwise as seen from behind the cart, when it “should” be turning counterclockwise, given the orientation of the blades. How can this be? Because the propeller is connected to the wheels in just such a way to make this happen when the cart moves forward.

This tells us that we should consider two competing influences on the cart: the force pushing it forward and the force trying to turn the propeller. Let’s start with the case shown in the video in which the cart is at rest with respect to the ground. On the one hand, the wind hits the propeller and imparts some force on the cart, caused by the air molecules bouncing backwards off the propeller blades. This acts to push the cart downwind (forward). On the other hand, the wind imparts a torque on the propeller, caused by the air molecules bouncing sideways off the propeller blades. This torque is transmitted to the wheels, and through them to the ground (assuming the wheels don’t slip), which in turn imparts another force on the cart. The cart is designed so that this force acts to push it upwind (backwards). Depending on the shape of the propeller, the air molecules bounce off in a combination of sideways and backwards (the net effect in the latter case is for the whole air mass to slow its forward velocity, not actually all move backwards; this “ballistic” model is very crude and we should really use fluid dynamics. But I’m a quantum mechanic, so this is as good as it gets for now.). The cart moves downwind because (apparently) the net force is in the forward direction. For short, the force is greater than the torque.

This condition persists even when the cart is moving at exactly the same speed as the wind. Absent the effect of the propeller, it sees no apparent wind. But the propeller is spinning and pushing air backwards and so there’s a forward force and opposing torque (slowing the propeller down). Presumably the force is still greater than the torque, and the cart accelerates to a speed greater than the wind.

When does this condition break down? It must at some point, or the cart will accelerate forever and build up infinite energy in the process. [Edit: no, this isn’t right. As long as the wind is moving relative to the ground, there’s the possibility of getting energy out, so there’s in principle an infinite energy supply.] Consider the flow over the propeller. When the wind is from behind, the propeller is in “propeller mode” since as it turns it pushes air backwards. It’s also in propeller mode when the cart is moving at the wind velocity. But given a really strong headwind, the propeller operates in “windmill mode”, meaning the wind is now trying to turn the propeller clockwise (as seen from behind the cart). Thus the torque acts in the opposite sense as in propeller mode, and so is now pushing the cart forward. At the same time, the force of the headwind acts to push the cart backwards. If the force was greater than the torque at zero velocity, the same should be true here as well, and thus the net force on the cart in a very strong headwind is backwards. So far, so good: The cart can’t continue to accelerate forwards forever. [Edit: this can’t be right for (at least) two reasons. 1. If the force is greater than the torque at tailwind speed v, then it should be at -v as well. So you should only be able to go twice the wind velocity. But the recent test runs show speeds of almost 3x wind velocity. 2. There aren’t two different “modes” to the blades; they always spin so as to push air behind the cart. Not sure what the fix is. The two situations of wind at speed v from behind and speed v from ahead are only different because the blades are spinning the latter case, but not in the former.]

In between these two situations is a certain wind and propeller speed combination such that the air flows parallel to the blades, at zero angle of attack. Seen from the perspective of the blade, the air is just rushing past it, not bumping into it at all (it’s very thin). Both the torque and force on the propeller are zero in this case, so absent drag on the cart itself and other sources of friction, the forces on the cart are balanced. This determines the ultimate velocity of the cart. [Edit: or not. This speed would have to be the wind speed itself (so that there’s no tailwind or headwind). But that’s wrong; the cart goes faster than the wind.]

This way of looking at things gives a pretty clear picture of what’s going on under the assumption that the forward force directly on the blades of the propeller is larger than the counteracting torque. To give a solid explanation one should come up with a realistic model of the propeller, calculate the forces and torques at various wind speeds using fluid dynamics, and show that the net force on the cart is in the forward direction, even when moving at the wind velocity. It also predicts that if the propeller were geared oppositely, the cart could only move with the wind and never faster since the zero angle of attack condition is met for some cart speed less than the wind speed. Moreover, it raises the question of whether it’s possible to change the gearing of the propeller, the size of the wheels, or the shape of the propeller so that the initial torque is larger than the force, and the cart moves upwind.

In the end, this is related to the physics of sailing! From the Wikipedia article on sailing:

The energy that drives a sailboat is harnessed by manipulating the relative movement of wind and water speed: if there is no difference in movement, such as on a calm day or when the wind and water current are moving in the same direction at the same speed, there is no energy to be extracted and the sailboat will not be able to do anything but drift. Where there is a difference in motion, then there is energy to be extracted at the interface, and the sailboat does this by placing the sail(s) in the air and the hull(s) in the water.

That’s similar to what’s going on here — the cart can manipulate the relative movement of the wind over the ground, since it is connected to both. Without the wheels there’s no way to transform the torque on the propeller into a force on the cart.

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A pithy explanation of evolution

Roger Ebert is a national treasure (src):

Evolution involves holding onto your winnings and investing them wisely. You don’t even have to know to how to hold onto your winnings. Evolution does it for you; it is the bank in which useful genetic mutations deposit themselves. There is a very slow rate of return, but it’s compounded. At the end of one eon, you get your bank statement and find your pittance has grown into an orang utan. At the end of the next eon, it has grown into Charles Darwin. Scientists, at least 99.875 percent of them, believe that in the long run only useful mutations deposit in this bank. Those mutations with no use, or a negative effect, squander their savings in a long-running bunko game, and die forgotten in the gutter.

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Happy Integral Day!

From Wikipedia:

According to Leibniz’s notebooks, a critical breakthrough occurred on 11 November 1675, when he employed integral calculus for the first time to find the area under the function y = x. He introduced several notations used to this day, for instance the integral sign ∫ representing an elongated S, from the Latin word summa and the d used for differentials, from the Latin word differentia.

int x, {rm d}x=frac{1}{2}x^2, for those not in the know.

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