When Pigs Fly

Imagine a world in which the threat of nuclear annihilation is a distant nightmare of the past and people wonder how their ancestors – that’s us – could have been so inhuman as to threaten genocide. That better world is so hard to imagine from our current vantage point that many people would say it will happen “when pigs fly.” When confronted with inconceivable ideas, a friend of mine used to have a similar expression: “That will happen when the Berlin Wall comes down.” Until shortly before the Wall was breached in 1989, that seemed as ridiculous as pigs flying, but after it occurred, my friend told me he was going to have to re-evaluate a number of seemingly impossible events. Maybe the inconceivable really is possible!

Something along those lines occurred to me today. Slashdot, the online news aggregator, had a story about a wind-powered cart that could travel downwind faster than the wind. Having sailed since childhood, that seemed as implausible as pigs flying. Once the cart was going faster than the wind, the sail would fill in the wrong direction (known as backwinding) and slow the vehicle, rather than push it forward. But, as I looked into the issue, I began to see how it might work.

So as not to bore non-geeks, I’ll defer the explanation of how it works to the appendix at the end of this post. Even without reading that, seeing is believing, so check out a video that shows the cart going faster than the wind. Disregard the first 20 seconds, which show another vehicle. Then note the streamers on the cart blowing forward, showing that the wind is coming from directly behind. The cart starts slowly, but three minutes into the video you’ll notice the streamers fall down – indicating that the cart is going the same speed as the wind –  and then stream backward as the cart exceeds the wind speed. The driver calls out the cart’s speed, showing the wind speed to be 15 mph when the streamers fall listless. The cart’s final speed, before running out of room, is 37.5 mph.

So imagine a world in which the threat of nuclear annihilation is a distant nightmare of the past. Perhaps that can happen, not when pigs fly, but after the Berlin Wall comes down and when a wind-powered vehicle traveling directly downwind goes faster than the wind.

ADDITIONAL READING

One of the handouts for my current seminar at Stanford on “Nuclear Weapons, Risk and Hope” has more on how the seemingly impossible can, in fact, occur. The other handouts for that seminar (Autumn 2010-11) have useful information on other aspects of Defusing the Nuclear Threat.

APPENDIX: How the wind-powered cart outruns the wind

The first key piece of data that helped me to understand how the cart could go faster than the wind, directly downwind, was to recognize that the propeller acts solely to produce thrust. It is not a windmill generating power. Rather, it is a propeller consuming power.

I obtained the second major insight from a web page authored by Mark Drela. Drela is an MIT Professor of Aeronautics & Astronautics, designed the human-powered aircraft Daedelus that set a world record by flying 72.4 miles from Crete to Santorini, and set a world speed record for human powered water craft in 1991. At first, Drela’s analysis appeared too complex for me to understand (yes, even Stanford professors can feel that way), but as I went through the equations step-by-step, I saw what Drela was saying. I also found a simpler way to express the basic idea and include that below. Once you have the basic idea, the full explanation is easy to get.

Let V denote the cart’s speed over the ground and W denote the wind speed. Because the wind is directly behind the cart, the apparent wind is V-W. For example, if the cart is going 20 mph and the wind is blowing at 10 mph from behind the cart, then the cart experiences an apparent wind of 10 mph coming from in front of the cart.

The trick to simplifying the analysis is to initially assume 100% efficiency in all elements of the system – no aerodynamic drag, no friction, and no losses in transmitting power from the wheels (where power is generated) to the propeller (where that power is consumed to provide thrust). If that simplified analysis indicates that net thrust can be generated at cart speeds that exceed the wind speed, then some of that net thrust can be used to overcome real-world losses.

In this simplified model, there is no drag due to friction or other losses so D, the drag on the wheels, is entirely due to the power being generated there to drive the propeller. Letting P denote the power generated at the wheels, and remembering that power is force times velocity,

P = DV.

Letting T denote the thrust generated by the propeller, and still assuming 100% efficiency, the power needed to drive the propeller is

P = T (V-W).

Note the difference in speeds: The wheels see the ground speed V, while the propeller sees the smaller apparent wind speed V-W. That difference turns out to be the key to how the system can work.

In this lossless model, the power generated at the wheels exactly equals the power consumed by the propeller, which is why I could use the same value of P for both. Equating the right hand sides of the above two equations, we find:

DV = T (V-W).

Again considering V = 20 mph and W = 10 mph, this equation is the same as

20D = 10T

which shows that the thrust will be twice the drag!

Because of the tailwind, (V-W) is always less than V and the thrust will always exceed the drag. In this simplified model, the cart therefore accelerates faster and faster. In the real world, inefficiencies will cause the cart to top out at some speed. But, if the system is well designed, this top speed will be significantly greater than the wind speed.

We saw that when the cart’s ground speed is twice the wind speed, the thrust was double the drag. A seemingly even more astounding result occurs when the cart is moving just as fast as the wind, so that V-W = 0. Then the thrust in the simplified model is infinite. That actually makes sense because an ideal propeller in still air needs no power to generate thrust. That’s because, neglecting aerodynamic losses (parasitic drag in aero-speak), all the power to drive the prop goes into accelerating the air from rest (which is how the air appears to the prop when V=W) to its final velocity when leaving the propeller. Since kinetic energy (proportional to the power required) grows like v-squared and momentum (proportional to thrust) grows like v, using a large prop allows a small value of v and significant thrust can be generated with very little power. That’s why the cart in the video has such a large propeller.