EXAMPLE 10 Flight

How does weight affect flight? After all, ostriches can’t fly.

Wouldn’t it be nice to be able to fly? Well, you have to be able to stay up. The power necessary for sustained flight is proportional to the wing loading, which is the weight supported divided by the area of the wings. We know that in scaling up, weight grows with the cube of the length of the bird or plane, and wing area with the square of the length. So the wing loading is proportional to the length of the flying object.

For example, if a bird or plane is scaled up proportionally by a linear scaling factor of 4, it will weigh times as much but will have only times as much wing area. So each square foot of wing must support 4 times as much weight. once you’re up, you have to keep moving. To stay level, an airborne object moving forward must fly fast enough to maintain the lift on the wings. It turns out that the minimum necessary speed is proportional to the square root of the wing loading. Combining this fact with the first consideration, we conclude that the minimum speed goes up with the square root of the length. A bird scaled up by a factor of 4 must fly times as fast. (Hovering helicopters, hummingbirds, and insects maintain lift by moving their wings directly rather than relying on aerodynamic lift from forward motion.)

Take, for instance, a sparrow, whose minimum speed is about 20 miles per hour (mph). An ostrich is about 16 times as long as a sparrow, so the minimum speed for an ostrich would be mph. Have you seen any flying ostriches lately? Heavy birds have to fly fast or not at all!

of course, ostriches are not just scaled-up sparrows, nor are eagles (nor are airplanes). Larger flying birds have disproportionately larger wings than a sparrow so as to keep the wing loading down.

The largest animal ever to take to the air was Quetzalcoatlus northropi, a flying reptile that lived 65 million years ago. It had a wingspan of 36 ft, weighed about 100 lb, and was as tall as a giraffe. Recent research suggests that it might have been able to “fly” 10,000 miles or more, nonstop, by taking advantage of air currents.

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You have to stay up, you have to keep moving—and you have to get up there. Here, basic aerodynamics imposes further limits. Paleontologists originally thought that Q. northropi weighed 200 lb and had a 50-ft wingspan. Even though that works out to about the same wing loading as for 100 lb and a wingspan of 36 ft, other considerations from aerodynamics show that a reptile of the larger size couldn’t have gotten off the ground.