EXAMPLE 9 Jumps

A flea can jump as high as 50 cm (20 in.) vertically, many times its own height. Some people believe that if a flea were as large as a person, it could jump 1000 ft into the air. Is that so?

Imagining—against our earlier arguments—that there could be so large a flea, we can deduce its limits: A scaled-up flea could jump about the same height as a small flea. The strength of a muscle is proportional to its cross-sectional area (see Spotlight 18.3). A jump involves suddenly contracting the muscle through its length, so it turns out that the ability to jump is proportional to the volume of muscle. But the volume of the flea and the volume of its leg muscles would go up in proportion as we enlarged the flea to a geometrically similar giant flea.

Let’s say that a real flea’s leg muscles account for 1% of its body. If we scale the flea up to the size of a person (without any change in its form), the enlarged flea’s leg muscles will still make up 1% of its body. For either flea, each bit of muscle has the same power: In a jump, it propels 100 times its own weight, and it can do so to the same height. Both the weight of the flea and the power of its legs go up proportionately. In fact, the maximum heights that people, fleas, grasshoppers, and kangaroos can jump from standing are all within a factor of 3 of one another.