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No. 1498:
Thoughts on Jumping

Today, can you jump as high as a flea? The University of Houston's College of Engineering presents this series about the machines that make our civilization run, and the people whose ingenuity created them.

Here's one for you: How high can a horse jump? Or a flea? How high can you jump? Variations among individual fleas, horses or people are large, but healthy members of any jumping species raise their centers of gravity about one meter when they leap.

The center of gravity of a high-jumper who clears a two-meter bar was already a meter or so high as he approached the bar. His body flattens out horizontally as it passes over the bar. So he jumps his center of gravity only an additional meter or so.

A horse's center of gravity is even higher than a human's. The horse also stretches its legs out horizontally to get over the bar, and he likewise rises only three or four feet. My dog gets her front paws higher than my head when she tries to chase a squirrel up a tree. And her center of gravity likewise rises a meter or so.

So what about that flea? It jumps hundreds of times its own height, but that also gets it about a yard off the ground. It seems spooky -- all God's jumping creatures reaching the same height. How can that be? Franklin Felber explains it in a letter to the magazine Physics Today. It comes down to modeling laws.

Here's how it works: Our jumping capability should increase with the length of our leg, and with its strength. Length is proportional to linear size, and strength is proportional to the leg's cross-sectional area. But area increases as the square of size. That means jumping ability should go up as the cube of linear size.

If that were the whole story, we'd be able to jump far higher than any flea. But we also need to consider the weight opposing the height of a jump. It too increases with the size cubed. (Go to the bookshelf and pick up a large book and one half its size. The large book will weigh eight times as much as the small one.)

Since both the opposition to jumping and jumping ability vary with the cube of size, that means size cancels out. The height any creature can jump is independent of its size. What a flea can do, so can a grasshopper, a monkey, or a lion.

There'll always be champions among cats, dogs and humans. There'll also be individuals who can come nowhere near a meter. The one-meter figure may be approximate, but it signals our kinship with other species. For we're all bound by the same rules of mathematics and of nature. Those rules keep us balanced in odd ways.

That message is pervasive and humbling. On the larger canvas it reminds us that size, strength, even brains, fail to erase a primary equity that exists among creatures. We try to make leaping into a metaphor that denies equity. Shakespeare cries,

... here upon this bank and shoal of time,
We'd jump the life to come.

We want our leaps to take us so much further than they really do. I think it helps keep things in proportion to remember that you and I can really leap no higher than a lowly flea.

I'm John Lienhard, at the University of Houston, where we're interested in the way inventive minds work.

(Theme music)

Felber, Franklin, Letter to the editor. Physics Today, March 1999, pg. 11.

Jumping kangaroo