On Jumping Height of Organism: Borelli's Law

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On Jumping Height of Organism: Borelli's Law

May 29 2011 12:43:36 pm EST

Topics: Morphology, Science,

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Giovanni Alfonso Borelli (1608-1679) was a Renaissance Italian physiologist, physicist, and mathematician. He was a pioneer in the field of biomechanics, publishing De Motu Animalium I and De Motu Animalium II, which explored the mechanical nature of biological systems.

D’Arcy Wentworth Thompson frequently sites Borelli’s work, and explores an interesting phenomenon in which a similarly constructed organism, regardless of size, will jump to the same height. In other words, a flea and a grasshopper, despite their differences in size, end up jumping to the same height. Consider the following passage in On Growth and Form (36-37):

Such problems as that presented by the flea’s jumping powers, though essentially physiological in their nature, have their interest for us here: because a steady, progressive diminution of activity with increasing size would tend to set limits to the possible growth in magnitude of an animal just as surely as those factors which tend to break and crush the living fabric under its own weight. In the case of a leap, we have to do rather with a sudden impulse than with a continued strain, and this impulse should be measured in terms of the velocity imparted. The velocity is proportional to the impulse (x), and inversely proportional to the mass (M) moved V = x/M. But, according to what we still speak of as “Borelli’s law,” the impulse (i.e. the work of the impulse) is proportional to the volume of the muscle by which it is produced, that is to say (in similarly constructed animals) to the mass of the whole body; for the impulse is proportional on the one hand to the cross-section of the muscle, and on the other to the distance through which it contracts. It follows from this that the velocity is constant, whatever be the size of the animal.

Putting it still more simply, the work done in leaping is proportional to the mass and to the height to which it is raised, W α mH. But the muscular power available for this work is proportional to the mass of muscle, or (in similarly constructed animals) to the mass of the animal, W α m. It follows that H is, or tends to be, a constant. In other words, all animals, provided always that they are similarly fashioned, with their various levers in like proportion, ought to jump not to the same relative but to the same actual height. The grasshopper seems to be as well planned for jumping as the flea, and the actual heights to which they jump are much of a muchness; but the flea’s jump is about 200 times its own height, the grasshopper’s at most 20-30 times; and neither flea nor grasshopper is a better but rather a worse jumper than a horse or a man.

As a matter of fact, Borelli is careful to point out that in the act of leaping the impulse is not actually instantaneous, like the blow of a hammer, but takes some little time, during which the levers are being extended by which the animal is being propelled forwards ; and this interval of time will be longer in the case of the longer levers of the larger animal. To some extent, then, this principle acts as a corrective to the more general one, and tends to leave a certain balance of advantage in regard to leaping power on the side of the larger animal. But on the other hand, the question of strength of materials comes in once more, and the factors of stress and strain and bending moment make it more and more difficult for nature to endow the larger animal with the length of lever with which she has provided the grasshopper or the flea.

Dates:

2011

2010