Answer to the Question 12/99

CREATURES GREAT AND SMALL

The question was:

Why is it that creatures of such different sizes as a flea, a grasshopper, a human, and a lion can jump (i.e., raise their center of gravity) 1 meter in the air at most (give or take a factor of two)?

(3/00) The problem has been solved (6/12/99) by Ansgar Esztermann (e-mail ansgar@thphy.uni-duesseldorf.de), who remarked that got his idea about scaling laws from a book by Isaac Asimov. (Additional list of people who suggested correct, but slightly different solution can be found below.)


The solution:

The height of a jump is proportional to the ratio between the energy which can be supplied to the body E and the weight of the body W.

The weight W is proportional to L3, where L is the linear dimension of the solid. The energy E is proportional to the product of the force produced by the foot multiplied by the distance along which this force acts. The latter distance is proportional to the foot size and therefore proportional to L, while the force is proportional to the cross-section of the muscles and therefore is proportional to L2. Thus E is proportional to L3, and the ratio E/W is independent of the size of the body L !



Mark de Graaf from Technische Universiteit Eindhoven in Netherlands (e-mail M.J.d.Graaf@tue.nl), Dan Peleg (e-mail dan.peleg@analog.com), and Louis Zammit Mangion from the Department of Physics of University of Malta (e-mail lzamm@phys.um.edu.mt) also correctly solved the problem. However, they directly claimed that the work performed by a muscle is proportional to its volume (which is correct - since the force is proportional to cross-section and distance is proportional to the length of the muscle). However, it seems to us that the argument presented above is "cleaner".


Comment: This and similar problems are discussed in the book of J.M. Smith Mathematical Ideas in Biology (Cambridge University Press, 1968). See also the discussion by F. Felber from Jaycor, San Diego, California in a letter to Physics Today, p.11, March 1999. See also the lecture: "Scaling: why giants don't exist" by Michael Fowler from Physics Department of University of Virginia. (We thank L.Z. Mangion for bringing this reference to our attention.)
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