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Scaling Argument on Jumping Heights of Creatures |
This proposition was first stated by D'Arcy Thompson. Thompson defined the height of a jump as being the amount by which the center of gravity of the creature is elevated. The analysis here is original but the idea is that of Thompson.
Let us assume that a creature has n legs of equal strength; its pedality* is n. If one of a creature's n legs exerts a force F over a distance k then the work work done by the creatures n legs is nFk and this is converted into kinetic energy, which is then transformed into a potential energy of mgh as the creature rises against gravity to a height h, where m is the mass of the creature and g is the acceleration due to gravity. Thus
If a muscle is attached to a leg bone with a lever arm L _{1} and exerts a force F _{1} then the product of force and lever arm is the same anywhere on the leg, including the foot; i.e., FL=F _{1}L _{1}. The distance k over which the force acts depends upon the lever arm L and a, the angle of flexure a of the leg; i.e.,
Thus,
Therefore
The force exerted by a muscle is proportional to its cross sectional area and the cross section area is proportional to the square of the size of the creature. Let S be some measure of the size of a creature such as height. Then
where b is a coefficient that depends upon what measure of size is used. The lever arm of the muscle is proportional to the size of the creature; i.e.
where a is the angle of flexure and c is a coefficient of proportionality. This means that
The body of the creature, as a result of the force nF_{1} acting over a distance k_{1} acquires a kinetic energy of
which at the top of the jump has been converted to potential energy. Thus
The mass of a creature is proportional to S^{3} and the mass density of p. There is a constant of proportionality v. Thus
Since
the height of the jump of a creature is given by
Thus, the jumping height h depends upon its pedality, the angle of flexure of the creature, the mass density of its body, the gravitational constant and three constants of proportionality that depend upon body shape, but it is independent of the scale S of the creature.
This is valid only for creatures which are exactly (mathematically) similar in shape. In actually big creatures cannot have a shape that is mathematically similar to small creatures. An elephant has to have thicker legs relative to its weight because the strength of legs are proportional to their cross section area, which is related to the square of size, whereas the weight is roughly proportional to the cube of size. Thus bigger creatures have to have relatively thicker limbs compared to smaller creatures. An analysis of this effect would lead to a second order level of analysis. For more on the topic of animal shape click here.
*pedality is used here to mean the number of legs (or feet) of a creature. There is another definition of pedality; i.e., to measure by pacing. Pedometry would be a better term for this latter concept.
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