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Froude's Analysis of Speed in Relation to Scale

William Froude (1810-1879) was a British engineer who formulated a law concerning the characteristics of ship design and the
speeds which can be achieved. The resistance R in a fluif to the motion of a creature or struction is proportional to the
square of its speed relative to the fluid and the area of resistance; i.e.,

R = C_{D}AρV²

where C_{D} is a drag coefficient, ρ is the mass density of the fluid and A is the area subject to resistance.

This resistance has to be balanced by the work achieved by the ship or creature. The work W which a creature or a
ship can achieve is proportional to the scale l cubed. On the other hand the area subject to resistance A is proportional
to the square of the scale; i.e., A=sl². Thus

R = W
or
C_{D}ρ(sl²)V² = kl³
which implies
V² = (k/(sρC_{D}))l and hence
V = Kl^{½}

This is Froude's Law, that the speed attainable by a creature or a ship is proportional to the square root of its
scale.

The Froude number F is defined as
the speed squared divided by the product of the acceleration due to gravity g and the scale of the object; i.e,,

F = V²/(gl)

This may be interpreted as the ratio of the kinetic energy of the object to its potential energy.

One interesting insight of Froude is that the energy generated by a steam engine is proportional to the square of the
scale of its boilers because the amount of heat energy which can be transferred into the boiler is proportional to it surface
area rather than its volume

. If the work W that can be achieved by a ship or a creature is proportional to
the square of its scale then the speed which can be attained is independent of the scale.
A similar limiting factor for
creature is that the work they can achieve is proportional to the surface area of their lungs.

R = W
C_{D}ρ(sl²)V² = kl²
and hence
V² = (k/(sρC_{D}))