Appendix to Chapter 5 from The Physics of Life by Adrian Bejan

Animal weight moves rhythmically, in such a way that it achieves a balance
between two expenditures of useful energy: lifting weight on the vertical, and
overcoming drag while progressing on the horizontal.

If the animal is modeled as a body with a single length scale( L ), then its
mass is of order of  . During each cycle the body performs work in
the vertical direction (  ) and in the horizontal direction (  ). The vertical
work is necessary to lift the body to a height of order L .

   (1)

The horizontal work is necessary so that the body penetrates the surrounding medium.

 ≈  ,      (2)

where  is the drag force,  is the density of the medium and  is
the distance traveled during one cycle. The drag coefficient  is essentially
constant, and is comparable with 1 . The work spent per distance traveled is

   (3)

The timescale of the cycle is the time of free fall from a height of order L , namely  . The horizontal travel during the cycle is , and Eq. (3)

  (4)

This sum is minimal when V reaches the speed

   (5)

Equation (5) is valid as a scale, i.e. in an order of magnitude sense. It is
obtained most directly by using the method of intersecting the asymptotes,
which means to set the two terms of Eq. (4) equal to each other. It can
also be obtained by differentiatig the right side of Eq. (4) with respect to V ,
setting the resulting expression equal to zero, solving for V and neglecting
factors of order 1 , in accord with the method of scale analysis. The frequency
of body movement is  , or

         (6)

The body force is determined by the work done vertically,  ,
which is the same as the potential energy at the end of the lifting motion,
 , therefore

     F ≈ Mg   (7)

The work per distance traveled is obtained by substituting Eq. (5) and
 into Eq. (4),

     (8)
The modifying factor  plays a role similar to the friction coefficient
μ during sliding or rolling, and depends on the medium. For flying, the air
density  is roughly equal to  , and the factor  is close to 1/10 .
For swimming, the medium density (water) is essentially the same as the body
density, and the factor is one. For running,  is between 1/10
and 1 , and depends on the running surface and  air drag. Running through
snow, mud and sand is represented by a  value close to 1 . Running
fast on a dry surface is represented more closely by a factor  that is
similar to flying.

In summary,  is of order 1 , and can be omitted in Eqs. (5), (6)
and (8). Important is that  differentiates between locomotion media
in a certain, unmistakable direction:

(a) If M is fixed, the speeds (5) increase in the direction sea → land → air.

(b) The work requirement (8) decreases in the same direction.

The history of the spreading of animal movement on earth points in the
same time direction: both time sequences, (a) and (b), are in accord with
the constructal law. The animal speeds collected over the 
(cf. figure 4.6) confirm the differentiating effect that the surrounding medium
had on the spreading of animal movement.

All these discoveries of design in animal movement can be expressed in
terms of the body length scale

  (9)

instead of the body mass M , or body weight M g . For example, by eliminating M between Eqs. (5) and (9) we obtain

     (10)

A larger animal or athlete ( M ) means a taller body ( L ), and a taller body means a faster body. Because the leading factor  is of order 1 for
swimming and running, the speed-height formula becomes

     (11)          

This is the same as Galileo Galilei's formula for the speed of an object
that hits the ground after falling from the height  . Stones dropped from
the Tower of Pisa hit the ground faster than stones dropped from my hand.
Equation (11) is the same as the formula for the speed of a water wave of
length scale (height) L . Bigger waves move horizontally faster. Compare the
speed of the waves in your teacup with the speed of a tsunami.