# Why does the gravity of Earth appear in this formula estimating speed?

I'm currently reading the wonderful book The Dinosaurs Rediscovered by Michael J. Benton. It contains the formula

$$v = 0.25 * g^{0.5} * SL^{1.67} * h^{-1.17}$$

where $$v$$ denotes velocity, $$SL$$ is stride length in $$m$$, $$h$$ is hip height in $$m$$ and $$g$$ is the gravity of earth. Not being a biologist (in fact not being really familiar with any natural science), I don't quite understand why gravity appears in this formula. I can't imagine this would hold in any meaningful way in circumstances where gravity where different from earth's, so why not simply express the term $$0.25 * g^{0.5}$$ as $$0.78 m^{0.5}/s$$, which I assume stems from some fitting of terms to observed speeds given stride length and hip height.

• I know nothing about this, but this formula appears to be the work of one Robert Alexander. If you can find some of his papers, he likely explains how this formula was developed. Alternatively, this website seems to go into the derivation. May 22 '20 at 13:30
• Reading a little bit about the derivation of the formula as well as the corresponding wikipedia article, I would assume the g stems from the origin of the equation in Froude's number. May 22 '20 at 18:46
• This question is about physics or engineering, not biology. The fact that the formula can be applied to biological movement is irrelevant. Jun 24 '20 at 18:20

Units

G is going to be different depending on what units you use, the original formula was derived using FEET not meters. As it stands the formula works both with metric and imperial units. With your change this would not be true. Paleontology is often uses both units so a formula that is not unit dependent is preferable.

This is because a leg can be modeled as an inverted pendulum (like a metronome). A pendulum is a function of gravity. This creates a simple relationship between hip height, stride length and speed. There is more to it but you would need to ask the physics community for details.

You can in facts build a mechanical robot that walks downward slopes by itself with nothing else than 2 inverted pendulums.

McGeer, T. (1990). Passive dynamic walking. I. J. Robotic Res., 9(2), 62-82. • Thanks, that answer also clarifies some things for me. I have to say, I didn't even know about the origin of this formula in physics when I asked the question. I just love dinosaurs (and mathematics) and this formula appeared to me quite beautiful and surprisingly universal. Jun 25 '20 at 6:31
• What McGeer work showed is that there is a speed at which walking is almost effortless because you just let your legs act as pendulums (which is function of g, SL and h). Alexander seems to have showed that there is indeed a fixed relationship between v, SL and h across animal species, as you would expect, and used it to predict the walking speed of dinosaurs. That's an interesting use of biomechanics.
– user37022
Jun 25 '20 at 18:41