# Scaling Laws and the Speed of Animals - Comments

• klotza
In summary: Hi everybody, this is the author. I hope enjoy what I have written, and let me know if anything needs clarification.Yes, the same is true for animals like spiders that don't use the same mechanisms to move as most animals... like if you lifted a spider the same as if you lifted a cat or dog.
klotza
Gold Member
klotza submitted a new PF Insights post

Scaling Laws and the Speed of Animals

Continue reading the Original PF Insights Post.

mfb, rikblok and Greg Bernhardt
Hi everybody, this is the author. I hope enjoy what I have written, and let me know if anything needs clarification.

Nice article @klotza!

Thanks for the insight! This is a very interesting topic for me, but I must say I was quite skeptical until I realized this is about TOP speed, more in line with physical limits. I'm fairly certain mobility is advantageous across most forms of beings so It certainly makes sense to me that evolution has pushed towards the limits across all scales.

Greg Bernhardt
This 1.4m "dominance zone" I guess I would call it keeps nagging at me. I'd be grateful for any elaborations you could offer there... as I'm not able to access the paper.

jerromyjon said:
This 1.4m "dominance zone" I guess I would call it keeps nagging at me. I'd be grateful for any elaborations you could offer there... as I'm not able to access the paper.
The additional constraint they apply on large animals, which oscillate their length L with some frequency f, is that there is a maximum angular acceleration that can be applied. The maximum torque depends on the muscle fibre force, the cross sectional area, and the length of the "lever arm," while the moment of inertia depends on the density, volume, and distribution of the shape. Comparing torque to moment of inertia they get a second-order ODE for the angle of the oscillating part (tail, leg, whatever) over time, which they integrate to find the time required to oscillate to a certain angle. This gives them an *absolute* maximum speed. So they compare their "maximum speed per body length" to the "absolute maximum speed" and solve for the body length at which the two are equal.

Hope that made sense. Picturing a cheetah, they are about 1.4 meters long, and plugging in their values for the "maximum speed" you actually get a fast jog, about 4 m/s. If you apply 10 bodies/second to 1.4 meters, you get 14 m/s=50 km/h which is about half a cheetah's top speed.

Greg Bernhardt
Congrats @klotza on this Insight making it to the first page of Reddit! Well deserved!

I couldn't help but be reminded of another similar study - the one finding (statistically) constant time of emptying the bladder across five orders of magnitude of animal mass (what the authors dubbed 'the law of urination'):
http://arxiv.org/abs/1310.3737

Greg Bernhardt
The second is that all motility is caused by the contraction of muscle proteins that have a similar structure across all life-forms.

Certainly all animals use the same types of muscle proteins for movement, but bacteria and other motile single-celled organisms use very different types of proteins for motion (for example, actinomyosin contraction in animals is driven by ATP hydrolysis whereas flagella are powered by the movement of protons across a membrane). Of course, the authors' estimate relies solely on considering the mechanical properties of proteins in general, so it seems to not be so dependent on how these proteins generate motion, just on the fact that proteins are generating force.

Greg Bernhardt
Bandersnatch said:
I couldn't help but be reminded of another similar study - the one finding (statistically) constant time of emptying the bladder across five orders of magnitude of animal mass (what the authors dubbed 'the law of urination'):
http://arxiv.org/abs/1310.3737

My father is a urologist and I went to a conference he organized and talked about this paper! The model they came up with doesn't quite make sense given their data. They read too much into their measured scaling, when I'm sure the error bars on it are huge to the point of making it insignificant. There was another paper using dimensional analysis to find a better model.

Greg Bernhardt and Bandersnatch
"metabolic rate per unit mass ... 2 kilowatts per kilogram of muscle. The paper doesn’t really explain where this comes from, citing controversy in the literature" Yes. This is controversial, but the argument is usually between a power law with an exponent of 2/3 or 3/4. A fascinating paper (The fourth dimension of life: fractal geometry and allometric scaling of organisms. West GB, Brown JH, Enquist BJ. Science. 1999 Jun 4;284(5420):1677-9) will definitely be interesting to readers of this post.

You said, "Above that it breaks down because the lifters generally get fatter without getting much more muscular."; do you have a source on that?

I'm slightly confused about the title of this... like I understand most parts but it just seems like you are saying all animals have the same rate at which they can move there own body length. Would just like clarifying on that

klotza said:
Hi everybody, this is the author. I hope enjoy what I have written, and let me know if anything needs clarification.
I'm wondering, is the same true for animals like spiders that don't use the same mechanisms to move as most animals do?

Some specific examples across species would help understanding this topic!

PiTHON said:
You said, "Above that it breaks down because the lifters generally get fatter without getting much more muscular."; do you have a source on that?
My source is mainly experience from when I used to be into powerlifting. However this paper also asserts the claim, with better data.

http://jap.physiology.org/content/89/3/1061.short

"Although it is possible that larger lifters activate less of their contractile filaments, the more likely explanation for their reduced strength per cross-sectional area is that they carry more of their body mass as noncontractile tissue. "

Last edited by a moderator:
Nice article! Adrian Bejan talks about this kind of thing in his book Design in Nature. He even extends the reasoning to the evolution of technology, like cars and planes. His idea is that the world is organized by flow. That structures that increase the flow of matter are selected.

This is related to the idea that energy from the sun flows through the surface of the Earth (coming in as yellowish light and leaving as infrared), and matter forms cycles (like the water cycle, the carbon cycle, etc.).

Tom Rayder said:
klotza said:
Hi everybody, this is the author. I hope enjoy what I have written, and let me know if anything needs clarification.
I'm wondering, is the same true for animals like spiders that don't use the same mechanisms to move as most animals do?
All I know is that ants and beetles appear to be on the image.

klotza said:
The additional constraint they apply on large animals, which oscillate their length L with some frequency f, is that there is a maximum angular acceleration that can be applied. The maximum torque depends on the muscle fibre force, the cross sectional area, and the length of the "lever arm," while the moment of inertia depends on the density, volume, and distribution of the shape. Comparing torque to moment of inertia they get a second-order ODE for the angle of the oscillating part (tail, leg, whatever) over time, which they integrate to find the time required to oscillate to a certain angle. This gives them an *absolute* maximum speed. So they compare their "maximum speed per body length" to the "absolute maximum speed" and solve for the body length at which the two are equal.

Hope that made sense. Picturing a cheetah, they are about 1.4 meters long, and plugging in their values for the "maximum speed" you actually get a fast jog, about 4 m/s. If you apply 10 bodies/second to 1.4 meters, you get 14 m/s=50 km/h which is about half a cheetah's top speed.

I wondered, isn't square-cube law only affects acceleration? Maintaining a speed at basic means acceleration has to overcome drag force of water or air. With bigger body, the drag force is higher, but not so high as the mass. With theese assumptions i don't find it strange that speed is rather independent from mass.

## 1. What are scaling laws and how do they relate to the speed of animals?

Scaling laws are mathematical relationships that describe how a particular physical or biological attribute changes as a function of body size. In the case of the speed of animals, scaling laws can help us understand how the size of an animal affects its maximum speed.

## 2. How do scientists measure the speed of animals?

There are several methods that scientists use to measure the speed of animals, including high-speed cameras, radar guns, GPS tracking, and even simple stopwatch measurements. Each method has its own advantages and limitations, so scientists often use a combination of techniques to get the most accurate results.

## 3. What factors influence the speed of animals?

The speed of an animal is influenced by a variety of factors, including body size, muscle strength, stride length, leg length, and overall body shape. Additionally, environmental factors such as terrain, temperature, and altitude can also impact an animal's speed.

## 4. How do scaling laws apply to different types of animals?

Scaling laws can apply to all types of animals, from insects to mammals. However, the specific relationships between body size and speed may vary depending on the animal's anatomy and evolutionary history. For example, larger animals tend to have longer legs, which allows them to take longer strides and potentially run faster.

## 5. Can scaling laws be used to predict the speed of extinct animals?

Yes, scaling laws can be used to estimate the speed of extinct animals based on their fossilized remains. By comparing the body size and proportions of extinct animals to living animals with similar characteristics, scientists can make educated guesses about their potential speed. However, these predictions may not always be accurate due to other factors such as diet and behavior that also play a role in an animal's speed.

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