# It's Body Length, Not Mass, That Lets the Cheetah Run So Fast

By Jennifer Ouellette on at

Cheetahs sprint at a blistering top speed of about 59 MPH, while the domestic cat runs at about 30 MPH, close to a blue whale’s 31 MPH, and a three-toed sloth runs less than one MPH. A new paper by a pair of French physicists concludes that it’s the body length, not the mass, of the animal that determines its top speed — at least for animals that can’t fly.

Using that metric, the physicists calculated that most animals should cover roughly ten times their body length per second at top speeds. And it applies across a very broad rang of size scales. That’s as true for a bacterium as it is for a cheetah, or an elephant, or a whale.

The analysis is essentially a scaling argument. Take, for instance, the square-cube law, first described by Galileo in 1638. Basically, it holds that as something gets bigger, its volume grows faster than its surface area. It’s the reason why it gets progressively harder to build ever-taller skyscrapers. If an animal grew twice as tall, wide and deep, for instance, you’d get an eightfold increase in volume and mass, but its surface area would only increase by a factor of four. As Alex Klotz writes at Physics Forums,

This means that as things get bigger, their own weight becomes more significant compared to their strength. (Ants can carry 50 times their own weight, squirrels can run up trees, and humans can do pull-ups.)

This also applies to an object’s terminal velocity, as J.B.S. Haldane described in his classic 1926 article for Harper’s Magazine:

You can drop a mouse down a thousand-yard mine shaft; and, on arriving at the bottom, it gets a slight shock and walks away, provided that the ground is fairly soft. A rat is killed, a man is broken, a horse splashes.

There may be a similar scaling law at work when it comes to determining an organism’s top speed.

For their analysis, Nicole Meyer-Vernet and Jean-Pierre Rospars looked at three features they believe all the organisms they included share: first, they all have roughly the density of water (since we’re all mostly water); second, all move by contracting muscles with similar protein structure; and finally, they assume a metabolic rate of 2 kilowatts for every kilogram of muscle. (Klotz notes that this is a controversial value.) That’s where they got their 10 body lengths per second number. They even suggest possible mechanical reasons why such a scaling law might be present.

There are a few caveats, of course. For instance, the medium through which an organism is moving sometimes matters, especially for very small things like bacteria. In fact, swimming through water for a bacterium is a bit like a human being swimming through molten asphalt. That may affect their calculations if Meyer-Vernet and Rospars didn’t factor that in when determining their bacterial speeds.

Also, this is really more of a rough approximation; physicists do love their spherical cows. Klotz notes that while the law seems to hold on average, there are a few outliers. However, “Even if you take the fastest bacteria and compare it to the slowest whale to try and shoot down their argument, you will have to explain a -.0.06 power law instead of a 0.0 power law,” he writes. “So their message stands: to zeroth-order, body traversal speed is independent of mass.”

So there.

References:

Haldane, J.B.S. “On Being the Right Size,” Harper’s Magazine, March 1926.

Meyer-Vernet, Nicole, and Rospars, Jean-Pierre. (2015) “How fast do living organisms move: Maximum speeds from bacteria to elephants and whales,” American Journal of Physics 83(8): 719.

[Via Mental Floss]

Top image: Stuart G. Porter/Shutterstock. Bottom image: Meyer-Vernet et al/American Journal of Physics.