Leg length and speed

A mouse and a horse both walk, trot and gallop. But the horse covers ground faster at each gait. Is there a simple rule that links an animal's size to the speed at which it changes gait? There is, and it comes from asking what limits a walking leg.

A walking leg is an upside-down pendulum. In a walk, the body vaults over a stiff, planted leg, like an upside-down pendulum. The hip moves along a circular arc whose radius is the leg length (about the hip height, h). A body moving in a circle needs a centripetal acceleration v² / h pointing down toward the hip. Gravity can provide at most g = 9.81 m/s². So in this simple model the walker stays on the ground only while v² / h ≤ g.

The ratio of the two is the Froude number:

Fr = v² / (g × h)

It has no units. In the model Fr cannot go above 1 while walking. Real animals change from walking to running well before that, around Fr ≈ 0.5.

A worked prediction. Solve Fr = 0.5 for the speed: v = √(0.5 × g × h).

• A person with a hip height of 0.9 m: v = √(0.5 × 9.81 × 0.9) = √4.41 ≈ 2.1 m/s (about 7.5 km/h). That is close to the speed at which people switch from a brisk walk to a jog.
• A long-legged animal with h = 1.8 m (assumed, about the height of a big horse's hip): v = √(0.5 × 9.81 × 1.8) = √8.83 ≈ 3.0 m/s.
• A small animal with h = 0.03 m (a mouse-sized animal, assumed): v = √(0.5 × 9.81 × 0.03) = √0.147 ≈ 0.38 m/s.

The speed grows with the square root of leg length, so doubling the leg length raises the transition speed by only a factor of √2 ≈ 1.41.

Dynamic similarity. Here is the big idea. Two animals of different size, moving at the same Froude number, are moving in a mechanically similar way: they use the same gait, and their stride length is about the same multiple of their leg length. Large and small animals are scaled versions of one another in motion. That is why a toy-sized horse gallops in much the same way as a real one, only more quickly. The same relation is used to estimate how fast an extinct animal moved from its footprints; the Dinosaur Dig lab shows how. Here we only need the principle.

Does this mean bigger is always faster? No. Top speed depends on more than gait changes: muscle power, how heavy the body is, stride mechanics and what the animal has evolved for. Across mammals, the fastest runners are medium-sized animals (tens of kilograms, like the cheetah). Very large animals are not faster. Studies of Asian elephants measured speeds of about 6 to 7 m/s (roughly 25 km/h). Even at that speed they kept at least one foot on the ground at all times, although their body motion resembled a run. Their huge bones and legs set the limit, as the last world will show.