MASS AND LOCOMOTION SPEED ESTIMATES OF DINOSAURS

Part 2: Locomotion Speed Estimates

        Animals move at different speeds by using different sequences of leg movements called gaits. There are two general classes of gaits; walking gaits, where each foot is in contact with the ground for more than half of the stride, and running gaits where each foot is in contact with the ground for less than half of the stride. Where each foot is in contact with the ground for exactly half of the stride is the transitional phase between walking and running gaits.

        Changes in gait are directly related to stride length (= distance from the toe of one foot to the toe of the same foot the next time it contacts the ground). During a slow walk in humans the stride is relatively short. As speed increases the stride length also increases. However, there is an absolute limit to the walking stride length, since the legs are of finite length. This limit is four times the leg length. However, the practical limit of walking stride length is considerably less than this because to reach the absolute limit a human would have to do a gymnastic split with each step!

        To increase speed above the practical walking limit it is necessary to literally leap between steps. These leaps occur when both feet are in the air at once; the rear foot having just pushed off of the ground and the front foot reaching out for the next contact. Consequently, running stride length can be viewed as the maximum walking stride length plus the additional distance from leaps. As running speed increases there is an increase in stride length. This increase is entirely due to increases in the length of leaps. Eventually, muscular and metabolic constraints limit the maximum length of leaps and, ultimately, running speed.

        Unfortunately, there is a further complication to interpreting walking and running speeds. Suppose we were observing locomotion in two humans of identical body proportions, but one a giant and the other a midget. As they walk progressively faster their stride lengths increase. With larger stride lengths there is also a progressive increase in the angle between the legs. At a slow walk there is a small angle between the legs, while at a fast run there is a much larger angle. At some particular speed the angle between the legs is 90o. But because the midget has shorter legs than the giant, his speed at a 90o leg angle will be significantly less than for the giant. If the giant is 2 m tall and the midget is 1 m tall, then the giant will have twice the stride length with a 90o leg angle as the midget.

        To assess locomotion speeds we need a size-independent measure of stride length. A simple, but realistic measure would be to divide the actual stride length by the leg length:

Consequently, when both the giant and the midget are moving with a maximum leg angle of 90o they both have the same relative stride length. This is referred to as dynamic similarity of locomotion because even though the giant is moving faster than the midget the two are using exactly the same leg movements.

        If we know the leg length of an animal and then measure the stride length we can easily calculate the relative stride length. The problem is that relative stride length is not a direct measure of speed. To determine the speed that corresponds to a given relative stride length we need a scaling factor.

        Engineers have used such scaling factors for many years. For example, assume an engineer wants to build a new hull design for a racing sailboat that produces a smaller bow wake. Since the bow wake is produced by lifting water against the pull of gravity this wastes energy that could be used to move the boat forward. Any reduction in the bow wake would translate into higher sailing speed. The engineer could draw the design, build it and then see how well it worked. This is a regrettably expensive way to find out if the design is a good one. It is easier and much less expensive to build a scale model. But to obtain valid results the model must respond like the full size boat. When the full size ship moves at its optimal speed it produces a bow wake of a certain size. But what is the optimal speed for the model, so that the boat and model are exhibiting dynamic similarity? This requires a scaling factor. The most widely used scaling factor in systems where work is performed against gravity is Froude number:

Froude number can be easily applied to the problems of scaling associated with walking and running. Graphing relative stride length of animals versus their Froude number produces a curved relationship. However, the square root of Froude number, called dimensionless speed, produces a straight-line relationship with relative stride length and is therefore somewhat easier to use.  Numerous studies have examined the relationship between relative stride length and dimensionless speed. Based on these studies, the dimensionless speed can be calculated from relative stride length:
Once dimensionless speed has been calculated, it can be easily (at long last!) converted to absolute speed:
To summarize the process, we must first get a size-independent measure of movement (= relative stride length), then use a scaling factor (dimensionless speed) to obtain an estimate of speed.

        Examine the two trackways available in the laboratory. Both of these trackways belong to the ichnogenus Eubrontes, and are believed to have been produced by a single species of theropod dinosaur (redrawn from M. Lockley. 1991. Tracking dinosaurs. Cambridge University Press, Cambridge, UK, 238 pp.). From these scale drawings you can obtain measurements (in meters) of foot size (measured from toe to heel) and stride length (measured from the tip of one toe to the tip of the same toe the next time it is in contact with the ground). In dinosaurs leg length is about four times foot length. Use leg length and stride length to estimate the speed these two dinosaurs were moving. To convert the speeds of these dinosaurs from m/sec to miles per hour, multiple by 2.25.

Questions

In terrestrial vertebrates the critical dimensionless speed is 0.7. Below this value vertebrates walk, above this speed they run. Were these dinosaurs walking or running?

Is the higher speed you calculated a reasonable approximation of the highest speed for the theropod that produced it?

Are there any inherent problems in reconstructing dinosaur speeds with this methodology?