## Hf lf FIGURE 14.7 Plot of relative stride length versus dimensionless speed for different animals based on data derived from living cursorial vertebrates in terrestrial environments. (After Alexander, 1976, 1989.)

of 1.4 would have moved slower than our exemplified large theropod. This is related to gravitational force (Eqn 1.1, Chapter 1) providing a boost for momentum (Eqn 6.4, Chapter 6) because of the greater mass of a larger animal. As a result, a formula that takes into account the acceleration placed on to a body by gravity helps to equalize the speed of small and large animals alike through yet another dimensionless quantity, imaginatively called dimensionless speed:

where vd is dimensionless speed. Because v is in m/s, l, is in meters, and gravitational acceleration is in m/s2, all of the units cancel out and a dimensionless number is left. When the walking or running speeds of modern animals such as dogs, cats, rhinoceroses, elephants, ostriches, kangaroos, and humans are measured, these measurements can be used with the leg lengths of the respective animals in the calculation of dimensionless speeds. When plotted against relative stride lengths, the derived line is usable as a general model for the similarity of animal movement on land (Fig. 14.7). This plot allows comparison of relative stride lengths, calculated from fossil trackways, to dimensionless speeds, which in turn correspond to the speed of the tracemaker.

For example, using our Jurassic theropod again, its relative stride length of 1.4 corresponds to a dimensionless speed of 0.45. Using algebra to rearrange Equation 14.6, the estimated speed of the theropod was