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Estimates of Speeds of dinosaurs

Gait describes the pattern and synchronony of footfalls made by an animal. Gait often changes with increasing speed to maximise energetic expenditure. The study of gait is well documented for many modern species including the horse. Gait can be studied in detail by using high speed photography. This is obviously not an option for examining gaits of non-avian dinosaurs.

Instead a predictive method must be used by examining trackways and skeletal reconstructions. However,
we can never know the **relative phase** (the timing of a footfall in relation to the other feet) or **duty factor** (the time the foot remains on the substrate).
This creates problems for interpretations of trackways, especially those of quadrupedal tetrapods. Gaits of extinct animals are
classified as **walk**, **trot** and **run**. Although quite arbitrary,
this classification does have advantages:

- "It can be applied to the evidence of trackways.
- It allows the gaits of dinosaurs to be compared to those of other animals.
- It provides some revealing insights into the locomotor abilities of dinosaurs." (Thulborn, 1989).

Gaits are therefore described by stride length, which can be measured directly from trackways, and by the relative
size of the animals.
This relationship between size and stride length is termed relative stride length "**l / h**, where
l represents length of stride and

Gaits are generally believed be l /*h* < 2.0 for walking, l /*h* = 2.0 for a trot and l /*h* > 2.9 for a run (Thulborn, 1984).
Hip height can be estimated using a number of methods; two are described below.

Geometric Method

Using the **geometric method** illustrated above, suceeding footprints ('X' and 'Y') are assumed to be the base of an isosceles triangle.
'H' represents hip height.

The **morphometric method** was suggested by Alexander (1976). Alexander assumed from estimates of foot joint features
that hip height was four times the length of the hind footprint
for dinosaurs, whether bipedal or quadrupedal. This can be regarded as a crude, but useful ratio.

Since it is likely that dinosaurs spent the majority of their time walking, rather than running we are more likely to find trackways of walking dinosaurs. When we do find a running dinosaur track, we cannot know whether it represents the top speed of that individual or species.

In 1976, R.McNeil Alexander published an exciting paper in *Nature*entitled "Estimates of speeds in dinosaurs" (Alexander, 1976).
He used a number of trackways for his calculations of stride length. Some represented bipedal dinosaurs, others
quadrupedal sauropods.

Alexander used mathematical principles to estimate the interaction
of proportion with speed and gait. Alexander wished to produce an equation that could be applied to animals of
many different
sizes. This meant he needed a "non-dimensional parameter to serve as a criterion for physical similarity"
(Alexander, 1976).
To this end, Alexander used the **Reynolds number**, which relates to the interaction of inertia and viscous
forces, and the **Froude number**.
The Froude number relates to interaction between inertia and gravity.
He used the following equation to estimate speed:

where u = velocity; *g*= the acceleration of free fall; d = stride length; *h*= hip height.

The fastest dinosaur that he recorded was moving at 3.6 ms-1. Alexander suggested this was more characteristic of the traditional lumbering image of dinosaurs.

Since the publication of this paper a number of other trackways have been examined. The equation developed by Alexander has
undergone various corrections and modifications by Alexander and other authors (Thulborn, 1981; Farlow, 1981; Russell & Beland, 1976).
Top speeds reported by authors have suggested some large theropods, such as *Tyrannosaurus*may have reached up to 10 to 15 ms^{-1}.

Alexander (1996) argued that based on the bone dimensions of *Tyrannosaurus*it is unlikely they could have travelled
at more than 8ms^{-1}.

**References**

Alexander, R.M. (1976) Estimates of speeds of dinosaurs. *Nature.*Vol.261, pp.129-130.

Alexander, R.M. (1996) Tyrannosaurus on the run. *Nature.*Vol.379, pp.121.

Avnimelech, M.A. (1966) Dinosaur tracks in the Judean Hills. *Proceedings of the Israel Academy of Sciences and Humanities,*Section of Sciences. Vol.1, pp.1-19.

Farlow, J.O. (1981) Estimates of dinosaur speeds from a new trackway site in Texas. *Nature.* Vol.294, pp.747-748.

Russell, D.A., Beland, P. (1976) Running Dinosaurs. *Nature.* Vol.264, pp.486.

Thulborn, R.A. (1981) Estimated spped of a giant bipedal dinosaur. *Nature.* Vol.292, pp.273-274.

Thulborn, R.A. (1984) Prefered gaits of bipdeal dinosaurs. *Alcheringa* Vol.8, pp.243-252.

Thulborn, R.A. (1989) The Gaits of dinosaurs. In *Dinosaur Tracks and Traces* (eds. D.D.Gillette, M.G.Lockley), pp.39-50. Cambridge University Press, Cambridge.

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