Jurassic Park and its sequels are best thought of as monster movies. But they do make dinosaurs look and act like real animals – which, of course, they were. For more than 100 million years, various groups of dinosaur were the largest predators and herbivores on the planet. There were many smaller species too, though we only know about a fraction of them, since fossils of them are rare, and we’re aware of many only through fragments.

Scientists have been able to answer the biggest scientific question posed by Jurassic Park in one of its most tense chase scenes: could a Tyrannosaurus rex outrun a Jeep? (Answer: no.) Knowing the top speed of an apex predator is vital as it tells us what sorts of prey it could catch. To better understand these creatures, scientists also want to know if a Stegosaurus’ fearsome spike-wielding tail could be used as a weapon, and what damage it could do. Another question is how pterosaurs (cousins of the dinosaurs) could evolve to become the largest flying animals.

Answering all of these questions involves understanding what forces and torques these creatures’ skeletons could withstand. It also involves estimating the strength of their muscles and the mass of their flesh. While some bones, and muscle fragments, have survived the last 65 million years, unfortunately, flesh was not preserved. This means that while these questions come from palaeontology, they must be answered using physics, force diagrams and multi-body simulations.

“I have been described as a physicist in denial,” admits Michael Habib of the Natural History Museum of Los Angeles County, whose work on pterosaur flight is informed by his training in fluid mechanics.

Many palaeontologists also study living animals, since bones and muscles have a lot of common features across the animal kingdom. Birds are even technically dinosaurs, having descended from a sub-order called Therapoda. But analogy can only go so far. “What we have in a dinosaur is a cross between a mammal and a bird and a crocodile and a monitor lizard,” says Heinrich Mallison of the Museum für Naturkunde in Berlin. While it’s tempting to use modern animals as stand-ins, “there is no extant animal that is a perfect model”.

Evolutionary biomechanic John Hutchinson of the Royal Veterinary College, University of London, agrees. “I’d rather model a T. rex as a T. rex. That’s the benefit of computational models: you can model the physics of that animal with its own anatomy.”

All about mass

Consider the ostrich, the largest living dinosaur. It has hollow bones threaded with air sacs that are connected to its lungs, which help it to breathe and keep its skeleton light compared with a mammal of the same size. Many of their extinct dinosaur cousins also had hollow bones and air sacs, which helped them to grow huge. But how massive were dinosaurs when they were alive, and how was that mass distributed through the body?

Hutchinson explains that you have to be careful to draw appropriate analogies with today’s animals, when body mass dictates so much of what an animal’s biology can do. “Ostriches [which have a mass of about 100 kg] are orders of magnitude smaller than a T. rex,” he says. “If you’re looking at a 100 kg dinosaur and comparing it to a 100 kg living animal, however, it’s probably okay.”

Using data from living animals, palaeontologists can estimate how much mass was in each part of a dinosaur or pterosaur body. They can then construct a model to determine the location of its centre of mass, the forces and torques involved when the animal took a step, and the stresses on the wing bones of flying reptiles.

However, since muscles are never preserved enough to reconstruct the entire animal, dinosaur weight estimates often vary by a factor of two or more. Estimating how much muscle and the distribution of air sacs dinosaurs had depends on a lot of assumptions. For instance, a fully grown T. rex could have weighed 5 tonnes, or 11 tonnes or anything in-between. Since heftier animals move more slowly than the svelte ones featured in Jurassic Park, for example, the assumptions going into the physical models strongly affect the results.

Run, T. rex, run

We know that the T. rex stood and ran on two legs, holding its body nearly horizontally. To balance its huge head and anchor its leg muscles, it had a huge tail. That means it had to support all its weight balanced on a single leg during each step while it walked or ran. The faster it ran, the more stress that single leg would have to support.

To model the mechanics of a running T. rex, Hutchinson and his colleagues examined the animal frozen in mid-stance, with its body supported on one leg (2002 Nature 415 1018). They considered the vertical forces only, since the horizontal forces on the leg nearly balance out. Using Newton’s first law, they calculated the minimum required muscle mass in both legs as a percentage of total body mass: 43%. “We had to estimate the moment arms of different muscle groups acting against gravity around each joint of the limb,” Hutchinson says (figure 1).

The ground-reaction force on the animal while running is higher than the animal’s weight while standing still. Hutchinson explains, “When a typical human sprints, you would exert a vertical force of two-and-a-half times body weight or more at a 15 miles-per-hour [24 km/h] sprint. A really good sprinter might have to sustain four or five times their body weight [to achieve] an even faster speed.” In other words, the faster a T. rex ran, the more load each of its legs would have to bear, and that limits its top speed. “Whether you’re a fish or a salamander or an ostrich or an elephant, it’s pretty constant how much force you can get out of the muscles that act to resist gravity,” Hutchinson says. “You get about 300 kN per square metre at best out of muscle when it’s contracting isometrically: not lengthening or shortening.”

Since muscles do contract while running, the 300 kN estimate is a generous one, providing an upper limit on how hard a T. rex could push itself. Hutchinson and his colleagues found that – based on their assumptions such as treating horizontal forces as negligible – T. rex could reach a top speed of around 40 km/h. With a Jeep’s top speed being far above that, it turned out that the chase scene in Jurassic Park got it right: the Jeep’s passengers got safely away.

Sting in the tail

Like elephants, some dinosaurs couldn’t run. Instead, they compensated by being heavily armed and armoured. For instance, Stegosaurus and its relatives had heavy tails tipped with long horn-covered spikes. One skeleton of the predatory dinosaur Allosaurus has a badly healed injury from a Stegosaurus tail spike lodged in its bone. But to know how effective the Stegosaurus tails could have been, palaeontologists need to use physics.

Mallison, of the Museum für Naturkunde, researched Kentrosaurus, a much smaller Stegosaurus relative from Tanzania that lived around 153 million years ago. The museum has a largely complete Kentrosaurus skeleton, which it reassembled in 2005 to reflect modern research on posture. Previously it had been assembled with sprawling limbs like a monitor lizard, but a correct construction showed it had upright, column-like hind legs and bandy front legs.

During reassembly, the museum laser-scanned each bone to create a 3D digital file. Mallison used these to build a complete model of the animal in a computer-assisted drawing (CAD) multi-body dynamics program, which is also used in sports medicine for gymnasts (2010 Swiss J. Geosci. 103 211). “[I]t’s kind of cumbersome to do that with real physical objects because they fall down and break,” he says. “If you do it in the computer, hey! no problem.” Another advantage to computer modelling is the ability to adjust parameters such as muscle mass, which – like the T. rex leg – tells us how strongly Kentrosaurus could whip its tail and how much inertia the whipping had (figure 2).

Mallison treated each tail bone as a separate mass body and strung them together to get the model of the entire tail, much the way physicists construct multi-body mechanical models for various systems. Using this model, he estimated that each joint between tail bones could only move about 4°, but together the joints allowed semi-circular swings. “The weight at the end of the tail was 8 kg, so the impulse that they can transfer is really huge,” he says. Even without using the tail spikes, they could crush the rib cage of a predator. Though it was smaller than Stegosaurus, “Kentrosaurus was a baseball hitter from hell.”

Staying aloft

Flight is another challenge for palaeontologists. Flying dinosaurs – birds, that is – can be understood through plenty of living examples (though even our understanding of the transition of modern-day animals to flight is contentious). Pterosaurs aren’t very bird- or bat-like, though, which means direct analogies with extant animals are of limited use to researchers who are trying to deduce how these animals achieved lift-off and maintained flight.

“[Pterosaurs] have a muscular skin-covered wing that is stretched between its giant fourth finger and the body,” says Habib of the Natural History Museum of Los Angeles County. “That kind of wing can generate a very high coefficient of lift.”

Coefficient of lift is a function of wing size and flight speed, neither of which we know precisely. However, Habib says, “the basic aerodynamic tricks required for staying aloft are not all that different from any other animal”. In other words: a wing is a wing to a certain extent, whether it’s on a bird or a hang glider. “A high lift coefficient helps them support their weight even at low speeds for a big flying animal,” says Habib. “The problem is getting going in the first place.”

Pterosaurs launched into the air differently than birds or bats, because on the ground they walked on their wings and hind legs. Habib and his colleagues think that means they also jumped into the air using all four limbs, “which gives you quite a bit more jumping power than if you just used your hind limbs”.

Pterosaurs were a diverse group, with some species living far inland and others hunting fish in the ocean. The different lifestyles were reflected in their flight abilities. The wing membranes aren’t preserved in many specimens, so we do not have precise wing shapes, but researchers have determined some had extremely long skinny wings, “like a super albatross”, as Habib puts it. These animals, like albatrosses, soared over open water. Others had shorter, wider wings, which made them able to manoeuvre at high speed.

Unlike birds, pterosaurs probably flew with their wings swept forward, a design only seen in a few rare aircraft. That’s because pterosaur heads were huge, in some cases several times the lengths of their bodies. The giant Quetzalcoatlus, for example, stood as tall as a giraffe with a wingspan of 11 m, but its body was only about 75 cm long. Its head, in contrast, was about 3 m long. As a result, “the centre of mass is a little forward of the shoulder,” says Habib.

Another type of pterosaur, the Pteranodon, had a long snout and a long crest on the back of its skull, which is impressive but looks tame when compared with many of its relatives. Some had gigantic axe-like crests or protrusions like old-fashioned TV antennas. However, these appendages had only a minor effect on flight, based on physical models. “A lot of these crests are really big, but they’re really flat side-to-side,” says Habib. “They add a bunch of drag mostly, but not a whole lot else, oddly enough.” They probably couldn’t turn their heads in flight, but neither do large modern birds, and it’s easy to see why. “These are animals that are travelling at speeds that are lethal upon contact with a surface, so it’s best to look where you’re going.”

Success story

When looking at prehistoric animal motion, it’s fascinating how little we know. For instance, how did they lie down to sleep? “We know pretty much nothing about how [living] animals lie down and stand up,” says Hutchinson. For that reason, he and his colleagues have been collecting data on living animals, which they haven’t published yet. Until they do, we won’t know much about how T. rex got down – except very slowly and carefully.

Despite the stereotypes of dinosaurs as failures, they and pterosaurs were remarkably successful animals, dominating the planet for more than 100 million years. No evolutionary failure could endure that long, and the way they lived had to be part of the secret to their success. The glory of physical models of extinct animals is that, with refinement and testing, researchers can fill in gaps in our knowledge to recreate some of the biggest and most interesting creatures that ever lived.