Topics
Newtonian mechanics, Rigid body dynamics, Rotational dynamics, Geologic time scale, Students, Educational aids, Vertebrates

Motion is a fundamental part of every introductory physics course. Here we present the exciting example of calculating the walking speed of sauropod dinosaurs via the inverted pendulum model1 of walking by using kinematics, Newton’s second law of motion (in rotational form), and simple harmonic motion. This methodology is first applied to humans, horses of various sizes, and the elephant Elephas maximus to demonstrate its validity and to provide an estimate of the uncertainties associated with this simple model of walking. The students then find that the enormous sauropod dinosaurs (the largest animals that have ever lived on land) walked at speeds very similar to modern animals. A human would have been able to walk alongside even the largest sauropod with relative ease.

While walking, humans alternate the support of their body from one leg to the other, as shown in Fig. 1. In quadrupedal animals, each pair of legs shows the same behavior as in bipedal humans and supports a fraction of the animal’s total mass. The support leg is kept almost straight, causing the hips to follow an arc that is nearly circular. Because of the symmetry of the motion, each leg spends an equal amount of time supporting and swinging. Studies of metabolism as a function of walking speed in horses2 have shown that animals choose a walking speed that minimizes the metabolic cost of the motion. This corresponds to a leg swing frequency that closely matches the resonant frequency of the corresponding physical pendulum. Since the time of free swinging and supporting is the same for each leg, the period of a swinging physical pendulum yields the time of support by the leg.
Fig. 1.
A walking human with leg of length L showing the motion while the left leg is in contact with the ground. The arc shows the path of the hip joint. (Graphic made by Jakub Zalewski.)
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A walking human with leg of length L showing the motion while the left leg is in contact with the ground. The arc shows the path of the hip joint. (Graphic made by Jakub Zalewski.)

Fig. 1.
A walking human with leg of length L showing the motion while the left leg is in contact with the ground. The arc shows the path of the hip joint. (Graphic made by Jakub Zalewski.)
View largeDownload slide

A walking human with leg of length L showing the motion while the left leg is in contact with the ground. The arc shows the path of the hip joint. (Graphic made by Jakub Zalewski.)

Close modal
The swinging leg is modeled as a uniform circular cylinder of radius R, length L (from its pivot point to the other end), and mass M, rotating about a pivot point near one of its ends, as shown in Fig. 2. The moment of inertia I of a uniform cylinder rotating about one of its ends is I = (1/3)ML2. The small part of the leg bone above the center of the hip joint is neglected in this work. The swinging leg bends a small amount at the knee while swinging forward, thereby lowering its moment of inertia, as discussed later. The center of mass is assumed to be located at a distance of l = L/2 from the center of the hip joint. Newton’s second law in angular form is
(1)
where α is the angular acceleration (⁠=θ¨⁠), τ is the torque, and I is the moment of inertia about the rotational axis. The force of gravity (acting on the center of mass) creates a torque on the leg, accelerating it toward the vertical position.
Fig. 2.
A simple cylindrical model of a leg of mass M and length L. The point of rotation (the center of the hip joint) is shown by the solid black circle at the top and the center of mass of the leg is shown by the ×. The distance between the hip joint and the center of mass is l, d is the lever arm, and θ is the angle between the leg and the direction of the force of gravity.
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A simple cylindrical model of a leg of mass M and length L. The point of rotation (the center of the hip joint) is shown by the solid black circle at the top and the center of mass of the leg is shown by the ×. The distance between the hip joint and the center of mass is l, d is the lever arm, and θ is the angle between the leg and the direction of the force of gravity.

Fig. 2.
A simple cylindrical model of a leg of mass M and length L. The point of rotation (the center of the hip joint) is shown by the solid black circle at the top and the center of mass of the leg is shown by the ×. The distance between the hip joint and the center of mass is l, d is the lever arm, and θ is the angle between the leg and the direction of the force of gravity.
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A simple cylindrical model of a leg of mass M and length L. The point of rotation (the center of the hip joint) is shown by the solid black circle at the top and the center of mass of the leg is shown by the ×. The distance between the hip joint and the center of mass is l, d is the lever arm, and θ is the angle between the leg and the direction of the force of gravity.

Close modal
As shown in Fig. 2, the force of gravity creates a torque on the leg
(2)
where M is the mass of the leg, g is the acceleration due to gravity, d is the lever arm, l is the distance between the point of support and the center of mass, and θ is the angle between the axis of the leg and the direction of the force of gravity. The angle θ (measured in radians) is assumed to be small:
(3)
Combining these three equations, we obtain
(4)
Second-order differential equations are often beyond the purview of introductory courses. However, the solution to Eq. (4) is provided to the students, and they verify the validity of this solution:
(5)
where θmax is the maximum value of θ, ω is the angular frequency (which the students show is equal to Mgl/I⁠), and θ0 is the phase angle. Since the angular frequency is equal to 2π divided by the period T, we have
(6)

Note that the mass of the leg cancels out. Only the leg length (and the acceleration due to gravity) determines the period of oscillation in this simple model.

As noted earlier, each leg supports the body for half the time. This time of support is half of the period T of the physical pendulum. Figure 3 shows the translational motion of the body with x (= L sin θmax) being the horizontal distance that the body travels during one-half of the support phase of that leg. The time for that displacement is T/4.

Fig. 3.
The relationship between the horizontal displacement x, the length of the leg L, and the maximum angle θmax.
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The relationship between the horizontal displacement x, the length of the leg L, and the maximum angle θmax.

Fig. 3.
The relationship between the horizontal displacement x, the length of the leg L, and the maximum angle θmax.
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The relationship between the horizontal displacement x, the length of the leg L, and the maximum angle θmax.

Close modal
Consequently, the preferred walking speed of the animal is given by
(7)

Only the leg length and θmax are needed to calculate the preferred walking speed of an animal.

Equation (7) permits the calculation of the preferred walking speed of the largest terrestrial animals that ever lived: the sauropods. Three different genera are examined: Diplodocus, Brachiosaurus, and Argentinosaurus, shown in Fig. 4. First, Diplodocus was one of the longest sauropods (24–26 m in length) and lived in North America during the Late Jurassic. Its barrel-shaped trunk was supported by sturdy legs, the elongated tail was kept horizontal, and the neck was slightly S-shaped. Second, Brachiosaurus lived in the same place and time as Diplodocus, but its skeleton is different: the tail is shorter and directed downwards, the forelimbs are longer than the hindlimbs, and the neck is positioned vertically, similar to modern giraffes. Brachiosaurus could reach 20 m in length and ∼9–10 m high. Finally, Argentinosaurus is one of the largest terrestrial vertebrates known, its body length is estimated to have been 30–35 m. Fossils of this dinosaur are known from the Late Cretaceous deposits in South America. Argentinosaurus is a representative of the dominant sauropod group during the Cretaceous: Titanosauria, which are the largest dinosaurs known, and the last surviving group of long-necked sauropods, which went extinct with the end of Mesozoic.
Fig. 4.
Side view of the three sauropods considered in the study: Diplodocus (left), Brachiosaurus (middle), and Argentinosaurus (right). Art by Jakub Zalewski. Scale bar: 5 m.
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Side view of the three sauropods considered in the study: Diplodocus (left), Brachiosaurus (middle), and Argentinosaurus (right). Art by Jakub Zalewski. Scale bar: 5 m.

Fig. 4.
Side view of the three sauropods considered in the study: Diplodocus (left), Brachiosaurus (middle), and Argentinosaurus (right). Art by Jakub Zalewski. Scale bar: 5 m.
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Side view of the three sauropods considered in the study: Diplodocus (left), Brachiosaurus (middle), and Argentinosaurus (right). Art by Jakub Zalewski. Scale bar: 5 m.

Close modal

Models of sauropod dinosaurs are brought to the classroom to begin this activity. The students are asked to note all factors of the anatomy that they think would affect walking speed. The class then discusses these factors. Equation (7) is derived with the students showing that Eq. (5) solves Eq. (4) with the verification that the angular frequency ω is equal to Mgl/I⁠.

In order to validate this methodology, the students first measure the maximum angle θmax for humans3 and elephants4 (see the Appendix5). Such data apparently does not exist for horses. Consequently, the value of θmax for humans is used for horses also. The students are given the leg length L and preferred walking speeds of humans,6 three kinds of horses,2 and elephants4 (see Table I). They also use images from the pioneering work of Eadweard Muybridge3 to measure the distance between the hip joint and the point of contact on the ground at different points in the support phase of walking to ascertain how constant that distance remains. The fact that the hip joint is not marked on these images requires that the students think carefully about how they should make such measurements. The students must discuss their protocol for this measurement with the instructor before proceeding. Problems requiring student creativity are an important part of their education. The results of these measurements are given in the Appendix.5 

Table I.

The species, leg length, θmax, calculated walking speed, and observed walking speed. Note that the observed human walking speed of Ref. 6 was determined from a sample of four females and six males. The mean of our calculation was determined using the same relative numbers of females and males. Heights were not reported in Ref. 6. The leg lengths for a female and a male of average height (1.63 and 1.75 m, respectively) were measured.

Species Leg Length (m) θmax (°) Predicted Speed (m/s) Observed Speed (m/s)
Human (female)6   0.92  30  1.17   
Human (male)6   1.00  30  1.22   
Human mean      1.20  1.296  
Miniature horse2   0.73  30  1.04  1.192  
Arabian horse2   1.24  30  1.36  1.422  
Draft horse2   1.41  30  1.45  1.542  
Elephant (front)4   1.80  25  1.38   
Elephant (rear)4   1.52  25  1.27   
Elephant mean      1.33  1.264  
Brachiosaurus (front)7   3.96  18  1.5   
Brachiosaurus (rear)7   3.84  18  1.48   
Brachiosaurus mean      1.49   
Diplodocus (front)8   2.19  18  1.12   
Diplodocus (rear)8   3.21  18  1.35   
Diplodocus mean      1.24   
Argentinosaurus (rear)9   4.39  18  1.58   
Species Leg Length (m) θmax (°) Predicted Speed (m/s) Observed Speed (m/s)
Human (female)6   0.92  30  1.17   
Human (male)6   1.00  30  1.22   
Human mean      1.20  1.296  
Miniature horse2   0.73  30  1.04  1.192  
Arabian horse2   1.24  30  1.36  1.422  
Draft horse2   1.41  30  1.45  1.542  
Elephant (front)4   1.80  25  1.38   
Elephant (rear)4   1.52  25  1.27   
Elephant mean      1.33  1.264  
Brachiosaurus (front)7   3.96  18  1.5   
Brachiosaurus (rear)7   3.84  18  1.48   
Brachiosaurus mean      1.49   
Diplodocus (front)8   2.19  18  1.12   
Diplodocus (rear)8   3.21  18  1.35   
Diplodocus mean      1.24   
Argentinosaurus (rear)9   4.39  18  1.58   
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The leg is bent during the swing phase, lowering its moment of inertia. Thus, our model is expected to give a period T that is too long, resulting in a walking speed that is too low. Furthermore, for humans and horses, the parts of the limbs closer to the body are wider than the farther parts, meaning that our uniform cylinder approximation yields a moment of inertia that is too large. This also lowers the angular acceleration. However, humans have a relatively large foot at the end of their leg, which counteracts this effect. Elephants and sauropod dinosaurs have/had legs that are/were close to cylinders, and smaller systematic errors are expected for their walking speeds.

Elephants and the sauropods have front and rear legs with different lengths, which will result in different predicted walking speeds. The results for both pairs of legs are averaged to determine the predicted walking speed for that species.

As discussed earlier, the results for humans and horses are expected to be systematically small. The results in Table I show that the ratio of predicted to observed speed for humans and horses is 0.926 ± 0.036, which is less than 1 at the 2σ level. This implies that the systematic errors in our model are less than 10% in size, though it must be noted that there are only four data points, which makes statistical arguments somewhat tenuous. As expected, the results for the elephant (Elephas maximus) are better, increasing our confidence in the predictions of this methodology for the sauropod dinosaurs.

Students use scale diagrams of the skeletons of Brachiosaurus,7, Diplodocus,8 and Argentinosaurus9 in order to determine the length of their legs. The methodology for determining θmax for these extinct animals is described in the Appendix and yields θmax = 18°. This is consistent with the expectation that these extremely massive animals (up to ∼75,000 kg) kept their legs straighter than smaller animals (like the modern elephant) during locomotion. The results of the calculated preferred walking speeds for Brachiosaurus, Diplodocus, and Argentinosaurus are given in Table I. Surprisingly, the results show that these enormous dinosaurs walked at about the same speed as modern animals. The result that we walk at about the same speed as these sauropods is fascinating to the students.

This activity provides an exciting application of the results from kinematics, Newton’s second law of motion, and simple harmonic motion to predict the preferred walking speed of sauropod dinosaurs. Using these different concepts on one problem helps to emphasize the unity of science to our students.

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Scott Lee enjoys interesting examples of physics among the fossil record from dinosaurs. [email protected]

Justyna Slowiak is a vertebrate paleontologist working at the Institute of Paleobiology, Polish Academy of Sciences. She is interested in Mongolian dinosaurs, especially the large predatory dinosaur Tarbosaurus bataar.

© 2025 Author(s). Published under an exclusive license by American Association of Physics Teachers.
2025
Author(s)

Supplementary Material

Supplementary materials- zip file