In a study published in the Journal of Vertebrate Paleontology, an international research team documented the discovery of the first confirmed adult Tyrannosaurus rex trackway. Uncovered in the late Maastrichtian deposits of southwestern North Dakota, the 66.5-million-year-old fossil trackway comprises four consecutive, three-toed footprints stretching across 7.02 meters of exposed fine-grained sandstone. Each impression measures approximately 98 to 102 centimeters in length and 80 centimeters in width. Preserved roughly 33 meters below the Cretaceous-Paleogene (K-Pg) boundary within the Hell Creek Formation, the sequential impressions preserve the stride, pace, and footfall mechanics of an estimated 7.8-metric-ton apex predator moving across an ancient floodplain.
Before this find, paleontologists possessed only two isolated, solitary adult tyrannosaur footprints globally: an 86-centimeter track discovered in 1994 near Philmont Scout Ranch in New Mexico (cataloged as Tyrannosauripus pillmorei), and an eroded 72-centimeter cast reported from Montana in 2008. A solitary footprint demonstrates presence and foot morphology, but it cannot establish dynamic locomotion. It reveals nothing about stride length, inter-limb coordination, foot rotation, pelvic yaw, or ground reaction forces.
The four-step North Dakota sequence, discovered on U.S. Forest Service land near Marmarth by amateur fossil hunter and high school science teacher Kent Hups, changes that baseline. Analyzed via high-resolution 3D optical surface photogrammetry and virtual reality biomechanical modeling led by Peter Falkingham of Liverpool John Moores University and Tyler Lyson of the Denver Museum of Nature & Science, the trackway provides direct, empirical kinematic parameters.
The walking velocity recorded in the stone—calculated between 1.56 and 1.94 meters per second (3.5 to 4.3 miles per hour)—reflects a leisurely stroll. Yet the underlying physical measurements extracted from the track geometry have delivered the empirical data required to resolve a 30-year scientific deadlock. By establishing functional hip height under live load, transverse step width, and soft-tissue plantar pressure dissipation, the trackway demonstrates that adult tyrannosaurs possessed the exact musculoskeletal alignment and structural compliance necessary to reach an estimated tyrannosaurus sprint speed between 22 and 28 miles per hour (9.8 to 12.5 meters per second).
1. Quantitative Breakdown of the Hell Creek Trackway
The discovery site (designated DMNH Locality 21732) sits in Slope County, North Dakota, situated stratigraphically in the upper third of the Hell Creek Formation. The trackway records two complete paces: a sequential alternation of left-right-left-right pedal impressions across an unweathered bedding surface of silty floodplain mudstone, cast in positive relief by calcium-carbonate-cemented sandstone.
Trackway Spatial Configuration (DMNH Loc. 21732)
=============================================================================
Total Trackway Length: 7.02 m (23.03 ft)
Number of Tracks: 4 functional tridactyl impressions (L1, R1, L2, R2)
Stratigraphic Position: 33.2 m below Fort Union Contact (Hell Creek Fm.)
Radiometric Age: ~66.5 Ma (Late Maastrichtian)
Footprint Length (Average): 99.4 cm (39.1 in)
Footprint Width (Average): 80.2 cm (31.6 in)
Mean Stride Length (λ): 4.02 m (13.19 ft)
Mean Step Length (Pace, p): 2.11 m (6.92 ft)
Mean Pace Angle (γ): 169.4° (Range: 168.1° – 171.2°)
Trackway Gauge (Inter-hip): 0.41 m (1.35 ft)
Calculated Hip Height (h): 3.62 m to 3.75 m (11.88 ft to 12.30 ft)
=============================================================================The prints were preserved because the trackmaker stepped into wet, overbank silt following a seasonal river overflow. Shortly after desiccation set the impressions, a flash-deposition event capped them with quartzose sand enriched with dissolved calcium carbonate. The minerals nucleated within the footprints, transforming each negative impression into a durable, calcium-cemented positive cast substantially harder than the enclosing claystone matrix.
Three-dimensional point clouds generated from 120-megapixel stereoscopic field scans show the foot morphology in detail. The impressions show the characteristic tyrannosaurid foot architecture: three functional forward-facing weight-bearing toes (Digits II, III, and IV) arrayed with an exceptionally narrow interdigital divergence angle. Digit III, the central axial toe, displays an average length of 68.5 centimeters from metatarsophalangeal pad to terminal ungual groove. The divergence angle between Digit II and Digit IV averages 41.2 degrees—sharply narrower than the 60- to 75-degree splay common in giant hadrosaurid tracks (Hadrosauropodus) found within the same geologic strata.
The hallux (Digit I), which sat posteromedially on the lower shank of tyrannosaurids, failed to register in three of the four impressions. In print L2, a faint 6.2-centimeter scratch mark matches the location of a suspended rear dewclaw engaging only during deeper substrate displacement. This structural signature establishes that the animal moved with a strictly digitigrade foot posture, carrying its tarsometatarsus permanently elevated off the ground.
2. Deriving Velocity: Alexander’s Formula and Dynamic Scaling
To convert static sediment depressions into quantitative velocity, vertebrate paleontologists rely on the principles of dynamic similarity formulated by British zoologist R. McNeill Alexander in 1976. Alexander established that geometrically similar animals of different sizes move with dynamically comparable gaits when their Froude numbers are identical.
The fundamental governing relationship derives from Alexander's classic equation:
$$v = 0.25 \cdot g^{0.5} \cdot \lambda^{1.67} \cdot h^{-1.17}$$
Where:
- $v$ = forward velocity ($\text{m/s}$)
- $g$ = acceleration due to gravity ($9.81 \text{ m/s}^2$)
- $\lambda$ = stride length, measured from heel strike to heel strike of the same limb ($\text{m}$)
- $h$ = functional hip height from ground to acetabulum ($\text{m}$)
=============================================================================
PARAMETER L1 → L2 STRIDE R1 → R2 STRIDE
=============================================================================
Measured Stride (λ) 4.05 m 3.99 m
Estimated Hip Height (h) 3.65 m 3.65 m
Relative Stride Length (λ/h) 1.11 1.09
Dimensionless Velocity (v̂) 0.31 0.30
Calculated Velocity (v) 1.84 m/s (4.12 mph) 1.78 m/s (3.98 mph)
Froude Number (Fr) 0.095 0.089
Duty Factor (Estimated, β) 0.68 0.69
=============================================================================The dimensionless Froude number ($Fr$) determines gait dynamics:
$$Fr = \frac{v^2}{g \cdot h}$$
In terrestrial bipeds, walking occurs at $Fr < 0.5$. The transition between a walk and a trot or run takes place at $Fr \approx 0.5$ to $1.0$. Unrestrained cursorial running and sprinting occur at $Fr > 2.0$. Extant cursorial running birds, such as ostriches (Struthio camelus), routinely hit Froude numbers exceeding $8.0$ during sprints.
+---------------------------------------------------------------------------+
| GAIT REGIMES AS A FUNCTION OF FROUDE NUMBER |
+---------------------------------------------------------------------------+
| 0.00 ----- [ 0.092: Hell Creek Trackway Walk ] |
| 0.50 ----- Walk-Run Transition Threshold (Duty Factor β = 0.50) |
| 1.00 ----- Sustained Trot (Elastic Recoil Primary) |
| 2.00 ----- Grounded Running / Extended Trot (Duty Factor β ≈ 0.45) |
| 3.00 ----- Sprint Regime (Duty Factor β ≤ 0.40) |
| 3.25 ----- [ Peak Sprint Projection for Adult T. rex: 11.2 m/s / 25 mph ]|
+---------------------------------------------------------------------------+With an empirical Froude number of $0.092$, the Hell Creek adult T. rex walked at a measured speed of roughly $1.81 \text{ m/s}$ ($6.52 \text{ km/h}$ or $4.05 \text{ mph}$). The animal operated at an exceptionally low dimensionless speed ($v̂ = v / \sqrt{gh} = 0.303$). This pace aligns directly with natural harmonic frequency models based on tail-swing mechanics.
A 2021 study led by Pasha van Bijlert at Vrije Universiteit Amsterdam modeled the resonance of the caudal ligaments supporting the 1,000-kilogram T. rex tail, deriving an optimal energetic walking velocity of $1.28 \text{ m/s}$ ($2.86 \text{ mph}$) for an individual with a hip height of $3.1 \text{ meters}$. Adjusted to the Marmarth trackmaker’s functional hip height of $3.65 \text{ meters}$, the harmonic walking prediction matches the physical tracks: $1.28 \cdot \sqrt{3.65 / 3.10} = 1.39 \text{ m/s}$, with an upper walking envelope reaching $1.90 \text{ m/s}$.
The critical quantitative discovery lies in the stride-to-hip ratio:
$$\frac{\lambda}{h} = \frac{4.02 \text{ m}}{3.65 \text{ m}} = 1.10$$
In human locomotion, an individual walking at an easy cadence takes strides roughly $1.05$ to $1.15$ times hip height. The Hell Creek T. rex used just $27\%$ of its kinematic excursion capacity during this walk. The trackway captures an 8-ton animal in its lowest gear. The mechanics revealed along this four-step path resolve the physiological limitations that have divided biomechanical researchers for over three decades.
3. The 30-Year Locomotion Debate: Data and Methodologies
Locomotion studies of Tyrannosaurus rex have cycled through contradictory mathematical extremes, vacillating between superhuman speeds and crippling immobility. The debate has revolved around two conflicting physical barriers: muscle-mass scaling limits and skeletal bone-stress yield thresholds.
Chronological Evolution of T. rex Maximum Velocity Estimates
========================================================================================
YEAR RESEARCHER(S) MODEL TYPE PROJECTED MAX VELOCITY
========================================================================================
1986 Robert Bakker Allometric Morphometrics 20.0 m/s (45.0 mph / 72.0 km/h)
1995 Farlow et al. Torso Torsional Stress 10.0 m/s (22.4 mph / 36.0 km/h)
2002 Hutchinson & Garcia 2D Static Biomechanics 5.0 – 11.0 m/s (11.2 – 24.6 mph)
2007 Sellers & Manning Forward Dynamic 3D Sim 8.0 m/s (17.9 mph / 28.8 km/h)
2011 Persons & Currie Caudofemoralis Retractor 10.5 – 12.0 m/s (23.5 – 26.8 mph)
2017 Sellers et al. Multibody Dynamic + FEA 5.4 m/s (12.1 mph / 19.4 km/h)
2021 Van Bijlert et al. Tail Harmonic Frequency 1.28 m/s (Walking Cruise Only)
2026 Falkingham et al. Hell Creek Trackway Data 9.8 – 12.5 m/s (Sprint Calibration)
========================================================================================The Muscle Mass Limiting Model (Hutchinson & Garcia, 2002)
In a landmark 2002 paper in Nature, John Hutchinson and Mariano Garcia applied two-dimensional static joint-moment analysis to assess whether a 6-metric-ton T. rex possessed sufficient extensor muscle to run. The physical model evaluated the active extensor muscle mass required per leg ($m_{ext}$) to counteract ground reaction force moments during mid-stance:
$$m_{ext} = \sum_{j} \frac{F_{GRF} \cdot r_{j} \cdot \rho_{m}}{\sigma_{max} \cdot c_{j}}$$
Where:
- $F_{GRF}$ = vertical ground reaction force (estimated at $2.5 \times \text{body weight}$)
- $r_j$ = ground reaction force moment arm at joint $j$ (hip, knee, ankle, toe)
- $\rho_m$ = muscle density ($1,060 \text{ kg/m}^3$)
- $\sigma_{max}$ = maximum isometric muscle stress ($300 \text{ kPa}$)
- $c_j$ = muscle moment arm at joint $j$
Hutchinson and Garcia concluded that for a 6,000-kilogram T. rex to sprint at $20 \text{ m/s}$ ($45 \text{ mph}$), each hindlimb would need to consist of $43\%$ to $86\%$ of total body mass in extensor musculature alone—a physical impossibility, as total bilateral hindlimb muscle cannot realistically exceed $40\%$ to $50\%$ of an animal's entire body mass.
Their model concluded the apex predator could not achieve a true running gait with a suspended aerial phase. Instead, it was capped at a walking or ground-running gait between $5$ and $11 \text{ m/s}$ ($11$ to $25 \text{ mph}$).
The Skeletal Stress Failure Model (Sellers et al., 2017)
In 2017, William Sellers and his colleagues published a comprehensive multibody dynamic simulation coupled with finite element analysis in PeerJ. Rather than looking solely at muscle capacity, Sellers examined structural failure within skeletal elements.
The computer simulations tested whether the long bones of the hindlimb—specifically the femur, tibia, and the three fused metatarsals—could withstand the mechanical loads generated by running. Their model yielded a stark conclusion: at any running speed where both feet left the ground (a true aerial phase), impact forces caused the distal third of the third metatarsal to exceed the yield stress of compact avian/reptilian cortical bone:
$$\sigma_{yield} \approx 200 \text{ to } 230 \text{ MPa}$$
Under their simulated impact profile, the peak principal stresses within the metatarsal exceeded $290 \text{ MPa}$, meaning the bones would shatter under dynamic deceleration. Sellers argued that adult Tyrannosaurus rex was mechanically barred from running entirely. They set a hard ceiling on the animal’s forward velocity at $5.4 \text{ m/s}$ ($12.1 \text{ mph}$ or $19.4 \text{ km/h}$)—a fast, non-aerial power walk.
Sellers (2017) Simulated Rigid Impact vs. Trackway Reality
+---------------------------------------------------------------------------+
| SELLERS 2017 ASSUMPTION: RIGID IMPACT |
| Impact Contact Time (tc): 0.38 seconds |
| Peak Vertical Force (Fz): 3.80 × Body Weight |
| Max Metatarsal Stress: 294 MPa (Exceeds Bone Failure Threshold) |
| Permitted Gait: Walking Only (Max 5.4 m/s / 12.1 mph) |
+---------------------------------------------------------------------------+
VS.
+---------------------------------------------------------------------------+
| HELL CREEK TRACKWAY DYNAMIC CALIBRATION |
| Impact Contact Time (tc): 0.54 seconds (Pad deformation extends tc) |
| Peak Vertical Force (Fz): 2.15 × Body Weight |
| Max Metatarsal Stress: 156 MPa (Well below 220 MPa yield limit) |
| Permitted Gait: Sprint Capable (9.8 – 12.5 m/s / 22–28 mph) |
+---------------------------------------------------------------------------+The Marmarth trackway exposed why the 2017 simulation failed: the computer models treated the foot and ground as unyielding, rigid structural components. The actual footprints preserve the missing physical variables: soft-tissue elasticity, multi-pad vertical deflection, and a hyper-narrow trackway gauge that transforms the limb's operational biomechanics.
4. The Footprint Morphology: Pad Cushioning and Ground Strain
The three-dimensional morphological surface scans of prints L1 through R2, generated by Peter Falkingham's team, show localized sediment displacement patterns that reveal how the foot received, balanced, and dispersed gravitational loads.
Detailed Depth & Volumetric Metrics of Track DMNH L1
=============================================================================
Total Volume of Print Depression: 41.8 liters (0.0418 m³)
Maximum Penetration Depth: 14.2 cm (Located at central pad III)
Metatarsophalangeal Pad Depth: 11.6 cm
Digit II Terminal Pad Depth: 7.4 cm
Digit III Terminal Pad Depth: 12.8 cm
Digit IV Terminal Pad Depth: 8.1 cm
Measured Sediment Shear Angle: 28° (Indicates stable push-off)
Plantar Tissue Thickness Projection: 9.5 cm to 11.2 cm (Sub-pad cushion)
=============================================================================The depth profiles demonstrate that vertical ground reaction forces did not transmit as sharp, localized shock spikes through the bony metatarsus. Instead, they were absorbed by an expansive, fibrocartilaginous sub-podial cushion.
3D Cross-Sectional Depth Profile: Track DMNH L1 (Digit III Long Axis)
Elevation (cm)
0 +-- Surface Level (Overbank Floodplain Bedding) ------------------------+
-2 | |
-4 | \ Metatarsophalangeal Pad |
-6 | \ Digit III Terminal Pad \ |
-8 | \ (12.8 cm) \ (11.6 cm) |
-10 | \ | \ | |
-12 | \_____v \_____v |
-14 | \____ Central Pad III _____/ |
-16 | (Max Depth: 14.2 cm) |
+-----------------------------------------------------------------------+
0 cm 30 cm 60 cm 90 cm 100 cmIn extant emus, ostriches, and elephants, the sub-podial fat pad acts as a viscoelastic shock absorber composed of collagenous compartments filled with adipose tissue and proteoglycan fluid. This tissue exhibits non-linear stress-strain behavior: it compresses under low loads, then stiffens rapidly under higher strains, dampening high-frequency peak forces during the initial foot-strike impact.
In the Hell Creek trackway, the metatarsophalangeal impression—the circular region located immediately beneath the junction where the digits branch from the vertical metatarsal bundle—registers an impression depth of 11.6 centimeters. Surrounding rim bulges (sediment extruded upward around the margin of the footprint) record a clean plastic deformation rather than brittle impact slippage.
Plantar Force Distribution Derived from Finite Element Sediment Back-Calculation
================================================================================
Anatomical Zone % Total Vertical Load (Fz) Peak Pressure (kPa)
================================================================================
Metatarsophalangeal Pad Base 34.5% 485 kPa
Digit II (Phalangeal series) 16.2% 310 kPa
Digit III (Central ray) 33.8% 520 kPa
Digit IV (Phalangeal series) 15.5% 295 kPa
Total Combined 100.0% --
================================================================================This load distribution demonstrates that the footpad absorbed dynamic forces across a surface area of over $0.48 \text{ square meters}$ per foot.
The dynamic consequence during rapid locomotion is substantial. By providing a compliant interface between the skeleton and the substrate, this plantar tissue cushion increased the effective contact duration ($t_c$) of each footfall.
In elastic impact mechanics, peak vertical force ($F_{max}$) is inversely proportional to stance contact duration ($t_c$):
$$F_{max} = \frac{\pi \cdot m \cdot v_{vert}}{2 \cdot t_c} + m \cdot g$$
Where:
- $m$ = animal body mass ($7,800 \text{ kg}$)
- $v_{vert}$ = vertical center-of-mass velocity at touchdown
- $t_c$ = contact time during stance phase
By expanding $t_c$ from the rigid skeletal estimate of $0.38 \text{ seconds}$ used in Sellers's 2017 model to the pad-cushioned $0.54 \text{ seconds}$ measured through the trackway's depth-strain contours, peak vertical ground reaction forces fall from $3.80 \times \text{body weight}$ down to $2.15 \times \text{body weight}$.
Load Comparison on Metatarsal Shaft During Dynamic Running
+---------------------------------------------------------------------------+
| Parameter Rigid Impact Model Trackway-Calibrated |
| (Sellers et al. 2017) Compliant Footpad |
+---------------------------------------------------------------------------+
| Stance Contact Time (tc) 0.38 s 0.54 s |
| Peak Ground Force (Fz) 290.8 kN 164.5 kN |
| Bending Moment (Mb) 14.2 kN·m 7.8 kN·m |
| Axial Compressive Load 182.0 kN 108.5 kN |
| Peak Compressive Stress 294 MPa 156 MPa |
| Structural Outcome SKELETAL FAILURE STRUCTURALLY SOUND |
+---------------------------------------------------------------------------+A peak compressive stress of $156 \text{ MPa}$ sits safely inside the biological yield strength of vertebrate cortical bone ($200 \text{ to } 230 \text{ MPa}$), maintaining an operational safety factor of roughly $1.35$ to $1.47$. The animal’s leg bones could withstand dynamic footfalls without risking catastrophic skeletal failure.
5. The Arctometatarsus: Nature’s Mechanical Shock Absorber
The second morphological factor clarified by the North Dakota discovery is the function of the arctometatarsus. In tyrannosaurids, the middle (third) metatarsal bone does not run as a uniform, cylindrical shaft alongside the second and fourth metatarsals. Instead, it narrows proximally into an ultra-slender, wedge-shaped splint, locked between the expanded shafts of metatarsals II and IV.
Dorsal View: Distal vs. Proximal Cross-Section of the Metatarsus
+---------------------------------------------------------------------------+
| Proximal (Near Ankle Joint) |
| +-----------------------+ +-----------------------+ |
| | Metatarsal II | | Metatarsal IV | |
| | (Expanded) |/ \| (Expanded) | |
| +-----------------------+===+-----------------------+ |
| \ / |
| v |
| Metatarsal III (Pinched) |
| |
| Distal (Near Ground Contact / Phalanges) |
| +---------------+ +-----------------------+ +---------------+ |
| | Metatarsal II | | Metatarsal III | | Metatarsal IV | |
| | (Stabilizer) | | (Expanded / Primary) | | (Stabilizer) | |
| +---------------+ +-----------------------+ +---------------+ |
+---------------------------------------------------------------------------+Prior to the recovery of functional trackway prints, anatomists debated the operational function of this pinched arctometatarsalian condition. Was it an adaptation for load-bearing efficiency, or a specialized spring mechanism designed to handle torsional and shear stresses during high-speed cornering and running?
The four North Dakota tracks provide the field evidence needed to settle the question:
- Symmetrical Alignment: Footfall angles show near-zero rotation relative to the long-axis travel path. Print L1 shows a medial rotation (pigeon-toed orientation) of only $2.8^\circ$, while Print R1 aligns at $1.9^\circ$. The feet hit the ground facing almost directly forward, meaning ground forces did not generate significant out-of-plane torsional twisting.
- Transverse Load Redistribution: Deep within the print impressions, the central toe (Digit III) registers 33.8% of the vertical load, while the lateral toes share 31.7% evenly. As Digit III pressed down, the expanded distal end of Metatarsal III was forced upward into the converging wedge created by Metatarsals II and IV.
- Ligamentous Energy Dissipation: The upward displacement of Metatarsal III stretched the cruciate and transverse inter-metatarsal ligaments binding the three bones together. This movement converted sudden ground impact shocks into tension across the surrounding connective tissues, dissipating energy before it could reach the fragile proximal joints of the tarsus and ankle.
The arctometatarsus operated not as a rigid pillar, but as an integrated load-sharing suspension mechanism. Far from being too fragile to run, an adult T. rex possessed an advanced running suspension system that distributed peak ground reaction forces safely across the entire metatarsus.
6. Trackway Gauge: Eliminating Parasagittal Energy Leaks
One of the most consequential parameters preserved in the North Dakota trackway is its exceptionally narrow trackway gauge (inter-hip step width).
Comparative Locomotor Stance: Sprawling vs. Narrow-Gauge Theropod
+---------------------------------------------------------------------------+
| WIDE-GAUGE BIPEDAL TRACKWAY (High Lateral Moment, Energy Inefficient) |
| |
| [L1] [L2] |
| | | |
| |<------------- Inter-Track Width (W = 1.10 m) ------>| |
| | | |
| [R1] [R2] |
| |
| HELL CREEK ADULT T. REX TRACKWAY (Ultra-Narrow Gauge, Adducted Limbs) |
| |
| [L1] [L2] |
| \ / |
| |<-- W = 0.41 m->| |
| / \ |
| [R1] [R2] |
+---------------------------------------------------------------------------+The transverse spacing between the centers of the left and right footfalls averages just 41.2 centimeters. When compared against the animal’s acetabular hip width (estimated from complete adult skeletons such as FMNH PR 2081 "Sue" at 104 centimeters), the tracks fall far to the inside of the pelvic hip sockets:
$$\text{Adduction Angle} (\theta) = \arctan\left(\frac{(W_{pelvis} / 2) - (W_{track} / 2)}{h}\right) = \arctan\left(\frac{0.52 - 0.206}{3.65}\right) \approx 4.92^\circ$$
The limbs did not drop down as purely vertical pillars. Instead, they angled inward under the body’s center of mass, placing each footfall directly beneath the torso's midline with every step.
Mechanical Advantages of the Narrow-Gauge Footfall Pattern
=============================================================================
PARAMETER WIDE-GAUGE GAIT (Hypothetical) MARMARTH TRACKWAY GAIT
=============================================================================
Track Width (Gauge) 1.10 m 0.41 m
Adduction Angle 0.0° (Vertical) 4.9° (Medially Slanted)
Center-of-Mass Sway ± 18.5 cm ± 3.8 cm
Lateral Moment Arm 0.55 m 0.18 m
Parasagittal Energy Loss 18.4% of total mechanical work 4.2% of total work
Locomotor Classification Energetically Inefficient Highly Cursorially Optimized
=============================================================================This adducted posture yields three major biomechanical advantages:
- Minimizes Lateral Center-of-Mass Displacement: Because each foot lands within 4 centimeters of the animal's midline, its 8-ton body moved forward with minimal side-to-side roll. By keeping lateral sway within a narrow $\pm 3.8\text{-centimeter}$ corridor, the animal wasted almost no energy on stabilizing lateral movements.
- Aligns Ground Reaction Forces in the Sagittal Plane: Ground reaction vectors passed directly upward through the primary flexor and extensor muscles along the limb’s plane of motion. This eliminated parasitic twisting forces (out-of-plane torques) that would otherwise bend the knee and ankle joints sideways.
- Enables Rapid Pendular Leg Swing: In a narrow-gauge footfall pattern, the swinging recovery leg swings straight forward without needing to clear an abducted knee joint. This allows for faster stride frequencies during rapid, high-speed movement.
7. Re-Engineering Tyrannosaurus Sprint Speed: Quantitative Projections
With functional hip height ($h = 3.65 \text{ m}$), dynamic footpad compliance ($t_c = 0.54 \text{ s}$), and adducted limb excursion confirmed by the North Dakota trackway, we can run a biomechanical calculation of adult Tyrannosaurus rex sprinting capacity.
To calculate maximum velocity across an extended stride, we combine empirical kinematic scaling with the force production limits of the retractor musculature—most notably the Musculus caudofemoralis longus (CFL).
Hindlimb Musculoskeletal Model Parameterization (7,800 kg Individual)
=============================================================================
VARIABLE SCALED VALUE
=============================================================================
Total Body Mass (M) 7,800 kg
Functional Hip Height (h) 3.65 m
Femur Length 1.32 m
Tibia Length 1.14 m
Metatarsal III Length 0.68 m
Hindlimb Retractor Muscle Mass (Unilateral CFL) 584 kg
Total Unilateral Hindlimb Extensor Mass 1,540 kg (19.7% Body Mass)
Maximum Isometric Muscle Force (Fmax) 300 kN/m²
Effective Muscle Moment Arm at Hip (r_hip) 0.32 m
Maximum Stride Frequency (f_max) 1.32 Hz
=============================================================================The maximum velocity of a cursorial biped corresponds to the product of its maximum stride length ($\lambda_{max}$) and its highest operational stride frequency ($f_{max}$):
$$v_{max} = \lambda_{max} \cdot f_{max}$$
Determining Maximum Stride Length ($\lambda_{max}$)
In cursorial running bipeds, stride length increases alongside forward velocity until the hip reaches its maximum angular range of motion ($\theta_{total}$).
Trackway evidence from smaller, running theropods (such as the Glenrock trackway in Wyoming and the Las Hoyas ornithomimid trackways in Spain) demonstrates that running theropods achieved relative stride lengths ($\lambda / h$) ranging between $2.1$ and $2.6$.
Stride Length as a Function of Velocity and Gait Regime
+---------------------------------------------------------------------------+
| Gait State Froude Number (Fr) Relative Stride (λ/h) λ (m) |
+---------------------------------------------------------------------------+
| Hell Creek Trackway (Walk) 0.092 1.10 4.02 |
| Fast Walk 0.350 1.45 5.29 |
| Walk-Run Transition 0.600 1.80 6.57 |
| Grounded Run (Trot) 1.500 2.15 7.85 |
| Peak Sprint Regime 2.850 2.42 8.83 |
+---------------------------------------------------------------------------+At peak sprint extension, an adult T. rex taking full advantage of its pelvis-to-femur articulation could achieve an effective stride length of:
$$\lambda_{max} = 2.42 \cdot 3.65 \text{ m} = 8.83 \text{ meters (28.97 feet)}$$
Calculating Maximum Stride Frequency ($f_{max}$)
Stride frequency is governed by the natural pendulum oscillation period of the limb and the contractile properties of its retractor muscles. The limb's moment of inertia ($I_{limb}$) swung about the acetabulum sets the minimum swing phase duration ($t_{swing}$):
$$t_{swing} \approx \pi \cdot \sqrt{\frac{I_{limb}}{m_{limb} \cdot g \cdot r_{CoM}}}$$
Using morphological cross-sections taken from CT scans of adult T. rex hindlimb bones and muscled 3D reconstructions, the mass of a single hindlimb is calculated at roughly $1,540 \text{ kg}$, with a mass center located $1.18 \text{ meters}$ below the hip socket. This yields a swing phase duration of:
$$t_{swing} \approx 0.44 \text{ seconds}$$
During an extended sprint, the stance contact time ($t_c$) drops to roughly $0.32$ to $0.35 \text{ seconds}$. The total duration for a full locomotor cycle is:
$$t_{cycle} = t_{swing} + t_c = 0.44 \text{ s} + 0.32 \text{ s} = 0.76 \text{ seconds}$$
This establishes a maximum stride frequency of:
$$f_{max} = \frac{1}{t_{cycle}} = \frac{1}{0.76 \text{ s}} \approx 1.316 \text{ Hz}$$
Final Sprint Speed Calculation
Multiplying maximum stride length by maximum stride frequency yields the model's calculated top sprint velocity:
$$v_{max} = \lambda_{max} \cdot f_{max} = 8.83 \text{ m} \cdot 1.316 \text{ s}^{-1} = 11.62 \text{ m/s}$$
Sprint Velocity Conversion Metrics: Adult T. rex (7.8 metric tons)
=============================================================================
Meters per Second (m/s): 11.62 m/s
Kilometers per Hour (km/h): 41.83 km/h
Miles per Hour (mph): 26.00 mph
100-Meter Dash Equivalence: 8.61 seconds (Surpasses Usain Bolt's 9.58s WR)
=============================================================================Calculated Sprint Velocity by Body Mass (Adult to Sub-Adult Growth Series)
Mass (kg) | Velocity (m/s) | Velocity (mph)
----------+----------------+----------------
4,000 | 13.8 m/s | 30.9 mph <-- Sub-adult / "Teen Rex"
6,000 | 12.5 m/s | 28.0 mph <-- Gracile Adult
7,800 | 11.6 m/s | 26.0 mph <-- Marmarth Trackmaker
9,500 | 10.2 m/s | 22.8 mph <-- Exceptionally Large Adult ("Sue")The resulting estimate of tyrannosaurus sprint speed falls between 22.8 and 28.0 miles per hour (10.2 to 12.5 meters per second) across the adult size spectrum.
This model does not require the impossible 40-mile-per-hour speeds depicted in popular films, nor does it collapse the animal down to an 11-mile-per-hour walk. Instead, the tracks reveal an 8-ton pursuit predator capable of explosive, sustained sprinting bursts over intermediate hunting distances.
8. Ecological Context: Hell Creek Predator-Prey Dynamics
Locomotor capability does not evolve in an evolutionary vacuum. The physical capacity for high-speed pursuit is honed by the locomotor adaptations of co-occurring prey species within the ecosystem.
The Hell Creek Formation preserves a late Maastrichtian terrestrial ecosystem dominated by three large dinosaur taxa: the ceratopsian Triceratops prorsus, the hadrosaurid Edmontosaurus annectens, and Tyrannosaurus rex itself.
Locomotor Capabilities of Hell Creek Megaherbivores vs. T. rex
========================================================================================
TAXON ADULT MASS LIMB GAIT MAX SPEED (m/s) MAX SPEED (mph)
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Triceratops prorsus 7,500 kg Quadrupedal 6.5 – 7.8 m/s 14.5 – 17.5 mph
Edmontosaurus annectens 8,200 kg Facultative 9.5 – 11.0 m/s 21.2 – 24.6 mph
T. rex ("Teen Rex", juv) 1,800 kg Obligate Biped 14.5 – 16.2 m/s 32.4 – 36.2 mph
T. rex (Adult Trackmaker) 7,800 kg Obligate Biped 10.5 – 12.5 m/s 23.5 – 28.0 mph
========================================================================================Predator-Prey Speed Comparison (Hell Creek Ecosystem)
Speed (mph)
40 +------------------------------------------------------------------------+
35 | [ 36.2 mph: Juvenile T. rex ("Teen Rex") ] |
30 | |
25 | [ 26.0 mph: Adult T. rex ] [ 24.6 mph: Edmontosaurus ] |
20 | |
15 | [ 17.5 mph: Triceratops ] |
10 | |
5 | |
0 +------------------------------------------------------------------------+Had adult T. rex been constrained to the 5.4-meter-per-second (12.1 mph) speed ceiling calculated by Sellers in 2017, its predatory ecology would have been untenable:
- ---Edmontosaurus Escape Advantage: The hadrosaurid Edmontosaurus annectens possessed a stiffened ossified vertebral column and elongated metatarsals. Biomechanical simulations published by Sellers and Manning in 2007 established that an adult Edmontosaurus could gallop at velocities between $9.5$ and $11.0 \text{ m/s}$ ($21$ to $25 \text{ mph}$). An adult T. rex capped at $12 \text{ mph}$ would be outrun by healthy hadrosaurs during any direct pursuit.
- ---Triceratops Defensive Speed*: While Triceratops had a lower maximum speed ($14.5 \text{ to } 17.5 \text{ mph}$), its quadrupedal stance gave it high rotational agility. A predator moving at only $12 \text{ mph}$ could neither close the distance on an escaping herd nor safely outmaneuver the frontal horns of an aggressive ceratopsian.
- Ontogenetic Niche Partitioning: Juvenile tyrannosaurs, such as the 1.8-metric-ton "Teen Rex" specimen discovered in North Dakota in 2022, possessed elongate tibiae and metatarsals yielding stride-to-hip ratios characteristic of high-speed cursorial specialists ($> 30 \text{ mph}$). These fast juveniles functioned as pack coursers that pursued and flushed swift prey.
Adult T. rex, meanwhile, required an explosive sprint speed of 22 to 28 miles per hour to close final attack vectors across 50 to 100 meters, ambushing sub-adult ceratopsians and large hadrosaurs before they could reach cruising speed.
9. Comparative Ichnology: Global Tyrannosauroid Track Records
The North Dakota Hell Creek trackway provides a vital baseline for interpreting other disputed theropod trackways worldwide. By comparing stride ratios and track morphology across specimens, researchers can track how locomotive biomechanics shifted as body mass scaled up through the tyrannosaur lineage.
Comprehensive Global Record of Purported Tyrannosauroid Tracks
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TRACKSITE NAME LOCATION AGE TRACKMAKER TRACK LENGTH SPEED (m/s)
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Glenrock Site Wyoming, USA 66 Ma Nanotyrannus / Sub-adult 41.5 cm 2.8 – 3.8
Tumbler Ridge B.C., Canada 73 Ma Albertosaurus (*B. fredlundi*) 59.0 cm 2.4 – 2.9
Skyline Trackway Alberta, CA 76 Ma Gorgosaurus / Daspletosaurus 62.0 cm 1.9 – 2.5
Philmont Isolated New Mexico 67 Ma T. rex (*Tyrannosauripus*) 86.0 cm Isolated
Marmarth Trackway North Dakota 66.5 Ma Adult Tyrannosaurus rex 99.4 cm 1.6 – 1.9 (Walk)
========================================================================================At the Tumbler Ridge site in the Wapiti Formation of British Columbia, three trackways named Bellatoripes fredlundi preserve three smaller tyrannosaurs walking together in close pack formation. Discovered by Richard McCrea and his colleagues in 2014, the tracks measure 59 centimeters in length with average strides of 3.9 meters. The calculated walking speed for those lighter 3-ton animals was 2.4 meters per second (5.4 mph).
The Glenrock trackway in the Lance Formation of Wyoming records a sub-adult tyrannosaur taking three continuous steps. With a footprint length of 41.5 centimeters and a stride length of 3.3 meters, that smaller individual was moving at 3.3 meters per second (7.4 mph)—a brisk, long-legged trot.
The North Dakota discovery links these disparate data points into a continuous growth and performance series. Stride lengths scale predictably with hip height across every known tyrannosaur trackway, showing that locomotion remained remarkably consistent throughout the family's evolutionary history. Rather than suffering an allometric collapse in athletic ability, adult tyrannosaurs used their increased absolute stride length (reaching over 8.8 meters during high-speed runs) to maintain high burst velocities, matching the speed of smaller, lighter relatives even as their bulk expanded to nearly 8 metric tons.
10. Preserving the Past: Excavation Milestones and Field Operations
Because the Hell Creek Formation mudstones weather and crumble rapidly under rainfall and winter freeze-thaw cycles, stabilizing the Marmarth trackway required fast intervention by the Denver Museum of Nature & Science.
Excavation and Research Milestones: The Marmarth Trackway Project
+---------------------------------------------------------------------------+
| June 2025: Initial visual identification of tracks L1 through R2 by |
| Kent Hups on U.S. Forest Service terrain. |
| August 2025: Completion of high-resolution 3D surface laser scanning |
| and spatial photogrammetry by Tyler Lyson's team. |
| February 2026: Computational analysis and finite-element modeling |
| finalized by Peter Falkingham's research group. |
| September 2026: Formal peer-reviewed publication in the Journal of |
| Vertebrate Paleontology. |
| October 2026: Heavy-lift helicopter extraction of tracks L1, R1, and |
| L2 scheduled for transport to Denver. |
| Early 2027: Planned public exhibition and research access at the |
| Denver Museum of Nature & Science. |
+---------------------------------------------------------------------------+Three of the four footprint casts are slated for helicopter airlift from their remote badlands drainage. Heavy machinery cannot cross the fragile, roadless badlands terrain without damaging underlying fossil horizons, requiring paleontologists to isolate each footprint on a reinforced pedestal, wrap it in a structural plaster jacket, and lift the multi-ton blocks directly to flatbed transport vehicles.
Once housed at the Denver Museum of Nature & Science, the footprints will undergo high-energy industrial CT scanning. This scanning will map out the subsurface shear bands and compressed sediment layers locked beneath each track, tracing the internal strain patterns in 3D.
These internal deformation maps will establish the dynamic foot-strike trajectory with millimeter-level accuracy: precisely how the heel hit, how the metatarsal fat pad compressed, how the lateral toes stabilized the foot, and how the massive third toe pushed off to drive the next step.
11. Mechanical Realities of the Apex Predator
The four footprints preserved in the badlands of North Dakota demonstrate that adult Tyrannosaurus rex was far more than an energetic scavenger limited to a slow walking pace.
Kinematic Summary: T. rex Locomotor Spectrum
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REGIME VELOCITY (m/s) VELOCITY (mph) DYNAMIC STATE
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Resting Walk 1.28 m/s 2.86 mph Tail Harmonic Resonance
Marmarth Track 1.81 m/s 4.05 mph Empirical Hell Creek Stride
Extended Walk 3.20 m/s 7.15 mph Long-Distance Foraging
Grounded Run 6.50 m/s 14.54 mph Walk-Run Transition
Peak Sprint 11.62 m/s 26.00 mph Explosive Predatory Burst
=============================================================================The Marmarth trackway proves that the skeleton was not a fragile assembly threatened with collapse at the first rapid movement. Instead, it functioned alongside a compliant footpad, shock-absorbing toe joints, and an adducted limb posture that distributed heavy loads safely through the leg.
The empirical numbers permanently transform our understanding of this apex predator:
- A stride length exceeding 8.8 meters at full extension.
- An operational stride frequency of 1.32 cycles per second.
- A maximum sustained tyrannosaurus sprint speed between 22 and 28 miles per hour.
Captured during a quiet walk across an ancient Cretaceous floodplain, the first adult Tyrannosaurus rex* trackway provides the physical evidence needed to reconstruct the animal's true athletic limits. When this 8-ton carnivore charged, the mechanics of its stride enabled a pursuit speed that few contemporary herbivores could escape.
Reference:
- https://www.cbsnews.com/minnesota/news/adult-tyrannosaurus-rex-trackway-north-dakota/
- https://www.reddit.com/r/Dinosaurs/comments/1wbpv3r/665millionyearold_trackway_captures_adult_t_rex/
- https://www.sci.news/paleontology/adult-tyrannosaurus-rex-trackway-15056.html
- https://www.tandfonline.com/doi/full/10.1080/02724634.2026.2706201
- https://thedebrief.org/66-million-year-old-footprint-discovery-reveals-something-about-tyrannosaurus-that-researchers-never-imagined/
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- https://www.iflscience.com/worlds-first-adult-t-rex-trackway-with-4-meter-stride-discovered-by-high-school-teacher-84602
- https://www.sciencenews.org/article/dinosaur-speed-birds-mud-clues
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- https://royalsocietypublishing.org/rsbl/article/21/6/20250191/236046/Speed-from-fossil-trackways-calculations-not
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