TYRANNOSAURUS REX RUNNING? ESTIMATIONS OF EFFICIENCY, SPEED AND ACCELERATION

Background. The estimations of maximum speed of Tyrannosaurus Rex vary from 5 to 20 m/s and higher and still are the subject of scientific discussion. Some scientists consider T. Rex the largest terrestrial superpredator that needed speeds greater than 60 km/h (17 m/s) to capture its prey. Some recent publications indicate that it wasn’t able to run at all due to its large mass and significant loads on the skeleton and limit its walking speed to 5–7.5 m/s. Objective. We will try to answer the question of whether large animal or robot sizes are an obstacle to rapid running and to evaluate the maximum possible speed of T. Rex. Methods. We will use: a) two energy efficiency indicators the drag-to-weight ratio or the cost of motion and the recently developed capacity-efficiency (connected with the power-to-weight ratio or metabolic rate); b) the vertical acceleration estimations; c) the available data about the speed, the stride and the leg length of human and animals. Results. The drag-to-weight ratio and the capacity-efficiency were estimated for running of different animals and humans. It was shown that the maximal running speed of T. Rex may reach the values 21–29 m/s. The values of its vertical acceleration are typical for bipedal running. Conclusions. Large dimensions of Tyrannosaurus Rex couldn’t be an obstacle to achieving rather high speeds during short intervals of fast running. Such conclusions allow us not to abandon the assertion that the dinosaur was a super-predator. Presented approach could be useful for studying locomotion in modern and fossil animals, human sport activity and for design of fast bipedal robots.


Introduction
The maximum speed of Tyrannosaurus Rex remains the subject of scientific discussion.Without going into detail, we note two opposite points of view.Some scientists consider T. Rex the largest terrestrial super-predator that needed speeds greater than 60 km/h (17 m/s) to capture its prey [1].Some recent publications indicate that it wasn't able to run at all due to its large mass and significant loads on the skeleton and limit its walking speed to 5-7.5 m/s [2,3].However, the speed of the African Bush elephant (Loxodonta africana, the herbivorous animal of similar weight -8 t) may exceed 11 m/s [4].
In this paper, we will try to answer the question of whether large animal or robot sizes are an obstacle to rapid running and to evaluate the maximum possible speed of T. Rex.For this purpose, energy efficiency indicators (the drag-to-weight ratio or the cost of motion and the capacity-efficiency, developed in [5,6] and connected with the power-to-weight ratio or metabolic rate) and the vertical acceleration estimations will be used.

Drag-to-weight ratio and cost of motion.
Since the aerodynamic drag can be neglected by running of humans and large enough animals [6,7], the total drag / X mg k ≈ is connected with supporting the animal weight .mg The drag-to-weight ratio 1/k can also be treated as the cost of motion, i.e. how much energy is used to move 1 N of weight to the distance of 1 m.Usually in literature, this characteristic is related to the 1 kg of mass - k To estimate the efficiency of running, we shall modify the approach proposed in [7], which as-sumes running as series of jumps and the energy of the vertical motion as wasted to support the horizontal motion.Then this wasted kinetic energy equals 2 0.5mv ( v is the vertical velocity in the be- ginning of the jump).By dividing this energy by the duration of the jump 2 / , v g we obtain the wasted capacity 0.25mgv and the drag coefficient where U is the velocity of a hori- zontal movement.
Since the aero-dynamical drag can be neglected, the vertical velocity just after (and before) the leg contact with the ground can be estimated from the simple relationship: 2 / , j l Uv g = where j l is the length of the jump, or running stride.Then the drag-to-weight ratio can be estimated as follows [6]: (1) Power-to-weight ratio and capacity-efficiency.The power balance for the steady horizontal anaerobic motion can be written as follows: where q is total available power per unit of weight (metabolic rate), and η is the propulsion efficiency (0 1).

< η <
Then the maximum velocity of running is independent of mass and equals .U kq = η That means that the large mass of T. Rex cannot prevent its fast running.It must be noted that a long time running can be supposed as an aerobic activity with the energy release proportional to the lung surface (or ).m 2/3 Then from the balance similar to (2), we can conclude that 1 3 , U m − / ∼ i.e. small animals are faster long distance runners.The same is valid for human athletes.For instance, the body mass of 10 000 m runner champion -Kenenisa Bekele (55 kg) -is only 58.5 % of mass of 100 m one -Usain Bolt (94 kg).

Estimations of the drag-to-weight ratio. The estimations of the cost of motion (1) for humans and animals are presented in
The red solid and dotted lines show the relationships (5) in the Figure .They attain their maxima at velocities 1.48 m/s and 1.06 m/s respectively.At greater velocity, the cost of running diminishes with the increase of the velocity.For bipedal running, the experimental values (shown by red triangles) are in good agreement with the relationships (5).Only very small animals such as cockroaches, lizards and quails are the exceptions, since their stride-velocity dependences sufficiently differ from the human ones (their stride frequency / j U l is much higher).
The real metabolic cost of human running is approximately 4 for 100 m male and 40 m female running respectively.
Solid and dashed blue lines show the relationships (5) in the Figure .For bipedal running, the experimental values (presented by blue triangles) are in good agreement with the relationships (6).Only very small animals such as cockroaches, lizards and quails are the exceptions, since their stride-velocity dependences sufficiently differ from the human ones.
It must be noted that real values of the power-to-weight ratio (or metabolic rate) are much greater than the capacity-efficiency.For example, the maximum metabolic rate of human athletes is approximately 2.9 m/s (28 W/kg) [8] and is 10 times greater than the capacity-efficiency of 100 m running and is in between the rate of standing 1.29 W/kg and walking 3.3 W/kg [10].Similar large differences occur also in the case of vehicles (see [6]) and can be explained by a small value of the locomotion coefficient .
η We can assume that the capacity-efficiency of T. Rex cannot exceed the value 0.33 m/s, which follows from second equation (5) and experimental data for bipedal running.If Tyrannosaurus Rex had the values E C > 0.33 m/s, it would mean a higher metabolic rate or/and a bigger locomotion coefficient η in comparison with existing animals with the same running gait.There are animals with higher values of capacity-efficiency (e.g., horses and kangaroos, see the Thus, to know its maximal speed, we need its maximal stride length.Unfortunately, information about T. Rex traces is very limited.Moreover, to use formula (7), we need the maximal length of the running stride, which is much larger and happens much less often in comparison with the strides corresponding to the comfortable running (see, e.g., the data about ostriches in [11]).We can use the information about the relative stride referred to the leg length.The maximum stride-to-leg ratio for humans and birds vary from 1.7 to 2.4 [12,13].
Then taking the value 3.3 m as the maximal length of T. Rex leg [14], we can obtain from (7) the estimations of its maximum velocity: U > 20.8- 29.4 m/s.Such high speeds correspond to the maximum values of the capacity efficiency, i.e., are close to the asymptotical values in relationships (6).Thus, we can conclude, if T. Rex needed to run fast, its capacity-efficiency could be around 0.33 m/s and its maximum speed between 21 and 29 m/s.To be sure that such high velocities are possible, we will estimate the values of the vertical acceleration a as a measure of forces acting in skeleton.
Calculation of the vertical acceleration.We can use the simple relationship: 2 / , c a v t ≈ where c t is the duration of a leg contact with a ground and can be taken from [14], where the information about running of many animals were summarized as follows: (the values of the leg length l l must be taken in meters, speed in m/s).Using also (4), we obtain: 0.87 0.84 8 10 .
The results of application of formula (8) are presented in the Table .For the bipedal running animals, the vertical acceleration doesn't exceed 3g.Bipedal hopping gait (kangaroo, hare) allows larger values of acceleration (5.2-5.6g),since the vertical load is distributed on two legs.The highest acceleration 8.5g occurs in cheetah, where four legs are in simultaneous ground contact.Putting in (8) values E C = 0.33 m/s and l l = 3.3 m, we can estimate the acceleration in T. Rex between 1.7g and 2.3g for speeds 21 and 29 m/s respectively.These values are typical for bipedal running and cannot destroy its bones.

Discussion
To be convinced of the correctness of our estimates, it is necessary to answer the question why the capacity-effectiveness of modern lizards is 6-7 times less than what is accepted for T. Rex (see the Table ).Application of these small values of E C to T. Rex could yield 6-7 times higher values of its speed (according to (4)) and 21-22 % lower values of its vertical acceleration (according to (8)).Such procedure looks incorrect due to the huge difference in sizes (e.g., the length of callisaurus draconoides is 69 times smaller than the length of T. Rex [14,17]).For very small animals, aerodynamic drag becomes essential and the presented formulae are no more valid.To be aware of this, we can use the critical value of k at which the aerodynamic drag on an elongated body of revolution with a laminar unseparated shape becomes equal to the drag associated with supporting the weight, [6].Here V is the volume of body; ρ is the density of air;

− ⋅ α ≈
Then, for callisaurus draconoides (the lizard of average mass 9.5 g, [17]), formula (9) yields *** k ≈ 322.This value is comparable with the estimation (4), presented in Table 1, k ≈ 91.It means that air drag influences the running of small lizards.The same conclusions can be drawn about cockroaches and quails.Relatively small values of capacity-efficiency don't mean that very small animals are bad runners.Their drag-to-weight ratio and E C was underesti- mated by ne-glecting the aerodynamic drag.In the case of T. Rex (U = 21-29 m/s, m = 8-14 t), eq. ( 9) yields values *** k ≈ 16000-35000, which are much lager than / E k U C = ≈ 64-88 (eq.( 3), E C ≈ 0.33 m/s).Therefore, the air drag in T. Rex running can be neglected and all presented formulae and estimations are valid.

Conclusions
The obtained results don't mean that T. Rex had to run at speeds 21-29 m/s.We claim only that its large dimensions couldn't be an obstacle to achieving such high speeds during short intervals of fast running.Such conclusions allow us not to abandon the assertion that the dinosaur was a super-predator.Such a popular animal couldn't only eat carrion.
Presented approach could be useful for studying locomotion in modern and fossil animals, human sport activity and for design of fast bipedal robots.
the value of gravity constant), we obtain the dimensionless criterion 1/ .

Figure
Figure and the Table), but they have different running gaits.Estimations of the maximal running velocity.InequalityE C ≤ 0.33 m/s and formula (4) yield the estimation of the maximum running velocity in Tyrannosaurus Rex ( / ) 3.71 ( ). j

Figure :
Figure: Drag-to-weight ratio (cost of motion) and capacity-efficiency for humans and running animals.Red lines correspond to 10/k values calculated with the use of formulae (2).Experimental values (formula (1)) are shown by red markers.Blue lines correspond to EC values calculated with the use of formulae(5).Blue markers show the experimental values (formula (4)).Bipedal running animals are presented by "triangles"; other running gaitsby "circles" the kinematic viscosity of air.If *** , k k >> then the aerodynamic drag can be neglected.Assuming the average density of animals to close to the density of water W ρ Table and in Figure by red markers.To estimate it dependence on the maximal running velocity, the simple relationships The results of the capacity-efficiency calculations with the use of equation (4) are presented in the Tableandshown by blue markers in the Figure.
[8] (e.g.,[8]) or 1/k = 0.41.This value is 10-15 times greater than shown in the Table estimations 0.027-0.042.It means that only a small part of the energy released in human body is transformed into the kinetic energy of the mass center movement, i.e., the propulsion efficiency is rather small.Calcualtions of the capacity-efficency.

Table :
Information about speed, lengths of stride and leg for humans and animals obtained from different sources.Estimations of drag-to-weight ratio, capacity-efficiency and vertical acceleration