2022年小美赛ABCD题
2022年小美赛ABCD题
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Certifificate Authority Cup International Mathematical Contest Modeling
Problem A (MCM)3
How Pterosaurs Fly
Pterosaurs is an extinct clade of flflying reptiles in the order, Pterosauria. They
existed during most of the Mesozoic: from the Late Triassic to the end of
the Cretaceous. Pterosaurs are the earliest vertebrates known to have evolved
powered flflight. Their wings were formed by a membrane of skin, muscle, and
other tissues stretching from the ankles to a dramatically lengthened fourth'
fifinger[1].
There were two major types of pterosaurs. Basal pterosaurs were smaller
animals with fully toothed jaws and long tails usually. Their wide wing mem
branes probably included and connected the hind legs. On the ground, they
would have had an awkward sprawling posture, but their joint anatomy and
strong claws would have made them effffective climbers, and they may have lived
in trees. Basal pterosaurs were inctivores or predators of small vertebrates
Later pterosaurs (pterodactyloids) evolved many sizes, shapes, and lifestyles.)
Pterodactyloids had narrower wings with free hind limbs, highly reduced tails,
and long necks with large heads. On the ground, pterodactyloids walked well on
all four limbs with an upright posture, standing plantigrade on the hind feet and
folding the wing fifinger upward to walk on the three-fifingered “hand”. The fossil
trackways show at least some species were able to run and wade or swim[2].
Pterosaurs sported coats of hair-like fifilaments known as pycnofifibers, which
covered their bodies and parts of their wings[3]. In life, pterosaurs would have
had smooth or flfluffffy coats that did not remble bird feathers. Earlier
gestions were that pterosaurs were largely cold-blooded gliding animals,
riving warmth from the environment like modern lizards, rather than burning
calories. However, later studies have shown that they may be warm-blooded
(endothermic), active animals. The respiratory system had effiffifficient unidire
tional “flflow-through” breathing using air sacs, which hollowed out their bones
to an extreme extent. Pterosaurs spanned a wide range of adult sizes, from
the very small anurognathids to the largest known flflying creatures, including
Quetzalcoatlus and Hatzegopteryx[4][5], which reached wingspans of at least
nine metres. The combination of endothermy, a good oxygen supply and strong
1muscles made pterosaurs powerful and capable flflyers.
The mechanics of pterosaur flflight are not completely understood or modeled
at this time. Katsufumi Sato did calculations using modern birds and concluded
that it was impossible for a pterosaur to stay aloft[6]. In the book Posture,;
Locomotion, and Paleoecology of Pterosaurs it is theorized that they were able)
to flfly due to the oxygen-rich, den atmosphere of the Late Cretaceous period[7].
However, both Sato and the authors of Posture, Locomotion, and Paleoecology
of Pterosaurs bad their rearch on the now-outdated theories of pterosaurs.
being abird-like, and the size limit does not apply to terrestrial pterosaurs,
such as azhdarchids and tapejarids. Furthermore, Darren Naish concluded that
atmospheric difffferences between the prent and the Mesozoic were not needed
for the giant size of pterosaurs[8].
Another issue that has been diffiffifficult to understand is how they took of
If pterosaurs were cold-blooded animals, it was unclear how the larger ones
of enormous size, with an ineffiffifficient cold-blooded metabolism, could manage
a bird-like takeoffff strategy, using only the hind limbs to generate thrust for
getting airborne. Later rearch shows them instead as being warm-blooded
and having powerful flflight muscles, and using the flflight muscles for walking as
quadrupeds[9]. Mark Witton of the University of Portsmouth and Mike Habib of
Johns Hopkins University suggested that pterosaurs ud a vaulting mechanism
to obtain flflight[10]. The tremendous power of their winged forelimbs woul
enable them to take offff with ea[9]. Once aloft, pterosaurs could reach speeds
of up to 120 km/h and travel thousands of kilometres[10].
Your team are asked to develop a reasonable mathematical model of the
