The Physics of Superheroes: Spectacular Second Edition (16 page)

The impact of air molecules too small to see can crush objects more substantial than a paper straw—fifteen pounds per square inch is a pretty significant pressure if it acts in only one direction. Boil about three quarts of water for ten minutes or so in a fifty-five-gallon steel drum by placing it over a flame, and then seal off the drum and remove the source of heat. The steam that has displaced the air and filled the drum will now condense back into the liquid state, taking up much less volume and exerting a much smaller pressure on the interior of the drum. With the can sealed from the outside world, no air can replace the condensing steam, and the pressure differential between the inside and outside of the steel drum will grow. In a few minutes time, the steel barrel will violently implode, as if struck by the fist of an angry, invisible giant. Such dramatic lecture demonstrations have practical consequences when railroad tanker cars are steam cleaned. When the tanks are filled with hot water and then closed up, the resulting pressure difference when the steam condenses back into liquid water can cause even a tanker car to suffer a violent implosion. If atmospheric pressure can do such damage, it puts into context how tough Aquaman, who can withstand the pressures of the ocean’s depth, must be.

Aquaman is not only known for his ability to breathe underwater, for his superstrength, or for his communication with his finny friends, but also for his great swimming speed. He has been known to swim at speeds of up to 100 mph, much faster than any mammal or fish. How does he do it? How do any of us swim?

The first rule of swimming is to avoid sinking. The best way to do this is to float, and the best way to do that is to have a density less than that of water. Imagine a box held underwater. If the box weighs exactly the same as an equivalent volume of water, then aside from possibly a different color, there is no difference between the box and an equal volume of water. If we drew imaginary lines around a volume of water, we would not expect that this mental exercise would cause the water enclosed to move either up to the surface or down to the bottom of the container. In such a situation, we say that the box is “buoyancy matched” to the surrounding fluid, and in still water it should remain at its original position indefinitely. If the mass of the box were greater than the mass of water occupying the same volume, then the box would sink, and if the box had less mass, it would rise to the surface and float. While humans are mostly water, we have about two quarts of air in our lungs, and some of our cellular material, notably fat, has a lower density than water. Consequently, we have, on average, a lower density (mass per volume) than water, and can float. Salt water is denser than fresh water, which is why it is easier to float in the ocean than in a freshwater lake or pool. A solid block of steel has a much greater density than water, and it thus sinks when placed in open water. We can lower the density of the metal by changing the volume it occupies. The same mass of steel, when fashioned into the shape of a hollow, open-topped boat, takes up a volume that has an outer wall of steel and an interior of air, so that its average density is less than water’s, and it will consequently float.

Aquaman can change his depth underwater presumably the same way fish do: by varying the air content in a special organ called, variously, a swim bladder, an air bladder, or a gas bladder. This organ holds a reservoir of air. When a fish swims to too great a depth, the larger water pressure squeezes the fish, and the fish responds through a complex chain of chemical reactions by, in essence, taking a deep breath, transferring air into its bloodstream, and injecting air into its swim bladder, thereby expanding it. The fish’s average density thus lowers, and it is pushed toward the surface. Similarly, it can extract air from the bladder, increasing its density, when it wishes to descend. Aquaman, being of unique half human, half-A tlantean parentage, has obviously developed an analogous organ that enables him to swim at any depth in the seven seas he chooses.

Being able to float in water is necessary in order to swim, but not sufficient. One must also be able to provide a backward force against the surrounding water, so that, by Newton’s third law, the water can provide an equal and opposite thrust that propels one forward. The hand motion of the breaststroke, (technically referred to as “scooping”) pushes the water backward and the swimmer forward. A question that has long puzzled those interested in the physics of swimming, and has recently been resolved experimentally, is whether one could swim faster if the viscosity, that is, the resistance to flow, of the fluid were increased. Clearly, thicker fluid, such as molasses, would provide more resistance to the swimmer’s stroke, and the harder the swimmer pushes against the fluid, the more a counterforce is provided that moves the athlete forward. However, it is harder to move through molasses than water, just as there is more resistive drag when one tries to run through a swimming pool compared with running through the air on dry land. Which effect plays a greater role, positive or negative, in determining a swimmer’s progress?

At my home institution, the University of Minnesota, just such an experiment was conducted in 2004 by chemical-engineering professor Ed Cussler. The swimming pool at the university was filled with nine hundred pounds of guar-gum powder so that the pool fluid had twice the viscosity of normal water. Athletes from the university swim team completed laps in the thicker soup, and their times were compared to normal conditions. The net result: zero. The swimmers were indeed able to increase the efficiency of their strokes, but this turned out to be exactly compensated by the increase in drag in the thicker fluid. The only effect was that the swimmers were significantly slimier when leaving the pool—but at least it was in the interest of science!

In order to achieve a velocitiy in the water of 100 miles per hour, Aquaman was making use of a miracle exception, along the lines of that invoked for the Scarlet Speedster, the Flash. But perhaps he also makes use of some of the adaptations of one of nature’s fastest swimmers, the dolphin. While not being in the Aquatic Ace’s league, the dolphin is able to achieve speeds of twenty to twenty-five miles per hour, so fast, in fact, that until recently, scientists thought that the dolphin was somehow violating the principle of conservation of energy. It was argued that dolphins would require more muscles than they actually possess in order to swim at such speeds. Of course, a dolphin’s speed is not in conflict with the laws of physics. Dolphins are nearly all muscle, which provides great strength for each of their strokes. Muscle is denser than water, but these mammals avoid sinking to the ocean’s bottom not through reservoirs of fat, but instead with blubber. Blubber has a lower density than fat and the spring- like resiliency of muscle, and it thus provides crucial assistance to the dolphin as it swims. It is as true in swimming as it is in comedy: Timing is everything. Dolphins change the shape of their flukes into arches at precise points in the stroke cycle in order to maximize their thrust. Furthermore, dolphins are extremely streamlined, with a bare minimum of appendages extending from their body that might contribute to water drag.

Interestingly, dolphins are continuously shedding their skin, at such a rate that their epidermis is completely replaced every two hours. What benefit could this provide that would justify the excess energy the dolphin must expend in order to be essentially always rebuilding its skin? Physical experiments and computer simulations suggest that the little flakes of skin that slough from the dolphin’s body as it swims perturb and break up turbulent vortices generated in the dolphin’s wake. By inhibiting the formation of these vortices the transition to turbulence is damped (in a situation not dissimilar to that of a dimpled golf ball, which will be discussed in Chapter 12), and lower turbulence translates to greater speed. Perhaps the key to the Aquatic Ace’s super swimming speed is not the speckled orange shirt we associate him with, but a severe case of underwater dandruff!

7

CAN HE SWING FROM A THREAD?—
CENTRIPETAL ACCELERATION

PRACTICALLY EVERY ISSUE OF
The Amazing Spider-Man
features scenes of him using his webbing to swing from building to building through the canyons of New York City. But is this realistic? Specifically, is Spider-Man’s webbing strong enough to support his own weight, as well as the weight of any falling criminal, victim, or innocent bystander whom he happens to catch mid-flight as he swings in his parabolic trajectory? As Spider-Man swings in an arc, there is an extra force in addition to his weight that the webbing must supply. Let’s now consider why.

Remember that Newton’s second law of motion,
F
=
ma
, told us that a force is needed to change an object’s motion. A change in motion, or acceleration, refers to either a change in magnitude (either speeding up or slowing down) or to a change in direction. If no force acts on the object, then it persists in “uniform motion,” which means constant motion in a straight line. Any change in motion, whether in magnitude or direction, can only come about if a force acts on the object. When an automobile negotiates a hairpin turn, an external force (friction between the tires and the road) changes the auto’s direction, even if the speed remains unchanged.

In order to change the direction of motion, an external force is needed—and the corollary of this is that a force can only produce an acceleration in the direction that the force acts. For example, gravity pulls an object toward the ground, regardless of its initial motion. More importantly, gravity can only pull an object toward the ground because that is the only direction it acts. If the Golden Age, pre-flying Superman runs off the edge of a cliff with a steady horizontal speed, he will start falling due to gravity. Since gravity does not act in the horizontal direction, his horizontal speed will not change as he falls! No force, no change, after all. His vertical velocity does increase, becoming greater the longer he falls, just as in the case of Gwen Stacy considered earlier, because there is a force acting on him in the vertical direction. The net effect of his constant horizontal speed plus an ever-increasing vertical speed is a parabolic trajectory that becomes steeper the longer he plummets.

Put another way, a 90-mph fastball, thrown without spin perfectly parallel to the ground, falls to the ground at the exact same rate as a ball simply dropped out of the pitcher’s hand at the same moment. Both balls would strike the ground at the same instant (if released from the same height), because the only force changing their motion is gravity, in the vertical direction. Any change in either the direction or magnitude of an object’s motion can only arise from an external force acting in the direction of the change.

As Spider-Man swings from building to building, his trajectory is a semicircular arc rather than a straight line. Therefore, even if the magnitude of his speed does not change during his swing, his direction of motion is continually being altered, which can only be accomplished by an external force. It should be obvious that this force comes from the tension in the webbing. The webbing, therefore, has to do double duty and supply two forces: (1) a force to support Spider-Man’s weight, which it would have to maintain even if he were simply hanging from the vertical line, and (2) a second force to divert him in a circular trajectory. If the webbing line were to snap in mid-swing, then the only external force acting on Spider-Man is gravity, and his motion at this point would not differ at all from that of a ball tossed with the same velocity that Spider-Man possessed at the instant the webbing broke.

The acceleration that this additional force in the webbing provides as Spidey swings in a circular arc is identical to the acceleration experienced by the moon as it makes its circular orbit about the Earth. In one case, the force arises from the tension in the webbing, while in the other it is Newton’s force of gravitational attraction, but for both, it changes straight-line motion into circular motion. Gravity is the moon’s “webbing,” causing its direction to change. If the tension in the webbing or gravity were suddenly to disappear, both Spidey and the moon would depart from their circular trajectories, and continue moving with the velocity they had at the moment the external forces were removed. With a little bit of geometry or calculus one can show that the acceleration a of an object being constantly deflected onto a circular orbit with a velocity v is
a
=
(v
×
v)/R
=
v
²
/R
, where R is the radius of the circle.

Spider-Man’s webbing has to supply a force mg, in order to support his weight, and an additional force mv
²
/R in order to change his direction as he swings. The faster he swings (the larger his velocity v) or the tighter his arc (that is, the smaller the radius R), the greater will be this centripetal acceleration v
²
/R. When Spidey swings from a web strand 200 feet long at a speed of 50 mph, the centripetal acceleration is 27 feet/sec
²
, in addition to the acceleration due to gravity of 32 feet/sec
²
. If Spider-Man’s mass in the metric system is approximately 73 kilograms, then his weight mg is 160 pounds, and the additional force the webbing must supply just to change his trajectory from straight-line motion into a circular arc is roughly 135 pounds. The total tension in the webbing is nearly three hundred pounds, and will be more if Spidey is carrying someone as he swings.

Three hundred pounds or greater of tension may seem to be more than a thin strand of fiber can withstand, but if Spider-Man’s webbing is anything like real spider silk, he has nothing to worry about. Dragline silk webbing, which spiders use for their webs and while rapidly fleeing predatory birds, is actually five times stronger per pound than steel cable and more elastic than nylon. The webbing’s properties result from thousands of rigid filaments only a few billionths of a meter wide (providing great redundancy so that no one filament is crucial for the integrity of the webbing), interspersed with fluid filled channels that distribute the tensile force along the length of the webbing. Spider-Man is able to alter the material properties of his webbing by adjusting its chemical composition as it jets from his web-shooters. Similarly, real spiders can control the tensile strength of their webs by varying the relative concentration of crystallizing and noncrystallizing proteins.