by Tom Rogers
When two-year-old Adam is scaled up by a factor of 50, the weight his leg bones can support will increase by a factor of 502,or 2,500. Unfortunately, his weight will also increase by a factor of 503 or 125,000. So his leg bones would have to be fifty times stronger. His neck and spine support compression loads and would also need to be fifty times stronger. But they contain numerous joints, which tend to be weaker than bones. Most likely, joints and bones all over his body would fail, and he would collapse into a very un-Disney-like mess.
King Kong
King Kong [PGP-13] (2005) features a giant gorilla that ends up rampaging around New York City. It’s one of the most enduring Hollywood movie images, but how realistic is its physics?
The question can be answered in part by calculating the scaling factor. A real gorilla is, at most, about 5 feet 7 inches high (1.75 m) and 396 pounds (180 kg) in weight. Increase its height to a King Kong–sized 25 feet (7.6 m) and the scaling factor is 4.5—a value within the design range of many types of animals, such as dogs (a Great Dane, for example, is at least four times taller than a Chihuahua), or for that matter humans. Carbon-based biological creatures in the size range of King Kong—namely dinosaurs—have existed. The great ape would have a weight of 18 tons (16,000 km) compared to a brachiosaurus weight of 84.8 tons (77,100 kg) or a tyrannosaurus weight of 7 tons (6,400 kg), again within the values for real (albeit extinct) animals. Such an enormous gorilla is conceivable, but could it behave as depicted in the movie?
The muscle strength to weight ratio would be less than onefourth its original value (assuming muscle strength scaled up with cross-sectional area.) Hence, the great beast would not have the leaping and climbing ability depicted in the movie. Many of the beast’s features, such as eye and nose size, would be way over designed, representing a poor level of optimization. A giant squid is about the only animal with such large eyes, and these are designed to be able to gather enough light for vision even in the near-lightless depths of the ocean. When standing, Kong would have about 410 times more gravitational potential energy stored in his body than a gorilla. However, his bones would only be about 20 times stronger (in tension or compression). If King Kong fell during a fight, he would be prone to injuries like broken bones. Could he successfully hold an uninjured blonde in one hand while fighting three tyrannosauruses simultaneously? Yeah right. These “terrible lizards” with massive jaws full of 12-inch- (30-cm-) long serrated teeth designed to tear flesh and penetrate bone, would be fully optimized oversized meat grinders. King Kong would end up looking like ground round and probably bleed to death even if he won. As for the blonde, she’d probably be shaken and bounced to death.
ANTS, ELEPHANTS, AND ELASTIC STABILITY
The analysis of scaling humans up and down has ignored something called elastic stability. A typical column fails in compression when the weight it supports is too high. Its maximum weight is directly proportional to its cross-sectional area but is not at all influenced by column length. However, make a column too long, and length becomes an issue. The longer the column, the more likely it is to fail because it lacks elastic stability. When it fails, it will suddenly buckle and collapse with much smaller loads than those needed to cause a compression failure in a shorter column. The bones in a person’s legs aren’t long enough to make elastic stability an issue, but obviously it imposes a limit on how long a creature’s legs can ultimately be.
Ants are famous for their ability to carry fifty times their weight. If scaled up from 3 millimeters to 3 meters Hollywood-style (a scaling factor of 1,000), they would indeed be formidable and terrifying. Of course, scaling up Hollywood-style means they could still carry fifty times their weight. To understand why this is nonsense, let’s make the same analysis as was just done for the two-year-old.
The ant’s leg strength in compression would increase by a factor of 1,0002, or 1 million. However, the ant’s weight would increase by a factor of 1,0003, or 1 billion. The ratio of weight to leg strength in compression would increase by a factor of 1,000. Each leg in the ant’s large-size version would have to be 1,000 times stronger than needed in the ant’s original size. Most likely, the ant would collapse under its own weight. However, the scenario gets even worse. Ant legs are very long and thin. If scaled up, they would have elastic stability problems as described earlier.
The effects of elastic stability can be demonstrated with a soda straw. Cut a 2-centimeter-long piece. Compress it like a leg bone by squeezing it between your thumb and index finger. It takes a lot of force to make it fail. Conduct the same experiment with a 12-centimeter-long straw and you’ll find that the longer straw buckles immediately with the slightest side force or misalignment. The longer straw has elastic instability.
The long, thin legs of ants, spiders, and other crawly little critters would be a complete disaster in an oversized version. The simple truth is that the design of creatures is dictated in large part by the physics of their size. They just can’t be scaled up or down by much without a design change.That’s why elephants have proportionally larger legs than goats.
Most of the incredible strength of critters such as ants and spiders is related to their small size. These amazing abilities simply do not scale up. If a magical spider bit a human and turned him into a man-sized spider-person with properly scaled-up spider strength, he would be lucky if he could get out of bed.
SCALING MACHINERY
In general, things get proportionately weaker from a mechanical standpoint and considerably heavier when scaled up. A multipleengine bomber will have proportionately larger wings than a fighter plane in order to provide the additional lift required to counteract the bomber’s much higher weight. Due to its smaller size, the fighter will be able to make maneuvers that would tear the wings off a bomber.
This table shows a comparison of a P-51 single-seat fighter aircraft to a fourengine B-17 bomber, both WWII aircraft. Actual scaleup factors are the bomber’s values—wing span, length, height—divided by the fighter’s values. In a Hollywood-style scale-up, every dimension on the aircraft would be multiplied by a constant scale-up factor.
Figure 8: fighter to bomber comparison ratio
As a result, parameters such as weight would scale up by the cube of the scale-up factor. Scaling up motor horsepower is more complex. If it scaled up with the engines displacement volume, motor horsepower would be increased by a factor of 22,952. Generally, Hollywood assumes that performance factors such as speed would also increase by the scale-up factor, following the principle that bigger has to be better. In this case, such an assumption would be especially silly since it would make the bomber supersonic, and supersonic aerodynamics are completely different from subsonic.
Note that actual scale-up factors are inconsistent with the Hollywood factors. In other words, a WWII bomber is not simply a scaled-up fighter. It has to be redesigned specifically for its mission. Some aspects of performance, such as maneuverability and air speed, have to be sacrificed to gain others, such as payload.
Likewise, the pilot of a bomber is not going to be interchangeable with the pilot of a fighter, and vice versa. While both men may be capable of flying the other’s craft, it’s going to take a lot of hours of flying to master it. The business of using fighter pilots to fly a bomber mission as depicted in Pearl Harbor (see Chapter 1) is a Hollywood fantasy.
This chapter has dealt with only a handful of scale-up issues. The problems extend into all areas of science and engineering. For example, physicists currently use quantum mechanics on the atomic level and general relativity on an astronomical level; the two are not interchangeable. Efforts have been repeatedly made to combine them into a set of universal principles. Eventually, this may be done. Meanwhile, even the principles of physics are subject to scaling issues.
Summary of Movie Physics Rating Rubrics
The following is a summary of the key points discussed in this chapter that affect a movie’s physics quality rating. These are ranked according to the seriousness of the problem.
Minuses [–] rank from 1 to 3, 3 being the worst. However, when a movie gets something right that sets it apart, it gets the equivalent of a get-out-of-jail-free card. These are ranked with pluses [+] from 1 to 3, 3 being the best.
[–] [–] [–] Scaling a living creature up or down by incredible amounts without killing it. This almost always tags a movie as comic-book-like when done as a major plot device.
[–] [–] Claiming that the incredible strength of small critters, such as spiders, can be passed on to a much larger critter, such as a human. Again, this tends to tag a movie as comic-book-like. Of course, if it is based on a comic book, then it’s not necessarily a problem.
[–] Portraying significantly larger or smaller inanimate objects as though they will behave similarly to their regular size.
CHAPTER 5
INERTIA AND NEWTON’S FIRST LAW:
Why Blowing Up Spacecraft Is a Bad Idea
NEWTON’S FIRST LAW—A BRIEF HISTORY
Newton’s first law could be called the bunny principle—it states that objects keep going and going, just like the Energizer bunny, until a net force changes their motion. The change could be speeding up, slowing down, or a change in direction.
Newton’s first law is more typically called the law of inertia, since inertia is the property of mass that resists a change in motion. The higher the amount of mass, the more difficult it is to get an object moving. Once moving, the higher the mass, the harder it is to stop or change the object’s direction.
It took geniuses to figure out this simple principle; Leonardo da Vinci was one of the first to do so. Human experience tells us that a force is required to keep an object moving. When the ox quits pulling, the cart stops; so it seems obvious that a force is needed for motion. But, what if there were always a force present that took over and caused stopping as soon as the pulling force ended? In our world, it’s essentially impossible to be totally free of friction or some other resistance force, such as air resistance. So, objects do tend to stop if nothing is pushing or pulling them in the direction of motion, but it’s because there’s virtually always a resistance force pushing in the opposite direction of the motion.
The difference between the resistance force and pushing force is called the net force. If the pushing force is larger than the resistance force, the object speeds up. If it’s the opposite, the object slows down. If the two forces are exactly the same size, the object stays at constant velocity. If it’s moving, it continues to move. If it’s stationary, it remains stationary.
Da Vinci most likely understood the ubiquitous nature of resistance forces like friction. He couldn’t create a frictionless environment, but could imagine one. He might have visualized a friction-free world with only a single moving object in it. In other words, there would be no other objects for it to collide with and no resistance forces to alter its motion. He would then have asked a key question: what happens to the object’s motion over time? The answer: nothing. The object would move in a straight line at the same speed forever. This profound principle was called Leonardo’s law.
Then along came Newton. He embellished Leonardo’s law with two additional related laws, invented calculus, and managed to get Leonardo’s law named after himself. It’s now known as Newton’s first law.
WWII NAVAL BATTLES AND THE PROTECTIVE NEWTONIAN SHIELDS
Both the concept that it takes a net force to change an object’s motion and the idea that there’s almost always some sort of resistance force opposing motion are profound. Let’s take a look at an example to see why.
Imagine a WWII battle, and a massive convoy is moving slowly through enemy waters with gun crews on alert. There are destroyers, cruisers, battleships, and aircraft carriers seemingly in every direction. Suddenly a lone aircraft swoops from the sky, diving straight for the deck of an aircraft carrier. Every gun crew on every ship that sees the aircraft opens fire. At the last moment the aircraft comes apart in a ball of fire and falls harmlessly into the sea. The aircraft carrier is safe.
Careful observation would show that the aircraft continued its forward motion even as it broke into pieces.The forces exerted by the bullets, even cannon shells, are not high enough to stop the aircraft’s forward motion. They do, however, punch it full of large holes that damage the airframe, disable the control systems, wipe out the engine, and rupture the gas tanks.
With the engine disabled and the aircraft falling apart, air resistance slows the pieces and gravity pulls them downward as though the ship were protected by a giant Newtonian shield. As long as the aircraft comes apart well before it reaches the aircraft carrier, the shield works and the pieces fall harmlessly into the ocean. But the shield has a weakness—if the aircraft gets too close before coming apart, it hits the ship and does a lot of damage.
Holding together just long enough to hit a ship was the principle behind kamikaze suicide attacks. Even if the airplanes were shot up, if they got close enough, they could still do damage. In WWII, antiaircraft guns often could not shoot a plane down at a safe distance, and so kamikaze attacks did indeed inflict lots of damage. Of course, the damage was not just caused by the hurtling aircraft’s mass but also by the explosives it carried.
If a kamikaze plane, or for that matter a bomb or torpedo, hit a ship and blew it up, the explosion usually did little damage, if any, to nearby ships. These explosions would typically send both large and small chunks of metal hurling through the air, but nearby ships would once again be protected by their Newtonian shields. The combined forces of air resistance and gravity forming the shields would cause the chucks to harmlessly fall into the ocean long before they could strike a nearby vessel.
Without air resistance, even the act of shooting at attacking aircraft would have been highly dangerous to other ships in a convoy. The 20-millimeter, 40-millimeter, and 5-inch (126 mm) cannon rounds fired by the thousands at approaching enemy aircraft were easily capable of inflicting serious damage to a navy ship if they struck it at full velocity. However, air resistance usually did a good enough job of slowing them down to prevent serious damage.
Typically, larger cannon rounds containing high explosives were fused to explode long before falling back to Earth.The shell would burst into hundreds of smaller misshapen pieces. Although still moving forward, they would be less aerodynamic. With higher air resistance, the pieces would quickly slow down. The lower velocity in combination with a smaller size would greatly decrease the damage if they rained down on nearby ships.
WWII antiaircraft gun crews wore helmets to protect themselves from such falling debris. While a helmet offered no protection from a cannon round at full velocity, it would protect against small pieces of falling shrapnel.
WWII-STYLE SPACE BATTLES
WWII naval battles have been the model for numerous movie space battles, such as those in Star Wars. Unfortunately, a good resistance force is hard to find in outer space, so fighting a space battle WWII-style would be highly dangerous to all participants. Blowing up an enemy spacecraft, regardless of size, could damage other vessels, even at great distances.
If the fighter craft were small and attacking at high speed on a collision course with a much larger battle cruiser, there would be no safe way to “shoot down” the smaller craft. Such a craft could easily be closing in at several thousand miles per hour— many times the velocity of a speeding bullet. If the fighter craft were blown up, its pieces would still continue forward with many, if not all, pieces smashing into the battle cruiser. These pieces would not be like shrapnel falling from the sky. There would be no air resistance to slow them or gravity to pull them “downward.” They would impact at high velocity and include not just large pieces of shrapnel but also gasses or plasma liberated by the explosion. These fluids could easily strike with velocities several times higher than the atomic blast wave at ground zero in Hiroshima (estimated at a mere 980 mph4).
Blowing up such a craft would be worse than turning a rifle shot into a shotgun blast. Okay, armor plating would work better
against numerous small high-velocity particles than against a single large particle, but many of the smaller particles would gain kinetic energy from the explosion when the ship blew up. Even small high-energy particles would be capable of knocking out sensors and weakening armor plating, not to mention sending shock waves into a battle cruiser’s interior.
If two large battle cruisers were flying side by side and a small spacecraft flew between them, the larger ships would face an added dilemma. If they shot at the small enemy ship and missed, they’d hit each other. Any projectiles they fired would go in a straight line. There would be no gravity to pull the projectiles downward into the water between them. (There would be no water either.)
A gunner could fuse his projectiles so they blew up in the space between the cruisers, but then he’d shower the opposite cruiser with high-velocity shrapnel.The fact that the shrapnel was misshapen and had poor aerodynamics would be no help.There’s no air to provide air resistance in outer space.
Lasers and high-energy particle beams would have similar problems. There would be no way to limit a laser’s range, and if one cruiser fired a beam and it missed the intended target, the beam would damage any other cruiser in the beam’s path. But then such a beam should never miss. Even so, the problems of nearby exploding craft would persist.
Compared to space gunners, WWII antiaircraft gunners had far less danger of injury from blowing up nearby enemy craft, but they also had far more difficulty doing so. WWII projectiles had a downward curving trajectory and would have been moving slowly enough that a fast-moving aircraft could get out of their way. The WWII gunner couldn’t aim directly at the aircraft. He would have had to estimate where the aircraft would be when the bullet arrived and how much the bullet would drop during its flight to the target—and then aim accordingly. Generally, thousands of rounds of ammunition were fired for every WWII aircraft shot down.
