by Tom Rogers
Is there any way the giant ship could have momentarily experienced a stopping acceleration high enough to send the heroes through the window? No. A high acceleration requires a high rate of change in motion, but Newton’s first and second laws say the ship’s huge mass will strongly resist any change in motion. If the velocities shouted by the first mate as the ship slowed down are plotted against time, remarkably, they fall in a straight line. The slope at any point along the line is equal to the ship’s acceleration at that moment. But since it’s a straight line, the slope is constant, and so is the acceleration. This steady acceleration is way too low to send the heroes through the windshield.
The force that the windshield’s glass would have needed to restrain a hero’s forward motion would have been equal to his mass times his acceleration. For a 150-pound man during the Speed II ship crash, this works out to a whopping 1.5 pounds. Conversely, Newton’s third law says that the man exerts an equal but opposite force on the glass. Surely a windshield could resist a 1.5-pound force. Surely a ship demolishing a dock and several buildings is exciting enough even without propelling the heroes through the ship’s windshield.
USING NEWTON’S SECOND LAW
As mentioned in the text, if a car traveling forty-five miles per hour (72 kph, or 20 m/s) hits a brick wall and stops almost instantaneously, in say 0.01 seconds, the acceleration is as follows (assuming acceleration is constant):
The acceleration just happens to be negative in this case. A negative sign only indicates direction such as forward or backward. Since the velocity was positive, the acceleration had to be negative or point in the opposite direction for the car to slow down. Note also that gs are a unit of acceleration, not force. The force this would create on a 220-pound (100 kg) bubba is as follows:
(Note: as long as the acceleration occurs on Earth, the force acting on a person can be calculated as the person’s weight times the number of gs.)
If the initial velocity and stopping distance are known but the time is not, the acceleration can be found as follows (assuming constant acceleration):
where:
x = stopping distance
If a ship slammed into a perfectly solid barrier and came to a complete stop in a distance of roughly 3.8 inches (10 cm) from a speed of seven miles per hour (11 kph, or 3.06 m/s), the ship’s acceleration would be as follows:
IT’S NOT THE FALL. . . .
It takes no imagination to understand that jumping from a tall building onto a sidewalk is going to do more than just hurt. It turns out that the stopping acceleration is inversely proportional to the stopping distance. When a person hits the sidewalk, it may crack but it is essentially not going to move. The stopping distance has to be entirely provided by crumpling the person. Assume the person is six feet (1.8 m) tall and lands feet first at the terminal speed of a sky diver: 120 miles per hour (193 kph). If the stopping force is provided by crumpling the person 25 percent—in other words crushing the person’s legs to about half their normal length—the person will be subjected to 327 gs. At such accelerations, blood flow to the head will stop, blood vessels rupture, and internal organs crash into the bones underneath them9. Such a fall is not survivable.
Is any fall from a great height survivable? Based on various unplanned experiments, the answer is yes!10 For example, Lieutenant I. M. Chisov, a Russian airman, was badly injured but survived when he fell nearly 22,000 feet without a parachute after his bomber was attacked by German fighters in 1942. He hit the edge of a snow-covered ravine and rolled to the bottom. In 1944 Nicholas Alkemade—tail gunner in a badly damaged British Lancaster bomber—discovered to his horror that his parachute was in flames after being ordered to bail out. He jumped anyway, landing in trees, brush, and drifted snow. He ended up with a twisted knee and a few cuts. The common element in survival is always an extended stopping distance provided by crumpling materials or objects other than the human who falls.
Water can extend the stopping time, but is of limited benefit. To enter water, a person has to push a volume of water equal to their body’s volume out of the way. Since water tends to have a lot of mass, it takes a lot of force to accelerate it out of the way, especially if done quickly. The water exerts a resistance force on whatever does the pushing. Naval academy cadets are forced to jump feet first into water from a ten-meter (32.5 ft) high platform in order to prepare them for the day they might have to abandon ship11. They are taught to hop off the platform rather than leaning forward until falling off, because the leaning could misalign them from a perfect vertical position. Any head movement can result in minor injuries such as a bloody lip. Needless to say, above a height of 10 meters, water is going to be a very dangerous landing pad—it only needs to knock a person out to kill them by drowning.
In the movie XXX State of the Union [RP] (2005), Darious Stone (Ice Cube) jumps from a train moving at 160 miles per hour across a tall bridge over water. Assuming that the bridge was 300 feet high (91 m) and neglecting air resistance, the hero would hit the water with a vertical velocity of about 94 miles per hour (152 kph). His horizontal velocity would be reduced by air resistance but would probably still be at least as high as his vertical velocity. To have any hope of survival, the hero would have to enter the water feet first to prevent head injury. He’d also have to be facing upward and hit at exactly the same angle as his velocity vector: about forty-five degrees, in order to prevent back and neck injury. Even then, the horizontal component of his legs’ velocity would slow down more quickly than the same component of his head and torso, creating a bending action on his body. If facing upward, the body would bend in a direction it’s designed for (the same direction as bent in toe touching). Otherwise, the body would be bent backwards, resulting in a back or neck injury.
If the hero entered in the traditional diving position—perpendicular to the water, arms then head—as soon as his arms and head hit the water, their vertical and horizontal motion would abruptly slow almost to a stop. Meanwhile, his legs and torso would continue with their same vertical and horizontal velocity. The result would be an even more extreme bending action on the body, easily enough to break a neck or even a back and slam the person’s torso onto the water’s surface, breaking ribs and damaging internal organs in the process.
So what does the hero do? He shoots the water with some type of handgun that looks like a sawed, off grenade launcher and foams up the water below him. Although it’s not likely to be effective, foaming the water with gunfire would lower its density and reduce the resistance force it would exert on the hero when he hits it—at least slightly. The hero dives into the foamy water vertically head first. Does he die? Does he suffer? Why no, it’s a miracle! He not only remains conscious but survives unscathed. After all, it’s not the fall; it’s the stop at the end of the fall that kills and, of course, this stop was perfectly safe by Hollywood standards.
DEATH BY RESCUE
Falling off a tall building is almost certain death unless one is miraculously rescued. So, when Lois Lane falls from the fiftieth floor and is inches from impact with the sidewalk, Superman must rush—faster than a speeding bullet—to save her by whisking her off in a horizontal direction. As she falls she will roughly reach the terminal speed of a sky diver and be closing with the sidewalk at 120 miles per hour (193 kph). On the other hand, the man of steel will be closing with her at a velocity in excess of a speeding bullet say around 1,400 miles per hour (2250 kph). When he catches Lois, he must increase her velocity from zero in the horizontal direction to match his horizontal velocity and stop her downward velocity almost as fast as if she had hit the sidewalk. If it takes 0.1 seconds to do this, Lois will be subjected to over 6,000 gs of horizontal acceleration, and Superman will end up with an armful of bloody mush. It makes no difference whether high acceleration occurs in the horizontal or vertical direction. It’s going to hurt.
Superman could stop—he is after all superhuman—the instant before he hits Lois, catch her, and then accelerate off in a horizontal direction a
t a rate that would not injure her. Just before stopping, he would have 10,000 times more kinetic energy than a 7.62 NATO machine gun bullet. The law of conservation of energy demands that something be done with the energy. About the only option is converting it to heat. When he stopped, Superman would become red hot and likely set Lois on fire—not in the romantic sense. Lois would be french fried. And since she’s inches from the sidewalk, Superman is still going to have to subject her to a high vertical acceleration to get her stopped—but certainly not 600 gs. There’s really no way he can save her, unless he stays close at hand so that he would not have to move so fast to catch her.
Acceleration Injuries
Human tolerance for acceleration depends on many factors, including age, physical fitness, direction of acceleration, and use of safety equipment. The following data is offered only as a rough indication12.
Blackout from prolonged exposure 4–6 gs
Chest acceleration limit during car crash at thirty miles per hour (48 kph) with airbag 60 gs
Head acceleration experienced by Princess Diana during fatal car wreck 100 gs
Chest acceleration experienced by Princess Diana during fatal car wreck 70 gs
CORNERING CALLING FOR A CORONER
The spacecraft from Earth (SFE) changes its direction to fight an enemy ship. It is traveling a mere 0.25c (25 percent of the speed of light) and makes a gentle 180-degree turn with a 1.0-mile radius (1.6 km). The enemy ship departs slowly without bothering to fire. Has it given up, surrendered, or retreated in fear? No, there is no need for any of these responses. The crew members of the SFE have turned themselves to bloody mush by making the 180-degree turn. They have subjected themselves to roughly 3.6 × 1011 gs of acceleration. Even making the turn at a mere 1,000 miles per hour (1,600 kph) would subject the crew to 12.7 gs of acceleration—enough to cause blackouts and even fatalities. The truth is that space battles would have to be fought at rather sedate speeds if the ships were supposed to make turns and keep the crew alive at the same time.
If a spaceship makes a turn, it has obviously accelerated because it has changed direction. In making the turn the ship will generally follow a circular arc. Anytime an object goes around in a circle, or for that matter even a part of a circle, it will be subjected to centripetal acceleration. And acceleration is acceleration, whether it is caused by traveling in a circle or some other type of activity. High acceleration leads to high forces that cause damage.
If centripetal acceleration is such a big problem for humans on spaceships, then how do air force jets flying at supersonic speeds engage in dogfights? Simple, they don’t. First, if they want to down an enemy aircraft, they will typically use a small guided missile, which travels much faster than the jet, locks onto the target, and destroys it when it’s several miles away. Small missiles can handle far more acceleration than humans and don’t blackout. If a jet fighter does engage in a conventional dogfight with blazing cannons, it has to drop to subsonic speeds (below 760 mph, or 1,200 km/s).
Centripetal acceleration has been a boogeyman for military pilots since WWII. Dive bombing took a special kind of courage. Not only did the pilot have to dive straight at a target that was likely shooting back, but once he’d released his bomb he had to pull up sharply to avoid slamming his aircraft into the target.This subjected him to multiple gs of centripetal acceleration at a most inconvenient time for a blackout. Blackouts were so common among dive bombers that the German Stuka bomber was designed with a type of autopilot that would pull the aircraft out of its dive as soon as the bomb was released.
For spacecraft, even slowing down would be a major problem. If a spacecraft traveling at 0.25c (25 percent of the speed of light) decided to stop and limited its acceleration to 1.0 gs, it would take 3 months to come to a stop. Even if spacecraft could attain incredible speeds, simply stopping would make space journeys lengthy.
If the spacecraft were rotating in order to create artificial gravity (see Chapter 15) and wanted to slow down, it would have to turn off the rotation and reorient all the spacecraft floors so that the downward direction in the spacecraft’s rooms pointed in the same direction as the forward motion of the craft. While not impossible, it would take a sophisticated design to pull this off. Stop in thirty seconds from a speed of 0.25c, and the crew would be bloody mush (acceleration = 2.6 105 gs), that is, if the spacecraft did not incinerate itself. Its kinetic energy would have to be converted into another form—most likely heat.
Star Trek dealt with this annoying little problem by creating “inertial dampers.” These fictitious devices operate on an unknown principle of physics and somehow dramatically reduce the effects of high acceleration. As stated earlier, Newton’s second law says that F = ma, and it’s actually the F (or force) that causes the damage. The m is generally referred to as the object’s mass, but really is its linear inertia or resistance to change in linear motion. Einstein theorized that m could be increased dramatically without adding a single molecule to it. All the object had to do was go fast enough to approach the speed of light and— presto—the object’s linear inertia would approach infinity. Surely, if inertia can be increased, then it can be decreased, hence, the creation of a fictitious inertia-decreasing device or inertial damper—a device that stretches physics like a rubber band. Decrease a person’s m to nearly zero, and people can tolerate high accelerations because the high accelerations will produce low levels of force on them.
Having the inertial dampers go offline due to damage by attacking enemy warships is standard practice in Star Trek episodes. When this happens, the starship’s interior jiggles about and crew members fall down. It’s all great fun. If such an event actually happened, the crew members would be puréed the first time the spaceship made a high-speed maneuver.
Star Trek is not consistent in its use of inertial dampers, and there is no real scientific basis for them. But the writers have avoided trying to explain them with useless scientific babble. The mere mention of inertial dampers makes it clear that the writers know there are serious problems with high-speed maneuvering. Without the dampers, storyline fallacies would abound. Inertial dampers border on comic-book science, but at times science fiction has to resort to pure fiction in order to have a story.
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.
[–] [–] Movie characters subjected to huge accelerations with no significant injury.
[–] [–] Movie characters riding in vehicles are thrown through windshields as the vehicle slows down with lowto-moderate acceleration.
[–] [–] Space battles with ultra-high-speed turns and maneuvers and no physiological effects on crew members.
[+] Sparing use of makebelieve devices that are not explained with a lot of scientific mumbo-jumbo but at least recognize that a scene’s physics would otherwise be impossible.
CHAPTER 11
HIGH-ENERGY FILMS:
Nuclear Firecrackers, Falling People, and Cars as Weapons
NUCLEAR FIRECRACKERS
Up close a nuclear bomb blast is deadly but merciful—instantly vaporizing breathing, feeling humans into shadows on the ground. At a distance, it’s pure cruelty. It can melt flesh off a face or cause prolonged torment through all kinds of trauma, not to mention radiation sickness, cancer, or genetic mutation. Few events can match its horrifying effects on people. It’s our ultimate boogeyman, yet our ultimate bodyguard. We can choose where, when, and how to unleash its incredible power. And when we do, anything or anybody that threatens us had better watch out.
So, of course, when giant spaceships, enormous asteroids, or huge masses of uncooperative molten iron
threaten Earth’s tranquility, it’s time to reach for the nuclear button. The problem is that on a planetary scale, our boogeyman bodyguard is a squeaky little mouse.
Oversized saucers 15 miles in diameter, sent by a mother ship, arrive from outer space in fiery clouds and park over major cities around the globe. At first, the inhabitants below the saucers respond with a mix of excitement, curiosity, and fear. But when the saucers start blowing up cities with single blasts from their death rays (Independence Day [RP]) the confusion vanishes—along with the White House. It’s show time for the nuclear bomb, our boogeyman bodyguard. But wait! David Levinson (Jeff Goldblum), always the devoted environmentalist, shouts “stop.” Nuking alien invaders over American cities would be an environmental disaster. How could anyone unleash such a boogeyman? On the other hand, letting alien ships incinerate entire cities complete with birds, trees, and nuclear power plants is—by golly—also an environmental disaster. Less emotional military officials (nitwits according to Hollywood) decide to nuke the aliens anyway, to no useful effect. It seems the saucers have protective force fields. As Goldblum predicted, Earth gets the boogeyman but not the bodyguard.
What to do? For every Goliath there’s a stone. An alien saucer must turn off the shield around its death ray just before firing it. Fly a jet aircraft up the death-ray port and voila: the alien ship explodes and falls down, all 15 miles in diameter of it. Krauss, in Beyond Star Trek, estimates a saucer would weigh about 100 billion tons and that dropping it from a height of about 1 mile would release over 10,000 times as much energy as the nuclear bomb used on Hiroshima.