Insultingly Stupid Movie Physics
Page 18
In a circular orbit, the force of gravity acting on an object acts as the centripetal force. The velocity required for a circular orbit is calculated as follows:
Where:
G = the universal gravitational constant
MP = the mass of the planet
r = the distance from the center of the planet
The first critical velocity VC1 would be found by substituting the radius of the planet rp as follows.
ESCAPE VELOCITY
Escape velocity is the second critical velocity value. This is the minimum velocity required to escape from the gravitational pull of the celestial body. If an object is moving with a tangential velocity between the first and second critical speeds, it will move in an elliptical orbit.
The escape velocity is derived from energy equations. The work required to move an object from a planet’s surface to infinity is set equal to the object’s kinetic energy. This results in the following expression:
Note that although the escape velocity is derived in a different manner, it’s simply the circular orbit velocity multiplied by the square root of 2. When a space vehicle slingshots around a planet or moon, it has to be going at or above the escape velocity to avoid being captured in an orbit or—worse—crashing into the surface (if the vehicle is below the circular orbit speed). Slingshotting allows a spacecraft to make a turn without using fuel. Normally, turns consume large amounts of fuel.
SUMMARY OF THE EFFECTS OF A TANGENTIAL VELOCITY AROUND A PLANET
First critical velocity—the lowest velocity possible for taking off from a celestial body with no atmosphere.
Second critical velocity—the lowest possible velocity for escaping a celestial body’s gravity.
If the booster motors were vectored with a downward angle, they would push the shuttle’s tail upward. Unfortunately, this would also rotate the shuttle’s nose downward. Yes, the shuttle would have small thrusters in its front, which in theory could counteract some of the nose’s downward rotation. But they would be low-powered devices used for positioning the shuttle when in orbit.They would not be designed for assisting a horizontal liftoff. Keep in mind that on the asteroid the gravitational attraction force on an empty space shuttle would be over 16,000 pounds (7,300 kg). With no air, there would be no lift generated by the shuttle’s wings. Getting the shuttle to rise off the surface using booster motors pointed in a horizontal direction would be a major feat.
There are two critical velocities associated with gravity: the velocity required for a circular orbit at the surface and the escape velocity. If the shuttle is moving horizontally with no lift force, it would have to exceed the velocity required for circular orbit at the surface before it could hope to lift off. As mentioned earlier in the rover discussion, this velocity is 1,800 miles per hour (2,900 kph)—a new land-speed record for humanity if not for Earth. Once off the surface, the shuttle is going to have to ramp up its speed to over 2,500 miles per hour (4,000 kph) in order to escape the asteroid’s gravity.
Even if the shuttle did miraculously manage to leave the spike-covered asteroid, it would likely not survive reentry into Earth’s atmosphere. The space shuttle Columbia burned up during reentry due to damage sustained when its wing hit a chunk of foam insulation that broke off its fuel tank during launch. How are a shuttle’s heat tiles going to survive scraping against and smashing into an asteroid’s rock formations?
Deep Impact was far more realistic than Armageddon Its 7-mile-long comet would have had only about 0.09 percent of Earth’s gravity level, and the movie portrayed it that way. Astronauts on the surface were attached to tethers to keep them from floating off. Even here, a person could not easily reach the escape velocity of 22 miles per hour (36 kph) simply by jumping. But escape velocity was still way too low for comfort.
The astronauts touched down on the comet in a specially designed lander rather than bulldozing a landing strip with a space shuttle. Naturally, nuclear bombs had to be drilled into the comet’s surface not at 75 feet, but at exactly 100 feet.This drilling had to be done at some distance from the landing craft so that the craft could almost run out of fuel as it raced to pick up the tardy drilling crew—just in the nick of time. Even then, the comet had rotated into the sunlight, causing an astronaut to be blown into space by a violent out-gassing of comet material instantaneously vaporized by the sunlight’s heat.
It’s not that the astronauts were dawdling. The digging device got stuck, and a brave astronaut had to selflessly climb down the 75-foot-deep hole then jump up and down on the digger to get it working. This took longer than the allotted time, but a bomb was successfully planted.
Back on the ship, the bomb was detonated. After all the bravery, the ingenuity, and the close calls, the mission failed. The bomb was supposed to merely alter the course of the comet. Instead, the explosion split the comet into two pieces—a big and a little one. Both of these parts continued on a path toward Earth, with the little guy racing ahead of the big one.
The split comet was a nice touch that illustrated one of the problems involved with trying to save Earth from an impact disaster. For the small piece to remain separated from the large one and speed ahead of it, the small piece would have needed to reach escape velocity, which would have been less than 22 miles per hour since the large piece had less mass after the small piece broke off.To reach such a velocity, the small piece would have needed to pick up the kinetic energy equivalent of about 10 megatons of TNT (assuming the small piece was 20 percent of the comet). The total explosive energy of the eight 2-megaton nuclear bombs carried by the spacecraft would have been marginal for providing it. Failing to reach escape velocity, the little piece would have been pulled back to the larger piece. Okay, it probably would not completely fuse back together. And maybe it would still be possible to shatter the big piece at the last minute by crashing the space ship into it as depicted in the movie. Still, if a nuclear bomb planted on the surface couldn’t effectively break up the comet, it’s doubtful that crashing a spaceship loaded with still other nuclear bombs of the same size would do much better.
KILLER TIDES
Let’s digress back to Armageddon and assume success in spite of all the space shuttle problems. It’s a momentous occasion. The Texas-sized asteroid on a collision path with Earth has just been split in half. The parts pass on either side of the Earth within a mere 400 miles of the surface. But why are people joyful? The oceans would have sloshed out of their basins and sent walls of salt water smashing across the world’s coastal areas. Places like Florida would be submerged. The water walls would be so heavy they would destabilize fault lines, setting off earthquakes and volcanic eruptions. When the oceans finally quit sloshing back and forth and the water receded, the sea in coastal areas would be filled with all kinds of sediment and contaminants. On land in coastal areas, dead marine life and people would be scattered everywhere, water systems contaminated, crops destroyed, and major cities demolished—not a cause for celebration.
Tides are normally created by shifting gravity forces that the Moon and, to a lesser extent, the Sun exert on the oceans.The mass of one-half of the asteroid would be about 3 × 1021 kilograms as compared to 7 × 1022 kilograms for the Moon. In other words, the moon is about 23 times more massive than the asteroid half, and gravity forces are directly proportional to mass. So what’s the big deal? Unfortunately, the gravity force is also inversely proportional to the square of the distance between the centers of mass of the objects causing them. The distance between the Earth’s center and the moon’s center is about fifty times longer than the distance between the Earth’s center and the asteroid half’s center. Taking into account all of the differences, the gravity force acting on Earth caused by half an asteroid is almost one hundred times higher than the gravity force on the moon. But there are two of these forces, one acting on each side of Earth. From the standpoint of tides these tend to reinforce each other. Keep in mind that ordinary tides caused by the Moon are around 10 feet (3 meters) and take several hours to rise. The
asteroid-produced tides would rise much higher in much less time.
Water, however, is only one source of devastation.There’s also wind. Normally the atmospheric tides created by the Moon are so small that they have almost no effect on weather. But increase these forces by a factor of 100 on opposite sides on the globe, and the result would be high winds over the entire surface of Earth. Underneath the passing asteroid halves, the winds could easily act like continent-sized hurricanes, adding major-sized storm surges to the already incredibly high tides.
The asteroid pieces would pass around Earth and collide back together on the other side in about half a day. They would then fly off into the cosmos. The gravitational pull of the asteroid mass would diminish quickly as it moved away from Earth. Within just a few hours the pull would be negligible. Unfortunately, the disastrous problems on Earth would persist for some time.The back and forth sloshing action of the ocean would last for hours if not days. High winds and erupting volcanoes with sloshing oceans could disrupt weather patterns for decades if not centuries, not to mention that such a large mass passing close to Earth and the moon could disrupt their orbits with unknown consequences. Earth would be a mess.
THE DIRECTION AND STRENGTH OF “INNER” GRAVITY
Imagine digging a shaft from the surface of Earth all the way through the center and out the other side (sounds like a movie plot already). Of course, there would be some minor problems such as extreme heat and pressure, but suppose these could be overcome. If a group of people descended into the hole in some type of elevator and stopped every so often to measure the force of gravity, they would find that it slowly decreased to zero at the center of the world and then increased back to its normal level as the elevator passed the center and ascended to the surface on the opposite side. If the density of Earth were constant, gravity force would be directly proportional to the distance from the center of Earth. When the inner core was reached, the gravity force would be one-sixth of its value at the surface. The Core does not depict this reduction in gravity force as the crew of the Virgil bores down toward Earth’s core. The inner core, however, is 4.5 times denser than the surface, and so the gravitational attraction force at the inner core would only be reduced by around 25 percent—not enough difference to mention.
Where the movie fails is in its depiction of gravity’s direction. Until the center of Earth was reached, the force of gravity would always be downward. Yet the Virgil is configured like a vertical subway train. Its crew walks around in their ship as though the gravity force were rotated by ninety degrees. They never have to climb ladders to go from the front of the ship (its lowest point) to the back (its highest point). As the ship passed its lowest depth inside Earth and started heading back out, the gravity force direction should have flipflopped, but this was also not depicted.
When the Virgil’s crew was in training for the mission, the movie indicated that the crew compartments could be rotated so that gravity remained pointed in the right direction. But this still did not solve the problem of moving from compartment to compartment. The design would have to be more complex than the simple subway-type design depicted in the movie. The movement between compartments would have to be done by climbing up and down ladders inside tubes connecting the pieces. The connecting tube would have to be located to one side of the compartments like a backbone so that each compartment could be rotated 180 degrees. Clearly, the moviemakers did not waste time working out such details. In a sense they had a point: why worry about minor details like the direction of gravity when the whole premise of the movie was ridiculous.
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.
[–] [–] Assuming that a near miss by a gigantic asteroid would cause no harm to Earth.
[–] [–] Space shuttles landing in impossible situations without damage.
[–] [–] Paying no attention to the direction of gravity.
[–] [–] Space shuttles taking off in impossible situations.
[–] [–] Underestimating or overestimating the gravity levels, surface orbit velocity, and escape velocity of an asteroid.
CHAPTER 15
SCENES WITH ARTIFICIAL GRAVITY:
The Good, Bad, and Ugly Space Stations
THE VOMIT COMET
The person who said, “crime doesn’t pay,” never watched a space movie. While some movies are upstanding, most are lawbreakers—laws-of-physics breakers, that is—especially when it comes to gravity. Trying to simulate the lack of or apparent lack of gravity takes creativity and hard work22—an anathema to the prospects of easy money.
Apollo 13 [GP] (1995) pulled the simulation off beautifully using NASA’s “Vomit Comet” (the KC-135A aircraft similar to the Boeing 707 commercial airliner). During a parabolic dive cycle, this aircraft provides about twenty-five seconds of apparent weightlessness but unfortunately cycles between the sensation of zero and elevated gravity—a cycle that drives the inner ear bonkers, producing mild to extreme nausea, hence, the nickname Vomit Comet. When the Apollo 13 movie crew needed to film a space flight scene, they didn’t do lunch, they risked redoing lunch. The result, however, was worth the indigestion: Apollo 13 was the third highest grossing film in 1995, with two academy awards and seven additional academy award nominations.
At the start of a parabolic cycle, the Vomit Comet climbs upward at a forty-five-degree angle. As it rounds the top section of a parabolic arc, the aircraft and all its contents are in freefall, which feels like zero gravity. The sensation is sometimes incorrectly referred to as zero-gravity, but, in reality, it’s not: the gravity force is just as high as it would normally be. The condition is also sometimes referred to as zero gs, but gs are a unit of acceleration and the acceleration of the aircraft is 1.0 g downward, even when simulating weightlessness. The sensation occurs only because the aircraft is in freefall.
As the aircraft heads toward the ground, the pilot has to pull it upward and the sensation of gravity returns with a vengeance. At the bottom of the dive a person will feel as though they are 80 percent heavier.
CREATING ARTIFICIAL GRAVITY
For filmmakers with finicky stomachs, artificial gravity is the cure. It requires no unsettling parabolic cycles, but to do it right requires some understanding of physics. Keep in mind that artificial gravity is not really gravity at all. To understand it, we first must understand which force causes the sensation of weight for a person standing on the ground. It’s not the gravity force. It’s something called a normal force.
Obviously, there’s a gravitational force pulling the person downward, but since the standing person is not moving or sinking into the ground, there must be another force of the same size pushing the person upward. The two forces cancel each other out. The upward force is called a normal force and is simply the upward force the ground creates on the person. Okay, the idea of the ground pushing upward may seem outlandish, but it does. The person’s feet push downward on the ground; the ground pushes upward on the person. It’s an actionreaction pair. The normal force always acts perpendicular (or normal) to the surface that creates it, hence the name normal force.
Since there are actually two forces acting on a person standing on the ground—a gravity force and a normal force—one, the other, or both must cause the sensation of weight. Let’s run a little mind experiment to find out which it is. First, let’s spring a trap door on a person and cause him to fall into a bottomless pit. In other words, remove the normal force but leave the gravity force. Other than the emotions from falling down a bottomless shaft, how does the person feel? Weightless. In fact, anyone in free fall will feel weightless. Clearly, the normal force
is required for having a sensation of weight.
THE BASIS OF ARTIFICIAL GRAVITY
Artificial gravity has nothing to do with gravity. In reality, it is the normal force acting as a centripetal force in a rotating cylinder, disk, or doughnut-shaped spacecraft that produces the effect. The normal force pushes upward on the inhabitants standing on the rotating floor. It is derived as follows:
FC = maC
Where:
FC = centripetal force
m = the mass of the object or person subjected to the artificial gravity
aC = centripetal acceleration
Centripetal acceleration is calculated as follows:
ac = v2/r
Where:
v = the object’s tangential velocity
r = the distance from the center of rotation to the object’s center of mass
The centripetal force or normal force must be equal to a person’s weight on Earth to make the person feel like he or she is in an Earth-like gravity field. Hence, the centripetal acceleration must be equal to 1.0 g (9.8 m/s2). In equation form:
g = v2/r