by Tom Rogers
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
[–] [–] Actors slamming their fists through car windows with no discernible injury.
[–] [–] Actors jumping through plate glass windows with no discernible injuries. (Note: plate glass would be found in older buildings or in ordinary residential windows. Sliding glass doors are an exception. They usually are made of safety glass.)
[–] Actors effortlessly jumping through large safety-glass display windows.
[–] Actors shooting highpowered firearms without any discernible recoil.
[–] The bullets from a strafing aircraft striking in nice neat evenly spaced rows.
CHAPTER 7
CREATIVE KINEMATICS:
Explosive Entertainment
WORLDWIDE EXPLOSIONS
An ancient asteroid impact triggers a sea of flame spreading about 6,000 miles (9,700 km) around the globe in seconds. It brings fiery death to everything in its path: trees, ferns, insects, and dinosaurs (opening scene Armageddon [RP]). Modern evidence indicates an ancient impact did indeed touch off a major firestorm and doomed the dinosaurs, but probably didn’t look like the movie version. First, there’s the flame-front speed depicted in the movie, calculated as follows:
SPEED = DISTANCE/TIME (EQUATION 7.1)
= 6,000 mi / (30 sec 1/3600 hr/sec)
= 720,000 mph or 1,160,000 kph
This speed is about 29 times faster than the minimum speed needed for an object to escape Earth’s gravity (escape velocity of Earth) and 950 times faster than the speed of sound: fast enough to blow away Earth’s atmosphere—impossibly fast.
A global-sized flame front spreading at a speed greater than the speed of sound is by definition a detonation. To maintain such velocities, the Earth would just about have to be covered in a thin sheet of explosive material like TNT. A rapidly expanding fire ball is like a jet airplane: even when it runs out of fuel, it will move forward, but not for hundreds and hundreds of miles. To keep moving it must have a source of fuel and the ability to consume it fast enough to maintain its speed, hence the need for the TNT. Even a flame front traveling at the edge of subsonic speed (just below 760 mph or 1200 km/hr) would have to consume combustibles in its path at an explosive rate to fuel its high velocity. Such high velocities could not be maintained without continual energy input.
This near-sonic-speed wall of fire would take about eight hours to spread 6,000 miles around the world. While such a front might be hundreds of miles wide, it would burn so fast that it would leave a large burned-out area behind it as it spread. From space, the flames would look more like a wide slow-moving ribbon than an all encompassing sea of fire—that is, if the flames were not completely obscured by smoke.
At the other extreme, forest fires release the energy stored in brush and trees much more slowly and generally travel at less than 4 mph (6.4 km/hr)—a speed that would take more than two months to spread 6,ooo miles around the world. By either estimate—high or low—an entire hemisphere is unlikely to be engulfed in flame all at the same time.
Still Armageddon’s opening depiction of an asteroid strike earns reasonably good marks. Its extraordinarily high flame velocity can be forgiven as a time-lapse effect, its sea of flame as over exuberance. The depiction could have benefited from scientific studies done with computer simulations but the overall visual effect was scary enough to make even a politician think about preventative action—a much needed activity. Unfortunately, the rest of the movie had a solution about as reliable as a campaign promise.
A realistic defense for preventing an asteroid strike could take decades and billions of dollars to develop. Sadly, it will probably also take a disaster—hopefully one in a lightly populated area— before humanity is willing to spend the money. Splitting a major sized asteroid in half with a nuclear bomb from our cold war arsenal will not be part of the plan. Such bombs simply do not have the required energy output (see Chapter 11).
How Explosions Propagate
An explosion’s effects are related to the ways its energy propagates, or spreads. As a rule of thumb these are:
1. Blast front: a wind-like movement of expelled materials including gasses, plasma, or debris such as shrapnel. Expelled materials can travel at very high, even supersonic, initial velocities. This material usually includes any fireball from fuel or explosive not consumed in the initial blast. Expelled material is often superheated and can cause secondary fires and burns.
On Earth, if expelled gasses travel at supersonic speeds they can compress air ahead of them enough to cause superheating and result in secondary fires. Expelled materials lose kinetic energy rapidly due to air resistance or, in the case of solid debris such as shrapnel, due to being pulled to the ground by gravity. In outer space expelled materials generally lose kinetic energy only when they impact another object.
2. Shock wave: a high pressure pulse traveling as a wave through air and caused by the blast front moving at or above the speed of sound. The shock wave can continue traveling considerable distances at the speed of sound long after the blast front has slowed down.
On Earth shock waves can do considerable damage. In outer space there are no shock waves because there is no matter to act as a medium for propagating the wave.
3. Electromagnetic (EM) pulse: a broad spectrum pulse of electromagnetic energy that can include everything from radio waves to gamma rays (for nuclear blasts). The pulse travels at the speed of light and can interfere with electronic equipment. It often includes a large amount of infrared radiation, enhanced by fireballs, burning objects, or superheated materials emitting infrared or thermal radiation for as long as they remain at elevated temperatures. This radiation can set secondary fires and cause burns.
In outer space, the blast front and EM pulse expand like two giant bubbles—surface area increasing with the square of distance from the blast—albeit the EM “bubble” expands much faster than the blast front. Since both kinetic and EM energies remain constant the intensity of the explosion (the total energy absorbed per unit of area for structures in the path of the blast) will decrease with the square of the distance from the blast. Doubling distance reduces intensity to one-fourth its original strength. The exceptions are chunks of solid debris or shrapnel which can be just as damaging at a distance as up close. On Earth, an explosion’s intensity will decline even faster due to air resistance. In general, an explosion in space can damage at much greater distances than its equivalent on Earth.
Armageddon concluded when the Texas-sized asteroid, on a collision course with Earth, was split in half by—you guessed it— a nuclear bomb, just in the nick of time to save humanity. The plume from the blast radiated outward in the shape of a disk about three times the diameter of the asteroid (a total comparable to the distance across the United States) in about two seconds (a speed of about 4,000,000 mph or 6,400,000 kph)—quite a blast for a device that normally produces a fireball a few miles in diameter and a shock wave traveling no more than a few times the speed of sound (760 mph or 1200 kph).
EXPLODING WORLDS
Hollywood recipes for planetary disasters are not just served with baloney, they’re made of it. When the Empire’s Death Star blows up Alderon (an Earth-sized planet) the pieces fly outward, amid a swirling orange-white fireball, a distance of about twice the diameter of the planet in about two seconds—a speed of 28,800,000 mph (46,400,000 km/hr), all the more remarkable because the exploding particles have to overcome the gravitational forces pulling the parts back into the form of a planet. Keep in mind that a typical detonation travels no more than a few times faster than the speed of sound (760 mph
or 1200 km/hr). While there is no law of physics that says a planet can’t be blown apart at such high velocities, certainly the numbers cast doubt on the notion that a death star could do it.
Okay, blowing up an entire planet is unlikely but if it did happen, would it look like the Star Wars depiction? Certainly the explosion would be like a gigantic nuclear blast and as a rule of thumb, about half of the energy in such a blast goes into heat and the rest into kinetic energy. The kinetic energy of a single 1 kg blob moving at the speed depicted in the movie would be about the equivalent of 20 kilotons of TNT, not quite twice the energy released by the 12.5 kiloton bomb dropped on Hiroshima. An equivalent amount of heat would be enough to vaporize a city, let alone a 1 kg blob—duh. Multiply the amount of energy for a single blob by the 6 × 1024 (more than a trillion times a trillion) similar blobs contained in an Earth-sized planet and the amount of energy released in the blast would be the equivalent of the Sun’s total energy output for about 800 centuries—released in a couple of seconds. Would the blast look like the movie depiction? Not likely. The planet would vaporize in a huge flash of blinding light. When the flash began fading, the vapor would start condensing into a gigantic slowly expanding dust cloud.
A less extreme exploding Earth-like planet would probably look like a balloon filling with liquid to its limits then popping, all in slow motion. The liquid in this case would be the glowing molten material of the inner planet. To fly apart the pieces would have to travel at least at terminal velocity (25,000 mph or 40,300 kph for Earth). Assuming a speed twice as high as terminal velocity, the debris from the planet’s surface would expand outward a distance of twice the planet’s diameter in about 19 minutes. On the other hand, if the pieces didn’t reach terminal velocity, they would fly outward and then collapse backwards into a turbulently swirling molten planetary blob. Since the mass contained in the planet’s solid crust would be small compared to the planet’s molten interior (assuming it’s Earth-like), the crust would essentially be swallowed up in a sea of brightly glowing lava. While not as dramatic as a sudden explosion, a more realistic explosion would have its own type of horror: the type that comes from the slow realization that a catastrophe is occurring and there’s nothing that can be done to stop it.
Filmmakers are well aware that blowing up a small scale model does not look the same as blowing up the real thing. Small explosions have similar velocities to large ones, but the flying pieces travel much shorter distances making the small explosions appear to happen much faster than the large ones being simulated. Cameramen compensate with slow motion photography. They film small explosions at higher than normal frame rates so that when the film is projected at its normal rate, the explosions are slowed down and look like they’re full sized.
Filmmakers should use the same principles in reverse to deduce that a planetary explosion—occurring on a gigantic scale—would look like it was happening in slow motion. Movies with exploding planets are rare, but it’s safe to say that the next one will probably look like the last one. It’s the law of Hollywood inertia: never alter the formula used in a successful movie.
THE SUPER SPEED OF SPACE TRAVEL
Space travel poses a different speed problem: it takes an incredibly high speed to get anywhere.The Apollo 10 mission holds the speed record for manned spacecraft at roughly 25,000 mph (40,300 km/hr). Double it and it would still take about 13 years to travel across the solar system (assuming that it is roughly circular with a diameter equal to the average distance from the Sun to Pluto of 2.8 billion miles). Travel to the nearest star outside our solar system, Proxima Centauri at a distance of 4.3 light-years or 2.5 × 1013 miles (4.1 × 1013 km) would take about 58,000 years—a little long to keep the kids in the back seat alive let alone entertained.
Decide to travel around our galaxy (the Milky Way) and the need for speed gets even more extreme. Traveling at the speed of light—about 1,300 times faster than the 50,000 mph used in the previous examples—it takes about 100,000 years to travel across the galaxy. To go much of anywhere, a spacecraft would need to travel about 1,000 times the speed of light, assuming that human lifetimes can be expanded to at least 200 years and suspended animation technology is available to facilitate return trips. But above all else, society would need the willpower to devote the major resources required for such journeys. Currently, we can’t even cough up the funds for a mere moon base. Of course, intergalactic travel could be done more cheaply with machines than humans, but there’s no movie in that.
Unfortunately, the speed limit for spacecraft is set below the speed of light, at least according to the famous galactic traffic cop, Albert Einstein. On a practical basis, it’s set far lower.
Einstein taught us that the mass of an object approaches infinity as the object approaches the speed of light—a puzzling statement that makes a lot more sense if the word inertia is substituted for mass. Recall: inertia is resistance to change in motion. Einstein is saying that it gets more and more difficult to change an object’s motion once it reaches speeds near the speed of light—making it infinitely difficult to actually reach such a speed. And infinitely difficult problems are rather hard to solve. To put the problem in perspective: at 25 percent of the speed of light (c) the inertia is 3 percent higher; at 50 percent c, the inertia is 15 percent higher; at 99 percent c, the inertia is 709 percent higher than at rest. Obviously, there’s no way to carry enough fuel to reach the speed of light.
The problem is further complicated by something called time dilation—exemplified by the twin paradox. Find a set of twentyyear-old twins, leave one on Earth, and send the other into space on a lengthy trip at a speed of 99 percent c. On returning, if she has aged by ten years, her twin sister—who stayed on Earth—will have aged by seventy one years. Start sending star ships on long missions at similar speeds all over the galaxy and no one will be able to keep their clocks synchronized. The result: a major breakdown in organized exploration.
Star Trek solved the problem for its spacecraft by warping space. Take a sheet of paper and ask a friendly ant to walk across it as you time the journey. It will take a while. Fold the paper so that it hangs in a loop with the ends touching each other at the top of the loop; the ant will need much less time to traverse from one end to the other. You have just warped space—as far as the ant is concerned. No one knows how to do it for a spacecraft but at least it’s conceivable.
Star Trek’s space ships can travel the galaxy at sublight speeds, keep their clocks synchronized, yet warp space for quick long distance journeys. According to Lawrence M. Krauss in The Physics of Star Trek there is even some theoretical support for it. Space could conceivably be warped by a super strong gravitational field. But don’t hold your breath waiting for the day. There are monumental problems in the way. As for more conventional thruster type technology, even short hops around a solar system will continue to be expensive, lengthy, and difficult.
OUTRUNNING EXPLOSIONS
Not all speed problems are galactic: take the problem of outrunning fiery explosions—a useful skill on the human scale. Sometimes the explosion occurs in the open and the object is to run and jump into the nearest body of water before the fireball hits the would-be escapist. Once underwater, the camera generally shows the deadly fireball sweeping overhead. If the explosion were far enough away—say, 1000 meters away—such an escape might actually be possible.
THE PHYSICS OF OUTRUNNING WHATEVER
When someone tries to outrun something like a car or an explosion he or she has to have a head start or it’s hopeless. To model the situation we’ll assume that the subject or person running away, as well as the object pursuing, both move at constant velocity. We’ll also assume that the object being outrun moves faster than the person. Otherwise, the subject is in no danger of being overtaken, and what drama is there in that? We’ll represent the head start as a distance dh. Distances can be calculated from the following kinematic equation:
d = v • t (EQUATION 7.2)
Where:
d = distance (note this is actually displacement or distance in a straight line)
v = velocity
t = time
We’ll use o and p subscripts to denote the object doing the chasing and the person chased respectively. The distance traveled by the object doing the chasing looks like this:
Substituting the two above equations into equation 7.3 yields:
Equation 7.4, as mentioned above, will work for any type of chase situation including either an expanding fireball or car chasing a person attempting to run away. For example, assume an explosion is 50 meters away traveling at the speed of sound and the person running from it can run 100 meters in 10 seconds (9.1 m/s, fast by any standard). The time such a person has to escape before being engulfed by the fireball is calculated as follows:
