by Tom Rogers
Although the hero had a mere 2.3 seconds to assess the situation and make his move, it’s no problem. He’s handled worst. But the car’s kinetic energy is not so easily dismissed. It is the equivalent of 254 .45-caliber bullets. This is more kinetic energy than a 154-pound (70 kg) person falling at terminal velocity (209 .45-caliber bullets). The stop isn’t as sudden as hitting the sidewalk, but rolling over the top of a car is no roll in the hay. It’s going to do some damage, and portraying it otherwise is going to do some damage to clear thinking.
In the real world a drunken imbecile weaving his car down the road at 30 miles per hour should be greeted with even more horror than a drunken fool emptying his six-shooter skyward in the middle of the street. Maybe it’s not Hollywood’s place to help us put the dangers in perspective, but it certainly wouldn’t hurt clear thinking if car-pedestrian interactions in movies were more injurious.
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.
[–] [–] [–] Using underpowered nuclear bombs to save humanity from certain destruction.
[–] [–] [–] Contrived happy endings that have no logical or reasonable scientific basis and that help obscure real problems facing humanity.
[–] [–] Having massive objects fall to Earth with only minor consequences.
[–] [–] Slapping together major engineering projects in ridiculously short periods of time.
[–] Humans falling from great heights and receiving no injuries.
[–] Moderate- to high-speed car-pedestrian accidents without so much as a bruise on the pedestrian.
[0] Fatalities from falling bullets on Earth (possible but unlikely).
[+] Fatalities from falling bullets on moons or planets with no atmospheres.
CHAPTER 12
MOVIE MOMENTUM:
The Attractive Force of Glass, Rail-Gun Recoil, and Cosmic Toyotas
SHOTGUN BLASTS AND THE ATTRACTIVE FORCE OF GLASS
Sergeant Martin Riggs (Mel Gibson) stands on the sidewalk as a sinister car approaches with a shotgun protruding from the window. Suddenly he sees it, but—blam!—too late. He’s blown violently off his feet and flies several feet backward through the nearest display window. Fortunately, he’s wearing his bulletproof vest and survives (Lethal Weapon [PGP-13] 1987).
If he were not on the sidewalk by a display window, then invariably he’d be blown into a rack of whisky bottles, a giant mirror, or some other large glass object. This happens so often in movies that Hollywood seems to have discovered a new principle of physics: the attractive force of glass. Fortunately, under normal circumstances, one needn’t fear. Although remarkably reliable, the attractive force of glass only works for shooting victims and then only in movies. Still, wouldn’t something as deadly as a shotgun blast at least blow a victim violently backward?
At first glance it looks like it would, and that we could prove it using the law of conservation of energy. We could calculate the buckshot’s kinetic energy before colliding with the victim and confidently predict that the victim would end up with this energy after being shot. If the energy were still in the form of kinetic energy, the victim’s velocity could then easily be calculated. Indeed, in an elastic collision this analysis would work perfectly. The trouble is that in an elastic collision the objects that collide don’t stick together.
A load of buckshot hitting a vest tends to stick; hence, its collision with the victim is clearly inelastic, meaning that the kinetic energy of the victim will always be significantly less than the original kinetic energy of the buckshot. The “lost” kinetic energy is not really lost but rather altered in form. Some energy becomes a shock wave in the victim, creating bruises and possibly cracked ribs. Some energy converts immediately into heat. Predicting where all the kinetic energy goes and what it does is a daunting task, and so a kinetic-energy analysis can’t determine whether a victim will be blown backward.
A quantity called momentum, however, is surprisingly helpful in these calculations because it cannot change form. Momentum is a measure of how hard it is to stop an object. Conveniently, when an object, such as a bullet, collides with another object, such as a shooting victim, the bullet shares its momentum with the victim. In other words, the bullet’s momentum immediately before the bullet strikes must be equal to the momentum of the bullet and victim immediately after, regardless of what happens to the kinetic energy. The phenomenon is called the law of conservation of momentum and as noted in earlier chapters, any law with the word conservation in it is about as close to absolute truth as we humans can get.
Unfortunately, friction messes up conservation of momentum. So, how do we handle this pesky little force? We simply ignore it. Okay, it sounds pretty sloppy, but in reality it isn’t. First, if the analysis is made immediately after the collision, friction will not have had enough time to mess things up significantly. Second, if the victim is blown off his feet, there will be little friction force present while he’s flying through the air. Third, if a friction-free calculation says a victim will not be blown violently backward, then accounting for friction will make the event even less likely: the friction force would resist backward motion.
ANALYSIS OF THE BACKWARD MOTION OF A SHOOTING VICTIM
We’ll assume there’s no friction to impede the backward motion of the victim. This would favor the event’s occurrence. To calculate the momentum of an object, we use the following equation:
p = mv (EQUATION 12.1)
Where:
p = momentum
m = mass
v = velocity
Before the buckshot collides with the victim, the victim’s momentum is zero, since he’s not moving. This means that we only have to consider the forward momentum of the buckshot. For simplicity we’ll treat the buckshot as though it’s a single object rather than calculating individual momentums for each pellet and adding them together. Both methods give the same result.
After the collision, the victim and buckshot stick together and so, again, we only have to calculate the momentum of their combined mass. We’ll assume that the combined mass of buckshot and detective is MD = 80 kilograms, and the momentum after the collision as PD even though it’s really the momentum of both the detective and buckshot stuck together. From conservation of momentum:
or
Substitution yields:
Note that the velocity of the detective is proportional to the ratio of the buckshot’s mass to the detective’s mass. This ratio is going to be tiny.
If we run a momentum analysis (see “Analysis of the Backward Motion of a Shooting Victim”) on the hapless Sergeant Riggs, we find he’s blasted backward at the momentous speed of 0.4 miles per hour. Keep in mind that humans can walk briskly at about 4 miles per hour. Since the analysis was done with assumptions that favor being blown backward, it’s clear that not just Sergeant Riggs but shooting victims in general aren’t going to be blown backward by the force of a shotgun blast.
Here’s another way to analyze the situation: apply the law of conservation of momentum to the shooter similar to the way it was applied to the victim. In other words, recoil from firing a weapon will give a shooter backward momentum equal to the forward momentum of the buckshot and hot gasses (from burning gun powder) exiting the shotgun’s barrel. (Note: buckshot will also include a light-weight, fibrous wad placed between the powder and buckshot.) Unless the muzzle of the shotgun is pressed against the victim, he will receive only the buckshot’s momentum. The magnitude of the victim’s backward momentum will be less than the magnitude of the shooter’s because the victim will not be hit by the hot gasses propelling the buckshot out of the gun barrel
. Also, thanks to air resistance, the buckshot will be moving slower and have less momentum than when it first exited the gun. If the recoil momentum from discharging a firearm doesn’t throw the shooter backward through the nearest window, then certainly the buckshot’s momentum won’t.
Sometimes Newton’s third law is incorrectly evoked as an explanation for why a shooting victim is not blown violently backwards. These explanations claim that the recoil force acting on the shooter and the buckshot’s force acting on the victim are an action-reaction pair. To qualify as an action-reaction pair the two forces must
Occur simultaneously
Be equal in magnitude
Be opposite in direction
The recoil force and the buckshot’s force fail the first two requirements. The recoil force begins as soon as the buckshot starts moving down the gun barrel. Recoil force is smaller, although it lasts a longer time than the buckshot’s force. The buckshot’s force does not happen until the buckshot hits its target. The force is very brief but also very large in magnitude compared to the recoil force. There’s no way that the force the buckshot creates on the victim and the recoil force acting on the shooter can possibly be an action-reaction pair.
There is one other possible explanation for a victim being blown backward through a window: involuntary muscle contraction. The victim could be so stunned by being shot that he involuntarily tenses his muscles, causing him to jump backward. But it’s nearly impossible to jump from a standing position without first bending the knees, which would require one to momentarily relax the leg muscles. Even after bending the knees, it’s normally not possible to jump backward by more than 2 or 3 feet (about 1 m). Such a puny jump does not even come close to the distances traveled by Hollywood shooting victims.
RAIL-GUN RECOIL
“They said the physics was impossible” (and it is), yet two bad guys lurk in the shadows watching through impossible x-ray vision scopes mounted on impossible rail-guns as they zero in on their prey, Lee Cullen (Vanessa Williams) and John Kruger (Arnold Schwarzenegger) in the 1996 movie Eraser [RP].The shiny new rail-guns come equipped with LED indicator lights on their sides and green beams of light emitted from their scopes— perfect for those stealthy sniper missions where one must not be seen. When their victims attempt to flee, the assassins send aluminum bullets zinging through the walls at nearly the speed of light, narrowly missing their targets who take refuge behind a refrigerator.
Are the assassins’ scopes out of whack? Did they jerk their triggers? Unlike ordinary bullets, bullets traveling near the speed of light would arrive on target almost the instant they were fired. There would be no time for wind, gravity, or motion of the target to create differences between the point of aim and the point of impact. The assassins are within 100 yards (91 m) of their targets. At these distances the misalignment caused by jerking the trigger or the inaccuracies from shooting a shoddy rifle would not usually cause a shooter to completely miss a human-sized target. Even if we imagine that the rail-gun physics make sense, the scene does not.
The rail-gun physics also makes no sense. First, there’s the issue of recoil. Yes, even rail-guns would have recoil. It’s not necessary to have exploding gunpowder for recoil. The hot gasses from burning gunpowder propelling a normal bullet do add to the recoil, but these gasses exit at about 1.5 times the velocity of the bullet and have far less mass. In a high-powered rifle cartridge such as the 7.62 NATO, the mass of the gunpowder is less than one-third the mass of the bullet. Propelling the bullet at the same speed without using gunpowder would reduce the recoil momentum by about one-third—a significant amount.With a handgun cartridge like the .45-caliber ACP, the gunpowder is only about 4 percent of the mass of the bullet. Here, propelling the bullet without gunpowder would make little difference in recoil. Recoil is caused by sending highvelocity mass out the end of the gun barrel. It does not matter if the mass is a bullet or the gasses from burning gun powder. Likewise, it does not matter whether the force propelling the bullet is from gas pressure or an electromagnetic field.
The big difference in Eraser’s rail-gun recoil comes from the claim that the bullet travels at nearly the speed of light. Einstein’s theory of relativity must be used to calculate the momentum of the bullet at such speeds, making the bullet’s momentum far higher than at lower speeds.
If we assume the aluminum bullet exits at 90 percent of the speed of light and weighs 0.26 grams or one-tenth as much as a .22-caliber rim-fire bullet, the backward velocity of the shooter (mass = 100 kg) must be a whopping 1,610 meters per second— about 4.7 times the speed of sound in air—for his momentum to be the same size as the forward momentum of the bullet. Obviously, such recoil is going to impart more than a sore shoulder. It’s going to be fatal.
CALCULATING RECOIL WITH BULLET VELOCITIES NEAR THE SPEED OF LIGHT
According to Einstein, when the speed of an object approaches the speed of light, the object’s momentum is calculated using the Lorentz factor as follows:
Where:
p = momentum
γ = Lorentz factor
m0 = mass at rest
u = velocity
Where:
c = speed of light
For the bullet in Eraser:
Substituting into equation 12.2 yields:
From conservation of momentum, the backwards momentum of the shooter (ps) must be equal to the forward momentum of the bullet as follows:
or
The bullet from the magical rail-gun in Eraser would also have an unmanageably high kinetic energy—equivalent to several kilotons of TNT. (Note again that Einstein’s equations must be used for calculating the kinetic energy.) The bullet is not going to zing through the walls. It’s going to demolish the walls, along with the assassins outside and everything else in the immediate vicinity. At distances under 100 yards, the shooter can have his choice: death by the rail-gun’s recoil or death by the bullet’s impact with the target. At long distance it’s only death by recoil.
An aluminum bullet is not magnetic—this point is often cited as a rail-gun physics flaw, but in fact it is not. Such a bullet only has to be conductive. Roughly speaking, a rail-gun works by passing a large current through the bullet from one rail to the other. In the process, a strong magnetic field is generated between the rails at a right angle to the current passing through the bullet. This creates a force acting on the bullet in the direction of the barrel points. Since the bullet is free to move, it accelerates down the barrel and exits as a projectile. The key problem with aluminum is its somewhat-low melting temperature. The electrical current used for acceleration can vaporize the bullet. If the aluminum bullet does make it out the end of the barrel, heat generated at ultra-high velocity by air resistance can also vaporize it or cause it to combust when in contact with air.
Excessive recoil problems can be solved and excessive kinetic energy problems reduced by dramatically lowering bullet mass. But even if the bullet’s mass in Eraser is reduced to 1/10,000 the size of a .22-caliber rim-fire bullet, the kinetic energy is still going to be the equivalent of over 1,000 pounds of TNT. An aluminum bullet smaller than a speck of dust going at 90 percent of the speed of light will overheat from air resistance and disintegrate the instant it leaves the rail-gun (assuming it doesn’t vaporize in the barrel). As it disintegrates, the bullet’s kinetic energy will be converted into heat, causing a rather nasty explosion a couple of feet (less than a meter) in front of the shooter’s face. There’s just not a happy ending available for this device.
In Eraser, the rail-gun bullet has huge amounts of momentum that hurls victims horizontally backward great distances. But even here the depiction is bogus. As shown in the movie, the rail-gun bullets are capable of zipping through walls. With such penetration, a human victim would offer almost no resistance to slow the projectile’s velocity. Since the bullet has almost exactly the same momentum when it exits the victim, conservation of momentum would still be satisfied even if the victim quietly drops to the floor where he or
she is standing. The force of a bullet striking its victim occurs in an extremely short period of time, so it has to be extremely high if it is going to send a person flying backward across an entire room. Such high forces would more likely tear the person apart. Getting hit by a high-energy projectile such as a .50-caliber machine-gun bullet has a gruesome effect on people. It can blow a person in half. It’s hard to imagine that a far more energetic projectile from a rail-gun would only throw a person backward.
So how is a victim actually blown backward through windows and across rooms in movies without injury? The actor wears a specially designed harness hidden under his shirt with a rope or wire attached to the back. He is then pulled slightly upward and backward through the window or across the room. The force required to do this is fairly low because it is applied continuously as the person flies backward.
THE COSMIC TOYOTA
It’s time to blast off. The hero has been visiting a planet in a distant galaxy (for some heroic purpose) and decides to leave. It’s clear that this is an Earth-like planet because the hero can breathe and walk around in just a jumpsuit. The gravity is comparable to Earth and, of course, the planet’s higher life forms all speak English. He steps into his spacecraft—about the size of a Toyota—fires the thrusters, and zips off into the cosmos.
