Mass, Momentum, and
the Martial Arts
“Ooooh…You think you’re so cool, with your
karate and your childlike reflexes.”
—Roy O’Bannon, SHANGHAI KNIGHTS
Buffy butts heads in “Phases” (B-2) with a chauvinistic, rogue werewolf hunter who snidely derides her physical abilities. The boys at Buffy’s high school who’ve had the misfortune to tangle with her know better, as do the various and sundry vampires and demons who cross her path. Buffy might be a little blond girl who looks like she could barely lift a crossbow, let alone fire one with deadly accuracy, but she is far from the stereotypical damsel in distress, and therein lies a large part of her appeal.
The plight of the Slayer is a poignant one, marked by loneliness, isolation, constant violence, and innumerable injuries, not to mention a sharply abbreviated life span: Most Slayers do not live past the age of twenty. But there are some consolations, not the least of which is a vast source of mystical Slayer strength and preternatural agility. But Buffy does not rely solely on her mystical gifts. She’s a well-trained fighter who logs relentless hours of physically grueling training to ensure that she’s in peak form when she encounters a vampire or demon. Small wonder, then, that the martial arts constitute a major component of every single episode in both the Buffy and Angel series. And all martial arts, at heart, are about effectively exploiting basic physics principles.
FINDING THE CENTER
In “What’s My Line?” (B-2), Buffy finds herself battling a series of assassins Spike has hired to kill her—or, at the very least, to keep her occupied so she can’t foil his nefarious plans. She makes quick work of the first assassin, but then finds herself trading blows with a young girl as strong and agile as she is. It soon becomes apparent that the girl is not one of the assassins. Instead, she turns out to be a second Slayer, named Kendra, who has mistaken Buffy for a vampire.
Both Slayers instinctively drop into traditional fighting stances when Buffy calls a time-out to assess this unprecedented situation. Their feet are spaced roughly shoulder-width apart to form a solid base, with one foot slightly forward, and their respective bodies are more or less erect, fists chambered—resting, just like cocking the trigger of a gun, on each Slayer’s hips in preparation for a strike should the other make a sudden aggressive move. Any well-trained Slayer could tell you that the key to a powerful martial-arts technique is a strong fighting stance that provides good control of her balance. It has to do with a person’s center of mass (also known as the center of gravity), typically located about one inch below the navel. Practitioners of Japanese martial arts call this area the hara, and consider it the seat of a person’s power.
From the perspective of physics, this isn’t far off the mark. Where Buffy’s center of mass happens to fall is directly related to how stable her body position is, which is in turn related to how much force she can generate with her techniques. She achieves the most stability when her center of mass is located vertically over the center of her base, because her weight is distributed evenly on both legs. This makes it much harder to unbalance her. If her feet are too close together, her base is significantly narrower, and Kendra can more easily knock her off balance. Similarly, if Buffy bends at the waist as she punches Kendra, she reduces her own stability and makes herself vulnerable to a counterstrike by essentially leading with her own face. The importance of body stability is even more obvious when Buffy fights Angelus—Angel’s evil alter ego—while she has the flu in “Killed by Death” (B-2). The high fever throws her off her game, and she swings wildly while punching, throwing herself off balance sufficiently for Angelus to briefly gain the upper hand.
Balance and stability are just two of the physics principles underlying even the most basic martial arts techniques. Let’s consider what’s involved if Buffy wants to throw a simple straight punch at Kendra. First, she must overcome her fist’s inertia by exerting some kind of force. According to Newton’s first law of motion, an object in motion tends to stay in motion, while an object at rest tends to remain at rest, unless an outside force intervenes. Every object—Buffy’s fist, Kendra’s body—has a given mass, and this mass determines its inertia: how much force is required to set an object in motion. Various muscles in the body must work together to produce movement. Energy is stored as a muscle contracts, and that energy is then transferred to another muscle when the contraction is relaxed, and so on. When Buffy throws a straight jab at Kendra’s face, she generates a force through a series of muscular contractions that causes her fist to accelerate. To phrase it another way, when Buffy chambers her fist, her muscles contract. She is storing potential energy, which is converted into the kinetic energy of motion as she relaxes those muscles to execute the punch.
How much force is generated depends in part upon Newton’s second law of motion: Force equals mass times acceleration (F = ma). The more force you apply to an object, the greater the rate of acceleration. So the harder Buffy throws her punch, the more her fist will accelerate. This is where proper stance comes into play. It’s less about her overall body mass, and more about how much of it she can involve in her strike. Buffy’s center of mass is located in her hip and abdominal region, which accounts for roughly one-third of her body weight. So more of her body mass is behind her punch when Buffy is stabilized than when she bends at the waist. And more mass translates into more force to accelerate her punch.
Buffy can increase the force behind her punch even further by stepping in, or snapping her hips forward as she punches. She might stun or disorient Kendra with short jabs to the face, thus buying herself time while the other Slayer is off-guard. But she can actually move Kendra backward if she puts as much of her body mass as possible behind her strike. There are other ways Buffy can exploit her body mass. She can bend her knees and sink into a low stance, storing potential energy like a coiled spring, and then quickly convert that into kinetic energy by thrusting her body upward to strike her target. Or she can lift her weight and then drop her entire body onto her doubled-over opponent to deliver the blow, getting a little extra help from the earth’s gravity to achieve a greater amount of force. In fact, at one point in her exchange with Kendra, Buffy leaps onto a nearby table before executing a kick, gaining just enough extra height (and potential energy) to put a bit more “oomph” behind her kick as she jumps off.
Of course, Buffy also has an extra dose of mystical energy at her disposal: her Slayer strength. Since energy and mass are equivalent, as we saw in the previous chapter, she can generate much more force than she would otherwise be able to generate using her mass alone. This comes in handy against weaker human opponents, but it won’t help her as much against Kendra, who has the same infusion of Slayer strength, or against the myriad of supernaturally strong demons she encounters on her nocturnal patrols. It just gives her a fighting chance.
The second element in Newton’s equation is acceleration. It’s a tricky concept, best described as how much velocity (an object’s speed and direction) changes over time. The mass of Buffy’s fist and its velocity combine to produce momentum. The more momentum that builds up as her fist accelerates, the faster its final velocity, and the more energy she can transfer into Kendra when she strikes. When Buffy’s fist is chambered, its velocity is zero. As she executes her punch, her fist accelerates, gaining momentum as its potential energy is converted into kinetic energy. Since kinetic energy increases with the velocity squared, this means that if Buffy’s fist is traveling twice as fast by the time she hits her target, it will have four times the kinetic energy, and she can transfer that much more energy into Kendra’s body.
How long it takes to transfer momentum is also critical. Newton’s third law says that momentum is conserved: It can neither be created from nothing, nor destroyed, but is passed from one object (Buffy’s fist) to another (Kendra’s body). We can combine this with the concept of work, which can be calculated by multiplying the amount of force generated by the distance Buffy’s fist must travel—and the time it takes to do so. There is a fixed amount of energy at Buffy’s disposal, so she has to make a choice: Either she can transfer a small amount of force continuously over a longer period of time, or she can transfer a large amount of force in the shortest possible time. Either way, the total amount of energy delivered to Kendra’s body remains the same, but unless Buffy plans a slow, prolonged siege against Kendra, she will try to transfer as much force as possible, in as short a time as possible, by executing her punch as quickly as she can.
The opposite applies to how Buffy lands when she somersaults over Kendra to avoid being struck by an ax. Here, the objective is to lengthen the time of impact. If Buffy locks her knees as she lands, her momentum drops to zero suddenly, and she feels a large force in her legs, possibly more than the bones and joints can handle. But if she lands with her knees bent and then rolls forward, she prolongs the time in which the impact takes place, so her momentum decreases more gradually, and the force of impact is smaller.
These same basic concepts apply to kicks. When Buffy throws a front kick at Kendra, she propels her body forward by pushing off her rear leg and bringing her forward leg into chamber position. The leg muscles contract and store potential energy. That potential energy is then converted into kinetic energy as Buffy executes the kick. The muscles contract and then relax, transferring the stored energy to the next muscle required to perform the kick. Buffy can generate even more force by pushing her hips forward, using her center of gravity to put more of her body mass behind the kick. It also slightly increases the distance her foot must travel before it hits the target—a seemingly small detail that can nonetheless translate into a stronger kick, since her foot accelerates more, and hence deposits more energy into her target.
What is happening at the point of impact, from the perspective of the target? If momentum is conserved in Newton’s third law, then for every action there is an equal and opposite reaction. That is, if one object exerts a force on another for a given amount of time, the second object reacts by exerting an equal but opposite force for the same amount of time. The force generated by Buffy’s punch creates a reaction force in the opposite direction when she hits Kendra. Kendra’s body gains exactly the amount of momentum that Buffy’s fist loses, barring any that is lost through conversion into other kinds of energy, like heat or noise.
Without a strong stance, Buffy’s body might be pushed backward by the impact, making her punch much less effective. That’s because part of the total energy is diverted from the target. If more of that energy is diverted than is deposited into Kendra’s body—or if Kendra adds her own countering energy by stepping in to block Buffy’s punch—Buffy may actually be forced backward. In fact, in “Revelations” (B-3), Buffy hurls herself into a double-leg flying kick at the demon Lagos. Lagos simply has too much body mass, and hence reflects the momentum in Buffy’s kick, so she bounces right off him, to comical effect.
Newton’s law of equal and opposite reaction applies not just to human or demonic bodies, but to every object—even the ground beneath our feet. Kung fu master Bruce Lee once famously observed, “Boards don’t hit back.” But in the strictest physics sense, they do. When a karate practitioner breaks a wooden board with his fist, he transfers momentum to the board, which accelerates in the opposite direction in response. If the part of the board that is hit—usually the center—is infused with more energy than its structure can handle, it will crack or break. A similar effect can be seen in “Smashed” (B-6), during an extended fight sequence between Buffy and a semireformed Spike, who have been inching toward consummating their long-standing love/hate relationship. The two former enemies trade blows and sarcastic taunts, tossing each other through the decrepit doors and into the sagging walls of an abandoned building nearby. The scene culminates in a spot of violent lovemaking that literally brings the house down around them.
Why doesn’t Spike simply bounce off when Buffy hurls him against a wall? After all, that’s what happens when Spike tosses her into a wall during their ferocious foreplay. But when Spike hits, the wall cracks instead. This has to do in part with their difference in mass. Just like Buffy’s fist executing a punch, when a body is in motion, it builds up a certain amount of momentum based on the body’s mass and velocity. Since Spike has more mass than Buffy, he gains more momentum, and thus more energy is transferred to the wall when he hits it.
Let’s assume that Spike weighs 140 pounds and is traveling at a final velocity of 10 mph when he hits the wall. At that speed, his body would have about 504 joules of energy, equivalent to how much energy it would have if Spike just ran at the wall as fast as he could. That’s actually not much energy, and were the wall made of brick, it’s more likely that Spike would break a bone or two, rather than cracking the wall. In contrast, a two-ton car traveling at 10 mph upon impact would easily crack a brick wall. Fortunately the wall in question seems to be made of wood and plaster, and is in serious disrepair.
But why, exactly, does the wall crack? The wall accelerates in response to the added momentum from Spike’s body, producing an equal and opposite reaction. But it doesn’t accelerate uniformly. The areas that took the brunt of Spike’s body accelerate much more than the surrounding wall, producing a localized strain. The fact that the walls are held in place by joints, foundation, and various supporting structures increases the strain, which eventually becomes too great. The effects are cumulative. The repeated infusions of energy brought about by Buffy and Spike’s amorous thrashings create so much strain on the entire building structure—already in an advanced state of decay—that it begins to collapse around them. Rafters and debris fall to the floor, cracking the floorboards sufficiently so that when Buffy and Spike fall backward in an embrace, they crash through and fall into the basement.
What about objects made of something other than wood and plaster? Different materials can withstand different amounts of deformation, a property known as elasticity. Most materials are elastic to some degree: when they are deformed or bent by an infusion of incoming energy, they will bounce back to their original shape. But elastic materials all have their limits. Metal springs and rubber bands are very elastic. Plaster and glass are not very elastic; instead, they are brittle, and snap with even a small deformation. Energy, like momentum, is conserved, but in a collision, it can turn into different forms of energy, such as heat or noise. How much of the energy is converted depends in part on both the relative toughness and elasticity of the materials involved in the impact. There is no such thing as a perfectly elastic collision, but if there were, all of the energy would be transferred to the target with nothing lost to heat or noise, for example.
The übervamp (Turok-han) sent by the First to take out the remaining Potential Slayers in “Bring On the Night” (B-7) has an unusual natural body armor protecting its heart, on a par with a Kevlar vest, rather than the flesh-and-bone breastplate of the average vampire. That body armor is both extremely tough and highly elastic. When Buffy drives a stake into the Turok-han’s chest, it becomes embedded, even though it clearly hasn’t pierced through to the creature’s heart. When the creature pulls out the stake, the body armor repairs itself. It rebounds from the impact and flows back into place, with no permanent rip or even an indentation where the stake hit. The same thing happens when Buffy shoots it with a crossbow in “Showtime” (B-7): The stake embeds itself, and the underlying breastplate is not pierced.
Anya compares the body armor to steel in “Empty Places” (B-7), but while the breastplate might share some properties with steel, the substance covering it has more of a doughy texture. The übervamp is like a very tough Pillsbury Doughboy or, to borrow Buffy’s labored metaphor for herself in “Chosen” (B-7), cookie dough that isn’t yet fully baked. Such a substance is considered “viscoelastic,” making it very difficult to pierce. It easily absorbs the impact of the incoming stake, flowing around the intruding object so that the stake becomes embedded before it can penetrate the underlying breastplate and do any fatal damage. In fact, it appears to be a subset of viscoelastic material known as an “anelastic” solid: The substance recovers its former shape fully after the offending stake is removed.
Back in the real world, scientists at the University of Delaware have adopted an equally novel approach to improving Kevlar vests. They treat the fabric with a “shear-thickening fluid”—basically a syrupy, viscous mixture (similar to hagfish slime) consisting of silica particles suspended in polyethylene glycol—making it strong enough to stop a bullet, yet flexible enough to wear comfortably. This turns the fabric into a kind of “smart material,” a class of materials that can sense and respond to changes in the environment—usually the presence of electric or magnetic fields, or changes in temperature.
In this case, the material responds to a change in pressure resulting from mechanical force. Under normal conditions, its molecules are weakly bonded and can move around with ease; that’s why the material is so flexible. But the shock of an impact causes those chemical bonds to strengthen so the molecules lock into place; once the force from the impact dissipates, the bonds weaken again. That’s why the fabric becomes rigid instantly when a bullet strikes, thereby preventing that bullet from penetrating, and reverts to its more flexible state once that force has ceased. A British company called d3o Labs manufactures “smart armor” using the same kind of material; it was used in the 2006 Winter Olympics to protect U.S. and Canadian skiers from injury.
Despite the ingenious design of its body armor, an übervamp can be staked if one stabs the region in just the right way so that the viscoelastic surface ruptures, enabling the wooden stake to pass through the breastplate to the heart—assuming that the stake doesn’t splinter upon impact with the steely breastplate. Buffy simply needs to transfer much more mechanical energy than usual in order to penetrate both protective layers. Physics can help her a little in this instance, because the striking target is so small and focused. The same amount of energy focused into a smaller area (just over the heart) delivers more energy per unit area. A larger target surface (the übervamp’s entire chest) would disperse the energy over the whole area and weaken the force of Buffy’s strike.
Buffy can increase her chances of penetration even further by choosing to whittle her stakes out of the right kind of wood. Any woodworker could tell you that certain woods are harder, and split less easily than others. Pine, for example, is quite soft, with broad grains, which is why pine boards are used so frequently in martial arts breaking demonstrations. Maple and ash are denser, with very fine grains, and don’t break very easily, which is why most bokken (long staffs used as weapons) in Japan are made out of ash. A stake carved out of ash would be less likely to splinter on impact with the bony breastplate of an ordinary vampire. Fortunately for the Scoobies’ stake-whittling needs, ash is among the species of trees common to the Southern California region, so the wood should be in plentiful supply near Sunnydale.
There’s another, less-well-known property of ash that makes it an excellent choice for staking the much-tougher übervamps. Since the 1880s, scientists have known that quartz crystals will produce a tiny voltage (an electrical field) when squeezed or pressed. They are another example of “smart” materials. This property is known as piezoelectricity, and it is an integral component of many modern technologies from sonar, radio, and television to the electric cigarette lighters found in cars and the portable sparkers used to light gas grills and stoves. The piezoelectric process also works in reverse. Applying an electrical field to quartz crystals will cause the crystals to deform ever so slightly, about one-billionth of an inch.
Piezoelectricity is not limited to crystals, either. While they would technically not be classified as smart materials, rubber, wool, hair, silk, bone, and certain kinds of wood nonetheless all exhibit (to a lesser extent) some piezoelectric properties: mechanical force, or stress, will produce varying amounts of voltage. It just so happens that ash is among those types of wood. A stake made of ash could conceivably spark-cut a hole upon impact, creating a breach in the Turok-han’s tough-yet-doughy body armor and allowing the stake to pierce the steely breastplate and hit the übervamp’s heart.
This might explain one of those pesky inconsistencies one encounters occasionally in the Buffyverse. The first time Buffy tries to stake an übervamp, she fails to penetrate the breastplate sufficiently to pierce the creature’s heart, yet by the series finale, every one of her newly activated Slayers is able to reduce übervamps to dust with a stake (although these are not the only weapons at their disposal). Buffy could have used a softer birch stake—which has a much smaller piezeoelectric effect, and hence produces a smaller voltage—in the first instance, then switched to ash for the final battle against the First’s vicious horde, in hopes that ash’s stronger structure and innate piezoelectric properties could even the odds a bit.
PUTTING A SPIN ON IT
What’s My Line?” (B-2) concludes with Buffy and Kendra joining forces against Spike and his minions, battling it out in an abandoned church. Making good use of the sacred objects at hand, Buffy takes out Spike by swinging a heavy metal incense burner on a chain (called a censer) around her head, like a lasso, then releasing it into the air. The censer hits the back of Spike’s head with sufficient force to knock him out cold. This is a classic example of the physics of circular motion. It occurs when an object like the incense burner rotates around a vertical axis at the circle’s center, such as Buffy’s body.
It’s a common misconception that what’s at work here is centrifugal force, but in reality such a force doesn’t exist. What keeps the chain taut as Buffy swings it overhead is the mass of the incense burner at the end of it, combined with the chain pulling the burner toward the center of the circle. This is known as a centripetal force. Without the chain pulling it inward to counter its natural inertia, the burner would simply fly through the air until gravity and friction from air resistance—or a collision with Spike’s bleached-blond head—caused it to fall to the ground.
The principles behind circular motion are similar to those for linear motion. First, there is rotational inertia: an object’s mass determines how much force is required to get it moving in a circle, or to change the motion of an object that is already rotating. Then there is an object’s angular momentum, which is determined by multiplying how much force is required to change an object’s motion (rotational inertia) by the rate at which it turns (angular velocity), for example, the number of revolutions per second. It is analogous to how mass and velocity combine to produce linear momentum.
Let’s apply these concepts to the battle in the abandoned church. When Buffy begins whirling the incense burner around her head, she is applying a force to set the censer in motion. The now-rotating censer has angular momentum. Angular momentum is conserved—that is, remains constant—unless an outside force acts on the rotating object. In this instance, the censer’s angular momentum increases because Buffy’s effort is adding energy to its rotation, causing it to spin faster and faster. All that pent-up energy is released quite suddenly as she tosses the censer at Spike. As with linear motion, the more it accelerates, the greater the final angular velocity of the censer when Buffy releases it, and the more angular momentum it possesses. The resulting energy is sufficient to send the censer flying through the air with enough force to knock Spike out, since a large part of that energy is transferred to his head.
Before she comes up with this ingenious method of taking out her opponent, Buffy throws a good number of punches and kicks at Spike, including a spinning back kick. Most spinning techniques in the martial arts rely on the same basic principles of circular motion. First, Buffy must generate sufficient force to overcome her leg’s inertia. The force that is applied to a rigid object, like Buffy’s leg, is called torque. It is simply a rotational (twisting) action, such as the force one would apply to tighten a bolt with a wrench. Buffy must press against the ground to start the rotation for the kick, pushing off hard to exert enough torque to propel her body around. In doing so, she transfers energy to the ground, which pushes back in response. She can generate even more initial torque by twisting her body in the opposite direction before throwing the kick. She literally “winds up” to produce more torque, much like an ice skater does before executing a jump. (On the downside, this will “telegraph” her intentions to her opponent, robbing her of the element of surprise.)
Once Buffy’s body is in motion, her angular momentum remains constant. Whatever energy she has managed to generate with her torque can’t be increased any further. But there is a way to increase the angular velocity (spin rate) of her kick. Unlike plain old inertia, Buffy can change the speed of a spinning kick, simply by changing how her mass is distributed. The farther her foot is from the center of the pivot point—for instance, if her leg is fully extended horizontally as she kicks—the more rotational inertia it has, and the more force is required to start or keep it moving. The closer to the center, the less rotational inertia her foot possesses. This also applies to the spinning incense burner: Buffy can increase the rate of its spin simply by choking up on the chain to make the circle smaller. The censer’s mass is now closer to the circle’s center than before, and therefore spins faster.
That’s why Buffy “coils up” as she spins into the kick, bringing her arms tight into her body and chambering her leg—again, just like an ice skater executing a jump. Buffy only opens up when she strikes. A closed position, with the arms and legs pulled in tight against the body, decreases her angular inertia and increases rotation speed, while her angular momentum remains constant. So Buffy’s angular velocity must increase to balance things out: She spins faster to make up for the lessening of rotational inertia. On the other hand, an open position, in which the arms and legs are allowed to swing away from the body, causes the speed of rotation to decrease. All that energy of motion is released when Buffy uncoils and throws out her leg for the actual kick. Because her foot has more distance and time to accelerate than with a linear front kick, it reaches a greater final speed at the moment of impact. And once again, the greater the final velocity, the more energy Buffy can transfer into Spike’s body when she kicks.
BALANCING ACT
As welcome as the sight of a high school self-defense class might be in “Phases” (B-2), it’s distressing that the gym instructor fails to correctly teach a basic hip throw. In fact, she inadvertently provides a useful lesson in how not to throw one’s opponent. When the school’s resident leering would-be Lothario, Larry, grabs Buffy from behind to simulate an attack, the gym teacher intones, “Bend over, using your back and shoulders to flip your assailant to the ground.” Not surprisingly, only Buffy is able to do so, flipping her much larger attacker over to the ground quite easily, thanks to her Slayer strength—plus a little extra incentive to violence when Larry lasciviously grabs her derriere.
What’s so bad about using your back and shoulders when executing a hip throw? Nothing, if you’ve got Larry’s linebacker build, or Buffy’s Slayer strength. Most high school girls lack these crucial attributes, however, and would probably hurt themselves if they tried to throw an attacker the way the gym teacher taught the technique. But with the aid of basic physics, any one of the teenage girls in the class can use an attacker’s size and strength against him to toss him to the ground.
Buffy’s gym teacher clearly didn’t understand that just like punches and kicks, the true power behind any hip throw comes not from the back and shoulders, but from a person’s center of mass: the point where gravity acts on the body as a whole. Think of the seesaw in Sunnydale’s playground. It has a triangular base that acts as a fulcrum, while the long board lying across it acts as a lever, pivoting back and forth above the fulcrum. The point at which the board is perfectly balanced atop the fulcrum is its center of mass: The lever is divided into two equal halves. Move that point ever so slightly in either direction, and one half of the lever becomes greater than the other half. The board will tip out of balance.
Balance is the essence of judo. Larry is balanced, and therefore stable, as long as his center of mass remains over his feet. If Buffy moves Larry’s center of mass outside of his stable base, gravity will make him lean or fall, and hence he will be easier to throw. This is where torque once again comes into play. A key element in determining torque is the length of the lever: the perpendicular distance from the pivot point (Buffy’s hip, Larry’s center of mass) to where the force is being applied. Once Buffy breaks her opponent’s balance forward, even if she does nothing more, gravity will pull on his center of mass. Since his center is now unstable, this gravitational pull creates a torque, causing rotational motion, and he can topple over all by himself. The more he is leaning forward, the longer a lever his body forms, and the greater the resulting torque. When Larry is standing upright, for example, his center of mass is stable, and his torque is therefore zero. If his center of mass is off balance, the resulting torque causes his body to rotate. Buffy can then reinforce this rotation by pulling on his arm to apply even more torque, while using her hip—placed just under Larry’s center of mass—to lift his feet off the ground.
The concept works even better if Larry is walking forward. Any kind of movement requires the body to be off-balance at some point. As Larry walks, his foot comes off the ground as he steps forward, and for that moment, he is off-balance—technically, he’s falling. If Buffy turns her body in toward Larry quickly as he walks, and fits her own hip just under his center of mass, she can redirect the energy from his forward movement, easily lifting his feet off the ground, then pulling him over and around, using her hip as the pivot point around which Larry’s body, as the lever, will rotate. The faster Larry is moving toward her, the more forward momentum he gains, and the easier it is for Buffy to throw him. In essence, she uses his own energy against him.
When Buffy initially tries to throw Larry in her self-defense class, he is stationary, so she can’t take advantage of any forward momentum he might generate by walking toward her. And his center of mass is not leaning out far enough for him to be sufficiently off-balance. His body mass is large enough to oppose the torque she is trying to apply by pulling with her back and shoulders. Buffy overcomes this through sheer Slayer strength. That might work for her against a human—powerfully built though he is, Larry’s strength is limited by his mortality—but she is unlikely to be able to throw Angel or even the lithesome Spike in this way. They have more body mass than she does, plus the same extra infusion of mystical strength to put behind it. The larger and stronger her opponent, the more force will be required to move him. That’s why good technique is so critical.
The good news for the diminutive Slayer is that hip throws generally work best against a taller opponent like Angel, because it’s much easier for her to get under his center of mass—assuming that she can catch him off-balance and execute the throw before he has time to counter it. (Angel can counter the hip throw simply by bending his knees to lower his center of mass, “planting” his feet into a solid wide base, and leaning slightly back to counter Buffy’s forward pull.) Bending her knees slightly as she turns her hip in toward Angel stores potential energy, which is then converted into kinetic energy when she straightens them. This extra burst of energy is just enough to set Angel’s body in motion for the throw. Once again, mass and velocity are the critical factors in determining how hard Angel falls. The faster Buffy throws Angel, or the heavier he is, the greater the force of impact when he hits the ground.
As with striking techniques, proper stance is critical for throwing. Buffy should stand with her feet shoulder width apart so that she is stable when she begins to execute the throw. But once the throw is in progress, her feet should be as close together as possible, heels together and toes pointing outward to form a V. She should become a human fulcrum. It might feel stronger to be standing in a wider stance—and indeed it is, if Buffy’s sole purpose is to maintain her own balance—but once the 200-pound Larry is hoisted onto her back, any space between Buffy’s legs provides a direct path to the ground via gravity’s pull. Her knees could easily buckle under the extra weight.
Buffy can always choose to let gravity do all of the work for her in a fight. This is how Illyria deals with Angel in “Shells” (A-5). As Angel lunges for her, Illyria sidesteps nimbly out of his line of attack, grabbing his jacket and continuing his forward momentum by tossing him through a plate-glass window. It’s not an actual throw, per se, but she does employ the judo principle of using her opponent’s momentum against him, and basic physics does the rest. Once through the window, Angel falls several stories to the pavement below. The higher the floor of Wolfram & Hart’s headquarters on which Angel is standing, the more potential energy he starts out with, the more momentum he gains as he falls, and the greater the resulting kinetic energy when he hits the ground—because he is falling over a greater distance, and for more time, than if he simply fell from the second floor.
All bodies fall at the same rate regardless of their relative mass (barring the presence of air friction), so Angel would speed up by 9.8 meters per second for each second that he falls. If Angel falls for five seconds, he will reach a speed of about 50 meters per second. He doesn’t fall quite that far, but even falling several stories, he would be traveling at a speed of 30 meters per second. When he hits the pavement below, the ground will respond with a counter force equal to the force Angel’s body exerted on it at the moment of impact. The force will transfer the energy of Angel’s fall into the pavement, minus any energy lost to heat and noise. If the energy transfer is large enough, Angel’s body might even crack the concrete pavement.
In this case, the energy would be about 40,500 joules, the same amount of energy contained in 1/50th of a stick of dynamite. That might not sound like much, but any human body falling that great a distance would be literally shattered by the impact. We can only assume that Angel’s vampiric powers allow him to survive the fall—that, and the fact that he lands in a forward break-fall position. Instead of one part of his body hitting first and bearing the full brunt of impact, his entire body lands horizontally, with his arms out to the side, palms down, and his face turned to one side. This disperses the impact over a wider surface area (and ensures that his nose isn’t crushed). Angel will still suffer broken bones, not to mention internal injuries, but these are trivial to an immortal vampire with a mystically rapid healing rate.
Hip throws aren’t the only weapon in judo’s impressive arsenal of techniques that have wormed their way into the Buffyverse. In “Lover’s Walk” (B-2), Spike returns to Sunnydale, only to be waylaid by a “committee” of vampire goons hired by Mayor Wilkins. An all-out brawl ensues. Amid the usual flurry of flying fists and feet, Spike deftly “clotheslines” an attacking vampire across the throat with his arm, throwing his opponent backward to the ground. As the vampire goon attacks, his entire body is moving forward at roughly the same rate. When Spike’s arm hits the goon’s throat, he exerts a force on the vampire that is far from its center of gravity. Specifically, Spike’s arm exerts a torque on the vampire that converts some of its forward motion into rotational motion. The goon’s feet fly out from underneath him while gravity does the rest, pulling his top half to the ground.
It’s similar to what happens to a seesaw if its pivot point is moved way off to one side. The seesaw is no longer balanced between equal halves, and the heavier end slams into the ground. Because all the energy comes from the goon’s forward momentum, Spike exerts almost none of his own energy. He generates just enough force to redirect the energy of his opponent’s attack. The more forcefully the other vampire hurls himself at Spike, the faster he accelerates, the more quickly he topples over, and the harder he hits the ground—and the less of Spike’s own energy is required to execute the technique.
SIZE MATTERS…SORT OF
The testosterone-ridden werewolf hunter who taunts Buffy in “Phases” (B-2) isn’t alone in his skepticism when it comes to a girl’s ability to fight. There’s an oft-repeated phrase in martial arts circles: “All things being equal, the bigger, stronger person will win.” That statement has some scientific merit. Someone who is heavier, like Angel, will be able to generate the same amount of force as a smaller person, like Buffy, using much less effort, even though both possess roughly equal amounts of technical proficiency and extra mystical strength.
The more force that acts on an object, the faster it will accelerate, yet the more massive an object is, the greater its inertia, and the more it will resist acceleration. If Buffy throws a punch at Angel, it won’t have as much impact as that same punch would have on the much smaller Kendra, because how much he can accelerate is inversely proportional to his mass. That is, it takes more energy to move him than it does to move the other Slayer. He has more mass than Kendra, and hence more inertia. Buffy can produce only a fixed amount of energy. She must transfer more of that energy into Angel’s body to move him than she would have to use against Kendra.
The only way to increase the force of Buffy’s punch is by increasing either her mass or the rate of acceleration. The average martial arts technique is executed in mere seconds. It’s physically impossible for Buffy to consume enough calories, or pump enough iron, to gain significant body mass over such a short period of time. So Buffy’s mass will be more or less constant, although she can gain some power by throwing more of her total body mass behind the technique.
Making matters worse, the faster an object accelerates, the more energy is required for it to keep accelerating, until it reaches a point where it can accelerate no more. Thus, although good technique can help increase the speed of Buffy’s punch, it can only do so to a certain degree. Sooner or later, she will reach the upper limit of a technique’s potential for acceleration, and she may still fall short of the final velocity necessary to overcome the substantial difference in mass between herself and Angel. Physics is innately unfair in this respect. Even in judo, which was designed to help the weak overcome the strong, size still matters when the competitors are equally skilled. Modern judo competitions are divided into weight classes for that very reason.
If that’s the case, why doesn’t Buffy simply bow to the inevitable when faced with an opponent who possesses superior strength: Glory, the exiled hell god, for example, or the misogynistic preacher-turned-serial-killer, Caleb, who derives his superhuman strength directly from the First? Anyone can do the math and determine that Buffy is destined to lose. Yet again and again, she prevails. When Buffy faces off against a former high school classmate-turned-vampire in “Conversations with Dead People” (B-7), he challenges her confident assertion that she will win their little death-match: “Two years of tae kwon do plus vampire strength—I think somebody’s counting their chickens.” While he is bigger and stronger, and has some formal training, Buffy has more practical combat experience, and hence knows something he doesn’t: All things are never equal. Size matters, but it’s not the whole story.
A popular physics joke describes how a farmer, a psychologist, and a physicist are asked for their advice on how to cajole a dairy cow into giving more milk. The farmer suggests revising the animal’s diet. The psychologist suggests painting the cow’s stall in soothing colors. And the physicist’s approach begins with the statement, “Assume the cow is a sphere…” Physicists will often make these kinds of simplifying assumptions. It makes their calculations a bit easier, and the results are sufficiently accurate for their limited scientific purposes. Nonetheless, it is an idealized construct. In the real world, a cow will never be a perfect sphere. And a street fight will never be fought under the same carefully controlled conditions as in a martial arts competition. There are no rules, no unspoken code of honor, and no mercy. In the Buffyverse, the combatants are truly trying to kill one another.
The Buffyverse, like the real world, is not a controlled environment. It is nonlinear and highly chaotic, and thus there are any number of other variables that can determine the outcome of a fight. For example, deprived temporarily of her Slayer powers on her eighteenth birthday (“Helpless,” B-3) in a sadistic coming-of-age acid test known as the Tento di Cruciamentum, Buffy must defeat an especially vicious vampire by relying on her wits, not her strength. (She tricks him into drinking holy water.) After being soundly beaten by Caleb the first time they trade blows (“Dirty Girls,” B-7), she switches tactics, using her speed and agility to dodge and roll until she gets hold of a mystical scythe that proves to be the equalizer between them (“End of Days,” B-7). She knows how to exploit the element of surprise, using the Buffybot as a decoy against Glory to catch the hell god unawares (“The Gift,” B-5). And she is resourceful in a pinch. Spike notes as much when studying a videotape of her battle with a nameless vampire in “Halloween” (B-2). Lacking a stake, she improvises, grabbing a nearby wooden fence post to reduce the vamp to dust.
The downside of all this unpredictability is that a single moment of hesitation, the smallest mistake, can cost the Slayer her life—and, sooner or later, the law of averages dictates that every Slayer will make that fatal error. As Spike fights an unnamed Slayer in turn-of-the-century China during the Boxer Rebellion in “Fool for Love” (B-5), the Slayer loses focus for one split second, right when she is about to stake him. That’s all it takes for him to gain the upper hand and sink his teeth into her neck. In fact, a Slayer’s margin for error decreases in inverse proportion to the difference in size and strength between her and her opponent. She will have less room for error against a vampire who outweighs her by 60 pounds than against one who outweighs her by 20 pounds. And sometimes a fight can be determined by something as intangible and unscientific as the relative mind-sets of the combatants. “It’s not about the moves, love,” Spike tells Buffy when she asks him to describe how he killed two Slayers. “Every Slayer has a death wish. Even you.”
Fortunately, Buffy’s alleged death wish is offset by her many positive attributes. She is smart, creative, willing to buck tradition and risk trying something daring and new instead of doing things the way they’ve always been done—an aspect of her personality that repeatedly brings her into conflict with the stodgy Watchers’ Council. She understands that it’s not always wise to play by the Council’s antiquated (and rather chauvinistic) rules, especially against demons who don’t adhere to human codes of conduct. She has the support of friends and family. And while she struggles with self-doubt as much as anybody, she doesn’t give in to defeatist thinking, even when she knows the odds are against her and she is likely to lose. Physics notwithstanding, that’s the hallmark of a true champion.
