The Mathematics of Hulk
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By Wil McCarthy
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The Incredible Hulk is a mysterious creature, to be sure. Part Mr. Hyde, part Romeo, part Running Man and part Paul Bunyan, this not-so-jolly green giant can't be sedated for analysis, and if there's a jail that can hold him, we haven't seen it yet.
Hulk is, to put it mildly, a strong fellow. Just how strong may remain a mystery; I suspect the answer will always be "exactly strong enough to meet the challenges of each new movie, comic book or TV episode." But we do know one extraordinary fact about Bruce Banner's viridian alter ego: his standing broad jump covers an incredible three miles, more than 1,200 times the human world record. At first glance, this would appear to make him 1,200 times stronger than an Olympic-grade human, but in fact there's a bit more to it than that.
The mathematics of Hulk begin with a figure familiar from high-school physics and junior-high algebra: the parabola. This is the path objects follow as they rise and fall through the gravity field of the Earth, and to make use of it we'll start with the time-honored tradition of converting everything to metric units, which are much easier on the brain than the confusion of "rods" and "hands" and "hogsheads" we call the English system. I'll spare you the actual equations, but three miles is 4.83 kilometers, and a jump that big requires an initial speed of 216 meters per second (483 mph). Hulk's time of flight is 31.4 seconds, and at the top of his arc he rises to a lofty altitude of 1,190 meters (three quarters of a mile). I'll bet his ears pop!
Superman would be green with envy
Anyway, these stats don't tell us very much by themselves, but we can use them to compute even more detailed information. Hulk's legs are about as long as a normal human being, so if he crouches all the way to the ground he's got about two meters of extension in them. This means he's got only that much distance to go from zero to 216 meters per second. Once his feet leave the ground, he's got nothing to push against, right? No way to add additional speed. So to get moving that fast in that short a distance, he needs to accelerate at almost 1,200 times the pull of Earth's gravity. This happens fast: From the time he starts the jump until the time he leaves the ground is only 0.02 secondsabout a tenth of the time it takes you to click your mouse. Cinematic effects aside, this guy does not move in slow motion.
If we take air resistance into account, he actually needs to be going a little bit faster, because the drag will slow him down. It will also tilt the far end of his parabola into a nearly vertical drop, but in the interests of convenience we'll ignore all that. Science can be surprisingly dependent on these little shortcuts and simplifications, so we needn't feel guilty about it. To take this math to the next step, we'll take still another shortcut, and model the Hulk as though he were a cylinder of solid muscle, some 4 meters tall and 1 meter wide. We know this isn't true, obviously, but it's close enough for our purposes here, and makes the numbers work out nicely.
Muscle tissue is mostly water, and weighs about the same as water, so this simplified Hulk-worm with a volume of just over 3 cubic meters would weigh 3,142 kilograms (6,926 pounds), which is slightly more than a high-end SUV. You do kind of have to wonder where all this mass comes from, since Bruce Banner doesn't appear to weigh more than one-fortieth as much. If it comes from the atmosphere, Hulk needs to inhale every single air molecule in a 14-meter (46-foot) radius while he's growing. That's quite an inrush of breath, and ought to cause serious problems all by itself.
Making a superhero muscle
Still, regardless of pesky questions like these, knowing both his mass and his acceleration, we can calculate the force Hulk's muscles have to generate: 37 million "newtons," or 8.4 million pounds. This is 18 times as much as a space shuttle main engine! If the leg muscles have a cross-sectional area of 0.79 square meters (which is what our worm model tells us), then the tension on the muscle tissue is 46 million newtons per square meter, or 6,670 pounds per square inch.
This is 190 times the pressure in your car tires, and about 90 times the failure strength of normal human muscle tissue. If you try this jump at home, you will literally explode before you leave the ground. You'd have better luck being shot from an artillery cannon! Hulk's muscles don't seem to suffer from the experience, though, any more than yours do from jumping a meter or two. This leads me to believe that Hulk's muscles, enhanced by nanotechnology, bioengineering and gamma-ray-induced mutations, are at least 200 times tougher than ours on a kilogram-for-kilogram basis. Given that Hulk has 40 times as much of the stuff as we do, this makes him 3,600 times strongeralmost three times our initial estimate.
Coincidentally, this comic-book-hero muscle tissue is about a third as tough as Kevlar, the ballistic nylon material used in flak jackets and police vests. The fact that Hulk is, indeed, at least as bulletproof as a solid block of Kevlar might make us suspect he's three times stronger still. If we had an X-ray machine with a really quick exposure time, we might snap a picture of his insides and confirm a few of these speculations, but I'll stop short of recommending this. You never know, eh? It might just make him angry.
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Wil McCarthy is a rocket guidance engineer, robot designer, science-fiction author and occasional aquanaut. He has contributed to three interplanetary spacecraft, five communication and weather satellites, a line of landmine-clearing robots and some other "really cool stuff" he can't tell us about. His short writings have graced the pages of Analog, Asimov's, Wired, Nature and other major publications, and his book-length works include the New York Times notable Bloom, The Collapsium and most recently The Wellstone and a related nonfiction book, Hacking Matter.
--------------------------------------------------------------------------------
By Wil McCarthy
--------------------------------------------------------------------------------
The Incredible Hulk is a mysterious creature, to be sure. Part Mr. Hyde, part Romeo, part Running Man and part Paul Bunyan, this not-so-jolly green giant can't be sedated for analysis, and if there's a jail that can hold him, we haven't seen it yet.
Hulk is, to put it mildly, a strong fellow. Just how strong may remain a mystery; I suspect the answer will always be "exactly strong enough to meet the challenges of each new movie, comic book or TV episode." But we do know one extraordinary fact about Bruce Banner's viridian alter ego: his standing broad jump covers an incredible three miles, more than 1,200 times the human world record. At first glance, this would appear to make him 1,200 times stronger than an Olympic-grade human, but in fact there's a bit more to it than that.
The mathematics of Hulk begin with a figure familiar from high-school physics and junior-high algebra: the parabola. This is the path objects follow as they rise and fall through the gravity field of the Earth, and to make use of it we'll start with the time-honored tradition of converting everything to metric units, which are much easier on the brain than the confusion of "rods" and "hands" and "hogsheads" we call the English system. I'll spare you the actual equations, but three miles is 4.83 kilometers, and a jump that big requires an initial speed of 216 meters per second (483 mph). Hulk's time of flight is 31.4 seconds, and at the top of his arc he rises to a lofty altitude of 1,190 meters (three quarters of a mile). I'll bet his ears pop!
Superman would be green with envy
Anyway, these stats don't tell us very much by themselves, but we can use them to compute even more detailed information. Hulk's legs are about as long as a normal human being, so if he crouches all the way to the ground he's got about two meters of extension in them. This means he's got only that much distance to go from zero to 216 meters per second. Once his feet leave the ground, he's got nothing to push against, right? No way to add additional speed. So to get moving that fast in that short a distance, he needs to accelerate at almost 1,200 times the pull of Earth's gravity. This happens fast: From the time he starts the jump until the time he leaves the ground is only 0.02 secondsabout a tenth of the time it takes you to click your mouse. Cinematic effects aside, this guy does not move in slow motion.
If we take air resistance into account, he actually needs to be going a little bit faster, because the drag will slow him down. It will also tilt the far end of his parabola into a nearly vertical drop, but in the interests of convenience we'll ignore all that. Science can be surprisingly dependent on these little shortcuts and simplifications, so we needn't feel guilty about it. To take this math to the next step, we'll take still another shortcut, and model the Hulk as though he were a cylinder of solid muscle, some 4 meters tall and 1 meter wide. We know this isn't true, obviously, but it's close enough for our purposes here, and makes the numbers work out nicely.
Muscle tissue is mostly water, and weighs about the same as water, so this simplified Hulk-worm with a volume of just over 3 cubic meters would weigh 3,142 kilograms (6,926 pounds), which is slightly more than a high-end SUV. You do kind of have to wonder where all this mass comes from, since Bruce Banner doesn't appear to weigh more than one-fortieth as much. If it comes from the atmosphere, Hulk needs to inhale every single air molecule in a 14-meter (46-foot) radius while he's growing. That's quite an inrush of breath, and ought to cause serious problems all by itself.
Making a superhero muscle
Still, regardless of pesky questions like these, knowing both his mass and his acceleration, we can calculate the force Hulk's muscles have to generate: 37 million "newtons," or 8.4 million pounds. This is 18 times as much as a space shuttle main engine! If the leg muscles have a cross-sectional area of 0.79 square meters (which is what our worm model tells us), then the tension on the muscle tissue is 46 million newtons per square meter, or 6,670 pounds per square inch.
This is 190 times the pressure in your car tires, and about 90 times the failure strength of normal human muscle tissue. If you try this jump at home, you will literally explode before you leave the ground. You'd have better luck being shot from an artillery cannon! Hulk's muscles don't seem to suffer from the experience, though, any more than yours do from jumping a meter or two. This leads me to believe that Hulk's muscles, enhanced by nanotechnology, bioengineering and gamma-ray-induced mutations, are at least 200 times tougher than ours on a kilogram-for-kilogram basis. Given that Hulk has 40 times as much of the stuff as we do, this makes him 3,600 times strongeralmost three times our initial estimate.
Coincidentally, this comic-book-hero muscle tissue is about a third as tough as Kevlar, the ballistic nylon material used in flak jackets and police vests. The fact that Hulk is, indeed, at least as bulletproof as a solid block of Kevlar might make us suspect he's three times stronger still. If we had an X-ray machine with a really quick exposure time, we might snap a picture of his insides and confirm a few of these speculations, but I'll stop short of recommending this. You never know, eh? It might just make him angry.
--------------------------------------------------------------------------------
Wil McCarthy is a rocket guidance engineer, robot designer, science-fiction author and occasional aquanaut. He has contributed to three interplanetary spacecraft, five communication and weather satellites, a line of landmine-clearing robots and some other "really cool stuff" he can't tell us about. His short writings have graced the pages of Analog, Asimov's, Wired, Nature and other major publications, and his book-length works include the New York Times notable Bloom, The Collapsium and most recently The Wellstone and a related nonfiction book, Hacking Matter.
