V IS FOR VECTOR
Making our way into Fantasyland, we find the King Arthur Carousel and the Dumbo-inspired flying-elephant ride, both excellent examples of rotation around a fixed axis. But it is the Mad Tea Party—usually called the spinning teacup ride—that provides us with an unusual illustration of vectors: motion in specific directions. A vector is technically defined as any quantity having both direction and magnitude. In physics, vectors typically describe force, velocity, acceleration, or similar three-dimensional properties. How the different vectors combine determines their net strength; one must take into account not just how strong a given force might be, but also in what direction it is pushing.
It is easiest to illustrate the concept in one dimension. Picture your standard number line. An object moving in a straight line has a direction, depicted by a small arrow above the number. If it starts at 0 and ends at 5, this is called vector (5); it’s the same as any other number along that line, except we have specified a direction. Because it’s moving left to right, it is a positive number. A vector pointing from right to left would be a negative number. Vectors can be added together or subtracted, just like regular numbers. Combine vector (5) with vector (−5), and the two cancel each other out completely; combine it with vector (−3), and you end up with vector (2); combine it with vector (4), and you end up with vector (9). And so on.
Frankly, vectors aren’t very interesting in the one-dimensional realm of the number line: There is no real difference between them and ordinary numbers. In two dimensions, vectors are pairs of numbers (Cartesian coordinates) that describe the direction of movement in a plane. In three dimensions, they describe directional motion through space using three coordinates.
Here is how vectors apply to the Mad Tea Party. Any rotating body’s motion has a vector that is constantly changing, because the direction shifts at each point in the turn. The teacup ride in Disneyland consists of a series of rotating circles, or turntables, each moving along its own vector that is constantly changing its direction because of the rotational motion. There is one big circular moving platform that rotates clockwise. Within that circle are three smaller ones that rotate independently, counterclockwise, and within each of those circles are individual teacups that rotate clockwise, independently from the two bigger circles.
The riders can spin their teacups as fast as they want by turning the metal wheel at the center of the cup, applying a torque to increase the teacup’s angular momentum, and hence the rate of spin. As Sean and I strain mightily to spin our teacup as fast as possible, I notice something intriguing. Every now and then we achieve an especially sharp, fast rotation, whereas at other points, no matter how hard we pull that metal wheel, we can’t achieve much rotation at all. Sean explains that this is because of dueling vectors. Sometimes the vectors work against each other, pulling in different directions and canceling each other out, to varying degrees. At other times in the rotation, they add together, all pulling in the same direction, so we spin that much faster.
Space Mountain—Tomorrowland’s main attraction—provides us with the quintessential example of a calculus problem involving vectors. When Walt Disney first designed Tomorrow-land, he noted that it would be out of date almost immediately. By twenty-first-century standards, the “future” it envisions is downright quaint, harking back to a more innocent era. Tomorrowland didn’t even have a roller coaster until Space Mountain opened in May 1977, after the original ride proved so popular at Disney World. Disney didn’t live to see it completed. Space Mountain took two years to build and cost upwards of $20 million, and the park set an attendance record the first weekend the ride opened. Six of the original seven Project Mercury astronauts were on hand for its inauguration.27
In the 1968 film 2001: A Space Odyssey, astronaut Dave Bowman (played by Keir Dullea) walks down a long white circular tunnel to the space ship that will carry him on his mysterious mission into deep space. It’s difficult not to recall Kubrick’s masterpiece while waiting in the long line for Space Mountain. The ride’s interior is eerily similar in design. We follow winding metal ramps down into the bowels of the coaster, encountering the occasional video screen showing famous astronauts talking about their missions. Finally, we reach the front of the line and take a seat inside our little rocket-shaped car.
We rise to the top of first one, then another lift hill, winding through a passage that features glowing red bars that seem to be rotating. At the top of the third and final lift hill, our rocket pauses briefly as we gaze out into the vast darkness of “space”—there appear to be thousands of stars and galaxies, when in fact it is simply a clever effect achieved with mirror balls scattered throughout the ride’s interior. A voice announces, “You are go for launch,” and pure gravity takes over as our rocket begins its rapid descent, accelerating through the remainder of the track. The sensation is enhanced by gusts of wind from strategically placed air vents as we careen and lurch through the darkness. When it is time for our “reentry,” we decelerate and return to the docking station.
For all its futuristic trappings, Space Mountain is a classic roller coaster, from a physics standpoint.28 Roller coasters operate on inertia, gravity, and acceleration—and the greatest of these is gravity. Our rocket builds up a large reservoir of potential energy while being towed up those three initial lift hills. The higher we rise, the greater the distance gravity must pull it back down, and the greater the resulting speeds. As our rocket starts down the first hill, all that accumulated potential energy is converted into kinetic energy and our car speeds up, building up enough kinetic energy by the time it reaches the bottom to overcome gravity’s pull and propel the car up the next hill. And so on for the rest of the ride.
Sean likes Space Mountain. A lot. Apart from the obvious fun factor, he declares that we can use calculus to determine our trajectory (the path we took) when all we know is our acceleration. Space Mountain has none of the intricate maneuvers that have become standard among more extreme coasters: fancy corkscrews, loops, and so forth. Instead, it relies on a series of shorter dips and sharp turns in near-total darkness. Because we can’t see the track, we can’t anticipate where we are likely to go next or prepare for the sudden shifts in velocity. The few visual cues we are given are deliberately misleading.
We can still figure out which path we took, because we can feel the physical effects of acceleration on our bodies and deduce our trajectory from that data. These are the g forces that describe how much force the rider is actually feeling; g is a unit for measuring acceleration in terms of gravity. Our rocket is constantly accelerating over the course of the ride: forward and backward, up and down, and side to side. Our inertia is separate from that of our rocket, so when it speeds up, we feel pressed back against the seat because it’s pushing us forward, accelerating our motion. When the rocket slows down, our bodies continue forward at the same speed in the same direction, but the restraining bar decelerates us to slow us down. All this acceleration produces corresponding variations in the apparent strength of gravity’s pull. For example, 1 g is the force of Earth’s gravity: what the rider feels when the car is stationary or moving at a constant speed. Acceleration causes a corresponding increase in weight, so that at 4 g’s you will experience a force equal to four times your weight.
That gives us an intuitive sense of our trajectory throughout the ride, but for a truly rigorous analysis, we should have had the foresight to bring along a makeshift accelerometer. As electronic components have continued to shrink, accelerometers became easier to embed. Our matching his-and-hers iPhones come with built-in accelerometers, which is how the device knows when to adjust the screen from a vertical to a horizontal view when you turn the phone on its side. If our little rocket came equipped with a built-in accelerometer—yes, there is an app for that—that accumulation of data would give us our acceleration function.
Let’s start with a simplified version of this standard textbook problem, assuming that we are moving in a perfectly straight line.
How can we figure out our trajectory—our position as a function of time—knowing just our acceleration? Our acceleration accumulates over time to give our velocity, Sean explains; we accumulate our increasing speed at each moment in time to determine our final velocity. So that means velocity is the integral of acceleration. Velocity in turn increases over time to give position, so position is the integral of velocity. “You just have to integrate the acceleration twice to figure out position as a function of time,” Sean concludes triumphantly—just like our free-fall problem.
However, there is a complicating factor: The rockets move left and right and up and down, not just forward in a straight line. So not only are the rocket sleds constantly shifting between potential and kinetic energy, but every time we shift direction, we also are shifting vectors—our direction of movement is constantly changing. Thanks to an unjustly obscure nineteenth-century British mathematician and physicist named Oliver Heaviside, we have the tool to solve this complex conundrum: vector calculus.
A product of the London slums that also produced Charles Dickens, the red-haired, diminutive Heaviside fell ill with scarlet fever as a child, which left him partially deaf. His social skills seem to have suffered as a result: He didn’t get along with the other children at school in Camden Town, although he was a top student in every subject save geometry. Perhaps traditional education couldn’t contain his eccentric genius: He dropped out at sixteen to continue his schooling at home. It helped that his uncle was Sir Charles Wheatstone, who co-invented the telegraph in the 1830s and was a recognized expert in the new field of electromagnetism. Within two years, young Oliver found himself working as a telegraph operator, quickly advancing to chief operator. It was the only full-time employment he ever experienced.
One could blame James Clerk Maxwell, the prominent physicist who first formulated the set of equations for electromagnetism that still bear his name, for Heaviside’s sudden shift into the ranks of the chronically unemployed. Heaviside discovered Maxwell’s seminal treatise in 1873 and was so enthralled by the work that he quit his job the following year to study it full-time, moving back into his parents’ home in London. (History has not recorded his parents’ reaction.) Once he’d grasped the essential points, “I set Maxwell aside and followed my own course,” Heaviside later recalled. In the end, he reduced Maxwell’s equations from twenty down to four vector equations and built upon that work to develop vector calculus.
Few objects move in a straight, flat line. We don’t drive down straight roads with no turns or hills, and a roller coaster would be a very dull ride indeed if it only moved in flat, linear motion. Vector calculus lets us solve the same calculus problem in three dimensions: retracing our path by determining our position at each instant over the course of the ride. We describe position in three-dimensional space with three Cartesian coordinates (x, y, and z), so there are now three numbers involved in our calculations. Nothing else has changed from the previous example: Our trajectory is still position as a function of time, and thanks to the data gathered by our accelerometer, we know our acceleration as a function of time. Acceleration builds up to give us our velocity, which in turn builds up to give us our position at any moment. It’s just more complicated because our movement has a constantly changing direction. We must keep track of three directions at once, and each has its separate position, velocity, and acceleration.
Heaviside never gained the recognition he deserved until after his death in 1925; he was justly bitter about this. He became quite eccentric in his later years, spending the last two decades of his life as a virtual recluse in Torquay, near Devon. He suffered bouts of jaundice and the tormenting of neighborhood children, who threw stones at his window and scrawled graffiti on his front gate. Neighbors reported that his home was furnished primarily with huge granite blocks. Otherwise scruffy and unkempt, he took to painting his impeccably manicured fingernails bright pink, and signing letters with the mysterious initials W.O.R.M. after his name—providing ample fodder for future armchair psychoanalysts as to what the letters might have meant to him. Perhaps he would have found comfort in the fact that, over a hundred years later, his method of vector calculus would one day shed light on a budding calculus student’s encounters with the rides at Disneyland.
MAKING A SPLASH
By late afternoon, we have worked our way over to Critter Country, where the skyline is dominated by the soaring peak of Splash Mountain. Splash Mountain is a “log flume” ride. Loggers used to transport logs down mountains to a sawmill by floating them down the river. Eventually someone had the brilliant idea of hollowing out those logs and using them as makeshift boats. The first artificial log-flume ride—called El Aserradero (The Sawmill)—opened in 1963 at the Six Flags theme park in Arlington, Texas. The Disneyland version is a large plaster mountain housing canals (or flumes) filled with water, and artificial hollow logs that can seat up to six people. The flow of water along the flumes propels the log boats forward, with a little help from mechanical chains and pulleys to hoist the logs up the hills. Just as with a roller coaster, good old-fashioned gravity does the rest.
Nothing says Disney like plaster facades and cheesy animatronics. This ride takes its inspiration from Song of the South, with scenes depicting the adventures of Br’er Rabbit. The robotic critters 29 lining the “banks” of the faux canal snaking through Splash Mountain serenade us as we float along, with a jaw-clenching ditty about positive thinking, finding your “laughing place,” and having a zip-a-dee-doo-dah day! Just as I am wishing I had a stun gun capable of overloading their circuitry with an electromagnetic pulse, we come to a sudden drop and plunge into the depths of the cavernous “briar patch.”
SPLASH! The front of the canoe hits the bottom and displaces a large amount of water. Sean is drenched from head to toe, and instantly regrets his chivalrous offer to take the front seat instead of me. Nor does the dousing end there. We soon experience another sudden drop with accompanying splash, and another, and then must endure the shrieking laughter of the animatronic animals reveling in our plight. They have gone from abrasively cheery to vaguely sinister; we even spot Br’er Rabbit on the bank, tied up and struggling, about to be eaten by Br’er Fox. And those mechanical vultures with glowing red eyes look eager to gnaw with abandon on our sodden bones. The animals have found their laughing place, and it is called Das Haus von Schadenfreude.
There is one last lift and one final, fifty-foot drop, accompanied by yet another dousing. This is one of the fastest rates of descent in the entire park. While we know from our exercise with free-fall rides that the collective weight of everyone in our log does not affect the rate at which we fall, it does help determine how wet we are likely to get on this final splash, because the amount of water displaced is proportional to that collective weight.
The good news is that despite being drenched, our log boat floated and didn’t sink, because our average density was less than that of water. Had we sunk, we would have faced a dilemma reminiscent of our old friend Archimedes.
When not drawing countless rectangles under curves, he was having spontaneous epiphanies in his bathtub. Legend has it that Archimedes accepted a challenge from a local tyrant, Hiero of Syracuse. Tyrants are not trusting by nature, and Hiero was no exception. He was convinced a local goldsmith he had hired to make a golden wreath as a gift to the gods had cheated, replacing some of the gold with silver. No self-respecting deity would accept a cheap alloy. But how could he prove dishonesty? Hiero turned to Archimedes for help, who promptly went to the public baths for a good long think. He noticed that the more his body sank into the water, the more water was displaced.
The weight of an object pushes water out of the way, Archimedes reasoned, and the water in turn pushes back. So the buoyant force exerted by a fluid, like water, is equivalent to the weight of the fluid displaced. This gave him an idea for how to test the golden wreath: Gold weighs more than silver, so a crown mixed with silver would need more bulk to achieve the same weight as a crown made of pures
t gold. He could weigh the crown and submerge it in water to measure its volume, and from that he could calculate the density. Archimedes had stumbled on a way to calculate the volume of irregular objects very precisely. Euphoric over this critical insight, he leaped out of the tub and ran stark naked into the street, shouting “Eureka! Eureka!”30 Once he determined the crown’s volume, then the ratio between its weight and its volume would indicate its density and answer Hiero’s question of purity.
So let’s imagine that, instead of floating, the log boat sank with all its passengers. We can ask everybody to hold their breath while we use Archimedes’ principle to determine the total volume and from that, to calculate their average density. But even had I convinced Disneyland (and my fellow passengers) to let me do that experiment despite the liability issues, I would still lack another crucial piece of the puzzle—I had failed to note the weight of all the other passengers. This is an object lesson in why it’s so important to carefully collect one’s raw data while doing the experiment.
If we know the combined weight of the passengers and the log we are riding in, the volume of our log, and the collective density (in units of grams per cubic centimeters), we can divide the total weight by the total density to get our volume in cubic meters. We also need to know the density of water; a quick Google search reveals that one liter of water has a density of 1 kilogram. Now we multiply the volume of our log and its passengers by the density of the water to find the volume of water displaced. Those hollow plastic logs hold six riders of varying weights. Assuming an average weight of 150 pounds per passenger (150 × 6, plus the weight of the log itself), that gives us a pretty substantial volume—and a substantial displacement of water when we hit the bottom of that final plunge. No wonder we’re completely soaked by the ride’s end.
The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse Page 10
