Jen-Luc Piquant should technically be wearing mouse ears in honor of today's post, but they clash horribly with her stylin' beret. Why? Because on Sunday, we dragged the Spousal Unit to Disneyland for a day of magical adventures! Okay, a day of standing in impossibly long lines for the privilege of experiencing a brief period of euphoria, followed by trudging to a different section of the park and a wait in another impossibly long line. Repeat until done. See, the Spousal Unit had never been to Disneyland, or Disneyworld for that matter, and it's a slice of Americana that really must be experienced firsthand -- although I have a love/hate relationship with the place. It's crowded, cheesy, and all about crass merchandising (motto: all headgear is improved by the addition of mouse ears!), but there's some nugget of goodness at its heart.
Or maybe that's just my childhood nostalgia talking. When I went with my family as a kid, it truly did seem like a magical kingdom. More practically, I have an entire chapter on amusement park calculus in my book-in-progress, and we were looking for interesting problems to add some rigor to all that candy-coated amusement. (Each chapter has a specific curve associated with it -- the "face" of a function, as David Berlinski puts it in A Tour of the Calculus -- and the idea is to have at least one problem for each in an Appendix showing the derivations of the concepts being discussed.) Amusement park physics is all the rage: high school physics teachers have been known to take classes on field trips to Six Flags, for example (except for one rather curmudgeonly school in Canada where the notion proved controversial). Six Flags is great for Newtonian mechanics; it's all about the pulse-pounding extreme rides. Disneyland is as much about artifice and storytelling as it is about adrenalin (and, of course, crass marketing!).
Everyone should go to Disneyland with a physicist at least once, particularly one as good-natured as the Spousal Unit, who let himself be dragged from one long crowded line after another, filled with over-excited youngsters, frazzled parents, and purple-haired, pierced hipsters doing their best to look bored and act like they were Really Just There For the Irony. Visiting Disneyland with a physicist is an entirely different way of looking at things. For instance, first I dragged him to The Twilight Zone's Tower of Terror in the adjacent California Adventure park. The schtick is that this is a cursed hotel in which four people got into an elevator one day, were zapped by lightning and mysteriously disappeared into... The Twilight Zone! Along with their luggage, so at least they had a change of clothes and toiletries after being transported into a nightmare dimension.
The interior is done up in vintage Gloria-Swanson-making-like-Miss-Haversham-in-a-home-gone-to-seed-style, and eventually you find yourself in a mock elevator, strapping yourself into your seat. You'll need that strap, too. We were lifted to the "top floor" of the faux-hotel structure, and then cursorily dropped into simulated freefall. Physics! As one would expect, this lifted us out of our seats slightly, as much as the straps would allow, and we got that one glorious moment of seeming weightlessness, before reaching a jerky stop and being raised back up for another drop. And another. After it was over, the Spousal Unit turned to me and gleefully exclaimed, "We made a parabola!"
Yes. Yes, we did. But until I started writing this book, it would never have occurred to me to think about the ride in this way. Score a point for calculus and hands-on education! Plot out our path point by point on a cartesian grid and connect the dots, and you do indeed get the telltale parabolic curve. The nice thing about the Tower of Terror's freefall, for book problem purposes, is that it forms a complete parabola, since it goes up and down several times. The part where we're falling is technically half the curve.
Years ago, when I still lived in NYC, I went to Six Flags New Jersey with my badass dojo cronies, and we all went on the Devil Dive. Forget simulated freefall, this ride was more akin to bungee jumping. Three of us were strapped into a big harness -- after signing a waiver promising we wouldn't sue Six Flags if we died horribly -- and lifted to the top of a 200-foot tower. That doesn't sound very high unless you're the person hanging precariously from a harness at the top of it -- then it becomes very high indeed. I remember Gabe nervously remarking, "Um, gee, maybe this wasn't such a good idea after all..." Then the signal sounded, they released us and we plummeted, screaming, towards the ground. One half of a parabolic curve, served straight, no chaser!
Even better, just before hitting ground, the harness catches and swings you outward over the park in a sweeping pendulum motion. And you swing back and forth, more slowly with each pass, until you've slowed down sufficiently for the ride operators to grab you and bring you back down to Earth. At least one person in our little party was convinced that having three people in the harness instead of two would result in a faster descent. Au contraire, mon frere! He had forgotten his high school physics class, namely, that objects fall at the same rate regardless of mass. (Granted, that's in a vacuum, and we weren't in a vacuum, but I doubt the extra person made an appreciable difference.) The Devil Dive is like a double dose of Galileo: freefall and the pendulum. It's understandably a hugely popular ride: there are homemade videos all over YouTube, like one:
But I digress: this is supposed to be about Disneyland. How to top the Tower of Terror? The obvious choice was Space Mountain. We braved the 45 minute wait, hopped in our little faux spaceship, and hurtled through the darkness at high speed, thrilling to the hairpin twists and turns we couldn't see coming. The Spousal Unit liked Space Mountain. A lot. As who could not? He didn't think it was a "conventional" roller coaster, in that it relied on pushing and pulling the cars mechanically, rather than simply exploiting gravity. (According to this article, that's not the case: Space Mountain's design does not incorporate cheats like "energy wheels" to keep cars from stalling at the top of hills -- except at the very end of the ride in the re-entry tunnel. Per the author, Bill Watkins, who was the ride's track engineer, "I submit that this does not violate the pure gravity ride principle because we are now down (nearly) at station level and it's important to fine-tune the vehicle speed before it enters the station." Discuss.)
There was no obvious moment of feeling suspended in true freefall, either. There were many sudden shifts in velocity, however, and we contemplated how to turn that observation int some sort of calculus problem involving vectors. Vectors also came into play at the Mad Tea Party. (*belligerence on*) Yeah, we went on a kiddie ride. What of it? Huh? HUH? (*belligerence off*) The teacup ride is actually more complicated, from a physics/calculus standpoint, than it initially appears; it's not just about angular momentum. (If that were all I was interested in, we could have just stuck with the King Arthur Carousel.) We noticed that every now and then, we'd get an especially sharp, fast rotation, whereas at other points, no matter how fast we tried to spin our little teacup, we just couldn't reach the same speed. Physics! See, there is one big circle that rotates. Within that circle are smaller ones that rotate independently. And within each of those circles are the individual "teacups", which rotate independently from the two bigger circles.
Technically, we're talking here about velocity, not merely speed. Velocity is speed with a direction, traveling along a vector. In the case of the teacup ride, each "circle" moves along its own vector. (That vector is constantly changing its direction, of course, since the "circle" is in rotational motion.) A standard textbook calculus problem would ask students to add together all those vectors at a given point to determine the total velocity. Sometimes the vectors work against each other, pulling in different directions and canceling each other out. That's when the teacup spun more slowly. Other times, the vectors line up in the same direction and add together, and you get those moments of faster rotation. (And you thought there couldn't possibly be calculus in a teacup. Hah!)
Next it was off to Splash Mountain in "Critter Country," in which one steps into a little canoe and floats down a deceptively innocent-looking canal filled with water. (Buoyancy! Every calculus book should mention Archimedes!) There are little animatronic animals along the "banks" of the faux canal, singing an incredibly annoying, jaw-clenching ditty about positive thinking and finding your "laughing place," and having a zip-a-dee-doo-dah day! Honestly, it's enough to make you wish you had a BB gun with which to wipe the smirks off their silly robotic faces. Clearly, the animatrons know what's coming: a sudden drop into the depths of a cavern and SPLASH! The front of the canoe hits the bottom, displaces a large amount of water, and whoever is seated in the front seat gets thoroughly doused. The Spousal Unit promptly regretted his chivalrous offer to take the front seat instead of me.
Oh, but the ride doesn't end there. We experience another sudden drop, and another big splash, and then must endure the shrieking laughter of the animatronic animals, which has gone from annoying to vaguely sinister, particularly in the eerie lighting of the fake cavern. (Is it me, or does that robotic raccoon look like it's ready to gnaw on our sodden bones?) They have found their laughing place, and it's called Casa de Schadenfreude. As a final ignominy, there is one final 50-foot water drop -- you know, because there might be one small piece of clothing not yet thoroughly soaked through.
Actually, Splash Mountain is a lot of fun, and quite refreshing on a very hot day. We were not there on a very hot day, however; it was surprisingly cool for Anaheim. We squished our way back over to New Orleans Square into one of the gift shops to buy replacement clothing. (According to the clerk, this happens all the time when people ride Splash Mountain.) This was the only point in the day where the Spousal Unit showed a hint of pique; sodden jeans are not a pleasant sensation when the sun is about to set.
There's definitely a lot of physics and calculus fodder in Splash Mountain, but the Spousal Unit was understandably focused on his slowly-drying attire. He explained that the rate at which this occurred formed an exponential decay curve -- as does the rate at which a hot cup of coffee cools to equilibrium. Specifically, the coffee initially cools very quickly, but the rate of cooling slows considerably as the coffee approaches equilibrium with its surrounding environment. Ditto for our clothing: the moisture evaporated quickly at first, then the rate slowed down, and down... would we forever be slightly damp? It was starting to look that way.
To kill some time before our dinner reservation, and give our clothes a little extra time to dry, we went to the Haunted Mansion (Pepper's Ghost illusion in action!), Thunder Mountain Railroad (another roller coaster), and the Mark Twain riverboat. Finally, we went on my all-time favorite ride, The Pirates of the Caribbean. Sure, there was water (buoyancy again), and one initial drop, but this time we didn't get wet. Huzzah! And our dinner reservation was at the Blue Bayou -- the only "fine dining" establishment in Disneyland, actually sited just inside the Pirates ride.
It's actually got a rather nice ambience, and good Cajun-inspired food -- served up in enormous Disney-sized portions -- even if they don't serve any alcohol. The Spousal Unit was very much in need of a drink by then, and the Tinkerbell Fruit Punch with the fairy light garnish just wasn't going to cut it. But at least he got the signature dessert -- a boat-shaped "cookie" with an edible sail featuring a piratical skull and crossbones motif: