The Calculus Diaries - Jennifer Ouellette [39]
Human beings have thrilled to the sensation of free fall for centuries, with occasionally dire results. Witness the enormous popularity of bungee jumping, which has its roots in the ancient Aztec ritual of the Danza de los Voladores de Papantla; the danza is still practiced today by “Papantla flyers.” In the 1950s, British documentary filmmaker David Attenborough took his BBC film crew to Pentecost Island in Vanuatu, where they recorded several young tribal men who jumped from tall wooden platforms with vines tied to their ankles as a test of courage. It was only a matter of time before extreme sports enthusiasts had the brilliant notion of harnessing themselves to bungee cords and jumping off tall structures for fun (and the occasional profit).24 Bungee jumping quickly spread around the globe, despite numerous accidents and the odd fatality.
For those (like me) who prefer a more sedate form of thrill-seeking, there are mechanical free-fall rides with, shall we say, more rigorous safety constraints. Six Flags Great Adventure introduced one of the first true free-fall experiences in 1983. The L-shaped structure featured a four-passenger car lifted via hydraulics to the top of a 130-foot tower and suspended for a few seconds. At the buzzer, the car would plunge down the drop track and onto the horizontal exit track to end the ride. The latter was necessary because coming to a sudden stop at the end of the drop would most likely cause serious injuries. The deceleration period dissipates all that kinetic energy over a longer period of time so it isn’t transferred all at once to the passengers.
The Tower of Terror is a variation of a “drop tower” ride that gradually has replaced the classic free-fall design since the 1990s, largely because it is closer to a true free-fall experience, and there is less mechanical wear and tear. A gondola or car—in this case, the mock elevator—is propelled upward toward the top of a large vertical structure and then falls back toward the ground. The brakes kick in before impact, slowing the ride, although the Tower of Terror essentially “bounces” its riders a few times before finally coming to rest.
Technically, we enter free fall when there is no longer any force (other than gravity) acting directly on us. Think of tossing an apple into the air. The moment it leaves your hand and you stop applying that upward force, it is in free fall. It continues traveling up, moving more slowly as gravity overpowers its upward motion, has a brief moment of hang time (that period of weightlessness), then begins its descent. Our car in the Tower of Terror follows the same trajectory. It receives an initial push from the hydraulics, but at some point that force is removed and we finish our ascent using pure momentum. That brief, exhilarating period of weightlessness occurs because riders fall at the same rate as their surroundings—in this case, their seats. NASA’s infamous “vomit comet” follows a parabolic trajectory while in flight, taking such extreme lifts and dips that it can achieve about twenty to thirty seconds of weightlessness for every sixty-five seconds of flight.
Thrills aside, the Tower of Terror provides an excellent example of calculus as it applies to classical mechanics. True, our physical motion is straight up and down. But if we plot our change in height (position) over time point by point on a Cartesian grid—both ascending and descending trajectories—and connect the dots, we end up with the telltale parabolic curve that so delighted Sean at the ride’s end. (The same is true for the apple.)
How does this work? We begin with our starting velocity. It is not 0, because we are specifying our starting velocity at the moment we enter freefall, not at the start of the ride (when it would be 0). We need a speedometer to tell us how fast we are traveling at that moment, and that number becomes our starting velocity (a constant). Let’s imagine the Tower of Terror tracks acceleration for us. We can take an integral of our acceleration to get our velocity, essentially