The Calculus Diaries - Jennifer Ouellette [40]
The result, when we graph it out, is a straight downward-sloping line. This is our velocity function. We can use that to determine our position (height) by taking another integral, adding together how far we traveled at each point in time. Plot each position as a function of a time and you get a pretty parabola. Now that we have a position function, finding our height at any given point in time is a snap.25
Many years ago, I went to Six Flags New Jersey with a group of friends, and we all went on the Devil Dive—a cross between bungee jumping and a really big tire swing. Three of us were strapped into one big harness and lifted to the top of a 200-foot tower. That might not sound very high unless you happen to be one of the people hanging precariously at the top of it; then terra firma seems a very long way down. One of my cronies had just enough time to nervously remark, “Um, maybe this wasn’t such a good idea after a—AUUGHH!” The buzzer sounded, the catch released, and we plummeted, screaming, toward the ground.
Just as we were about to hit the ground, the harness caught and swung us outward in a sweeping pendulum motion, moving through space along the trajectory of an arc of a circle. We swung back and forth like a three-person pendulum, until we slowed down sufficiently for the ride operators to grab us and release us from the harness.
The Devil Dive gives us a double dose of Galileo. First, there is the free fall. It is roughly the same problem outlined above, except in this case our position as a function of time forms only half a parabola, because we don’t enter true free fall until we begin our descent. Our acceleration is −32 feet per second per second at any time (t) after our drop begins. (The sign is negative because we are falling and our height is decreasing.) We can take an integral to get our velocity, and then integrate the velocity to get our position function, just as we did before.
Second, there is the pendulum motion at the end of the ride. An oft-told anecdote from Galileo’s youth tells of the seventeen-year-old future scientist growing bored during Mass in a drafty cathedral in Pisa. He noted a chandelier hanging from the ceiling swaying in the breeze. Sometimes it barely moved; other times, it swung in a wide arc. This proved more interesting to the teen than the priest’s sermon, and he began timing the swings with his pulse, with a surprising result: It took the same number of beats for the chandelier to complete one swing, no matter how wide or narrow the arc. Granted, the chandelier moved faster during wider arcs, but it completed its arc of motion in the same amount of time. The same motion can be seen in playground swings and the arc we make at the end of our Devil Dive. But there is a twist: Don’t be misled by that arc-like motion. If we plot our changing position with respect to time during this portion of the ride, we get a periodic sine wave. The fact that the pendulum swings in predictable periods is why it became the basis for the pendulum clock.
There is another relevant curve called the Witch of Agnesi, named after eighteenth-century mathematician Maria Gaetana Agnesi. The eldest of twenty-one children,26 Agnesi was known in her family as the Walking Polyglot because she could speak French, Italian, Greek, Hebrew, Spanish, German, and Latin by the time she was thirteen. Agnesi had the advantage of a wealthy upbringing; the family fortune came from the silk trade. And she also had a highly supportive father, who hired the very best tutors for his talented eldest daughter and insisted she participate in regular intellectual salons he hosted for great thinkers hailing from all over Europe.
The young Maria delivered an oration in defense of higher education for women in Latin at the age of nine; she translated it from the Italian herself and memorized the text. Contemporary accounts suggest that Agnesi loathed being put on display, even though her erudition earned her much admiration. One contemporary,