The Calculus Diaries - Jennifer Ouellette [43]
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