The Calculus Diaries - Jennifer Ouellette [23]
A physicist who blogs anonymously at Gravity and Levity describes physics and calculus at the high school level as a kind of game. “It was like a little logic puzzle where the rules of the game were given to you (usually on a formula sheet) and you were asked to use them cleverly to come up with a solution,” he says. “A friend of mine once put it succinctly: ‘Physics is all about finding out which variables you know and which variable you want, and then searching through your formula sheet for an equation that has all of those letters in it.’ That, more or less, was the physics game. You rearrange some symbols on a paper and you come up with an answer. Instant gratification.”
Some students take to the game quite naturally; others, like me, do not. But none of us will realize the full power of calculus until we move beyond treating it as a game and learn how to use it creatively to solve real-world problems.
YOUR MILEAGE MAY VARY
Even if we lose at the poker tables, I’ve gained something tangible from our weekend excursion: a valuable insight into the fundamentals of calculus. The derivative and integral are two different ways of looking at the same situation, namely, our Prius driving down a straight, level road. I can use the derivative to find our speed from our position and use the integral to figure out how far we’ve traveled based on our speed.
The speedometer and odometer in the Prius do these sorts of calculations all the time. It was quite ingenious of human beings to build these handy little devices whose primary purpose is to determine the exact information about speed and position that early mathematicians so meticulously calculated by hand. What is their secret? They have much more real-time data at their disposal. Both the speedometer and odometer are designed to collect every possible data point (for speed and distance, respectively) that it can in real time. The speedometer gives us a velocity function; the odometer gives us a position function. We can pretty much find out anything we need to know with this information, with no need to resort to calculus.
Speedometers measure the speed of a car by counting every single rotation of the tires. In older cars, they are mechanical, connected to a drive cable snaking its way from the transmission to the dashboard instrument cluster. The drive cable is basically a cluster of tightly wound coil springs wrapped around a center wire. When the wheels of a car turn, the gears in the transmission turn, and their rotational speed is sent down to the speedometer, where it can be measured and displayed.
The Prius speedometer is electronic (as is the odometer) and gets its rotational data from a vehicle speed sensor mounted to the crankshaft, rather than a drive cable. The sensor is little more than a toothed metal disk and a simple detector covering a coil that emits a magnetic field. The teeth interrupt the magnetic field as the disk rotates past the coil, creating a series of pulses, which are sent to the car’s computer via a single wire. The computer counts the magnetic pulses as each tooth of the metal disk passes by the coil. The real-time speed is displayed on the speedometer, so we can keep track of how fast we are traveling. The speedometer is linked to the digital odometer, so for every forty thousand pulses, the odometer adds one mile.
In fact, the Prius onboard computer goes even further: It combines the data on speed and distance with data collected from sensors monitoring gas usage to determine how many miles the car is traveling for each gallon of gas it consumes—both in real time, and on average over a given period. All this information