Hi Ho, Hi Ho, Part Ho

So at a moment when I have a much appreciated pause in my workweek (no class tomorrow or Thursday), I want to continue my Calculus discussion from last time, just because I'm bored and cranky and have nothing else to talk about (aside from some crazy baseball trade action).

In talking about real-life motion in Calculus terms, the curious thing is that the harder it is to conceptualize something, the easier the math is. For example, people understand the idea of speeding up and slowing down pretty naturally, but the math of that is harder than the math of velocity, which can be positive or negative and can therefore be confusing.

Even more confusing is displacement, where positive and negative totals mean that from the time you get out of bed in the morning until the time you go to bed at night your displacement is exactly the same as someone who didn't get out of bed in the first place. In fact, the major reason we use  distance instead of displacement is because the Puritans valued industriousness and wanted to reward people who got out of bed and did stuff, so they invented something called distance to give them and the idea caught on. But it's still nice to get back into bed and have your net displacement equal 0.

Mathematically, distance is just like displacement except it values all motion as positive. So driving 6 miles to work and 6 miles back is counted as 6 + 6 instead of 6 - 6. Unlike displacement, distance cannot be undone, as Ferris Beuller and friends learned.

As for velocity, where the 35 miles per hour I travel to work is undone by the -35 miles per hour I drive home, we have a similar thing called speed. Speed was invented by Asa Whitney, cousin of Eli Whitney who invented cotton gin and cotton martinis. Asa Whitney was a proponent of the transcontinental railroad who believed the achievement would be diminished and use of the railroad limited if people thought that the east to west velocity was negative. Speed treats all velocity as positive, whatever direction life might take you.

The math of distance and speed is more complicated than displacement and velocity because going 6 miles one way and then returning to the same spot takes you back to the beginning, thank you, not 12 miles away. We segregate this type of behavior into something called absolute value, which is useful but annoying in math.

But now back to acceleration, which is the change in velocity. You can't take direction away from acceleration. If your velocity is positive and your acceleration is positive, your velocity is increasing. If your velocity is negative and your acceleration is negative, then your velocity is negatively increasing (or increasing in magnitude, if you must. Magnitude, by the way, was invented by Magnus Pompey, the Roman General, who just liked naming stuff after himself).

Whatever the direction, we call this state "speeding up." If your velocity is positive and acceleration is negative, or vice versa, you are "slowing down."

Of course, everyone since Asa Whitney has known what speeding up and slowing down are. But Calculus gives us tools to determine what's going on and a series of somewhat confusing techniques to measure it. And that's as much math as I'm going to do tonight.