In a world of internet, air conditioning, and suburbs, it might seem like all the dragons have been slain and there are no adventures to be had. But frontiers exist still, and when you journey into the wild beyond, you want to travel properly armed. I was out there once, and I got caught short. I want to share this story because it is deeply instructive.
The Registan Desert south of Kandahar is hard. There are no roads, no buildings, and no surface water. The only people, if you can find any, are nomads. Ancient hand-dug wells are the only means of survival, but a drought caused those to collapse around the turn of the millennia. We launched an expedition to rebuild one such well, about 100 miles south of the city.
Our project succeeded, but others had not. We came across a failed well that three different companies had tried to rebuild. Three times government contracts had been awarded for this well, three times success had been reported, three times companies got paid, but here was the well, only a dry pit in the ground. I was there myself; I saw this first hand.
Wondering just how deep this pit was (our own well, once finished, reached down a little over 300 feet), I dropped a pebble and timed its fall. I no longer remember how long it took—four seconds, maybe? Whatever it was, I wrote the number down and tried to work out the pit’s depth.
Now, understand, we weren’t at the office. I hadn’t solved a physics problem of this sort since college, and my physics books and notes were a hemisphere away. Out here in the desert, there was no internet or wi-fi; truth is, even our satellite phone wasn’t working. I had pebbles, a wristwatch, a notepad and a pencil. And the thing to understand is, sometimes this happens. You don’t have to go all the way to Afghanistan to be stripped of your usual resources.
I remembered the formula for position as a function of time—every first semester physics student learns it—but somehow, I kept tripping over myself, and I just couldn’t solve this stupid thing. In this, I’m not alone. Just last night an engineer friend of mine—degree from Notre Dame and works for one of the leading defense contractors—was getting flummoxed by an equivalent problem. Solving these things in school is one thing, trying to do it on the fly, when you’ve got a Soviet pistol in your pocket and a turban on your head, is another thing.
Which brings me to the second thing I want you to understand: when you learn these things in the right way, you can solve them handily. This pit depth problem you can solve in your head. Galileo did it without calculus.
Galileo showed that falling objects exhibit uniform acceleration. Consequently, their velocity as a function of time, though not constant, is linear. Forget about time, though; take a look at height as a function of velocity. That turns out to be linear too! When we plot it, we get a triangle!
Now, you can reach the bottom of the pit by means of this uniform acceleration– the triangular path– but we can also find an average velocity which, had the pebble moved at that average speed for the whole time, the bottom would be reached in the same amount of time. There is a single velocity that is equivalent to the sum of the infinitesimal velocities that the pebble actually experienced. It is like comparing a triangle to a rectangle; at what velocity will the rectangle have the same area? At half the terminal velocity of the triangle scenario.
If the pebble fell for four seconds, and acceleration is 32 ft/sec^2, then 4*32 gives us 128 ft/sec. That’s our terminal velocity, so the average velocity was half that—64 ft/ sec. Multiply that average velocity by four seconds and we have a pit depth of 256 feet. As I say, you could do this in your head. We could reduce it to the compact formula of d=(t^2)*a/2 that I once learned in school, but such formulas become hard to remember with lack of use. A better way to carry this knowledge is to understand how the triangle and rectangle relate. Memorable is more important than compact.
It might also be pointed out that knowing a little of the history behind the math seems to be a big help. I am at a loss as to why this is, but to understand that Galileo conducted experiments that disproved the science of Aristotle, to know that Galileo was working without calculus, and knowing that Newton invented calculus to understand the movements of stars and planets, seems to aid clear mathematical reasoning. Indeed, I have noticed that one difference between the really top schools—Yale, University of Chicago, and so on—and lesser-ranked universities is that in any lecture at a top school, you are always told who developed such-and-such idea, and when, and why. Those little bits of historical context are usually missing at other schools. I don’t know why a bit of history helps us to understand math and physics, but there it is.
I regret that I could not calculate the pit depth at the scene. Doing so would have helped. Fortunately, I know how to calculate it now, and I won’t be losing the knowledge a second time. I believe it is important to be able to solve such problems under sub-optimal conditions, because the dragons still exist, they are not as far away as we think, and what good is all that superior human intelligence if we can’t use it when we most need it?