Testing the numbers, on the fly - part 2

Let me give another example of testing the numbers on the fly. And this time, I really mean "on the fly"!

Some years ago, I was traveling by air and looking out of the window when I saw a plane flying parallel to us but going in the other direction. In a few seconds it was gone. Now we were nowhere near an airport when this happened, so this was out of the ordinary. It made me think how much distance there had been between the planes, and whether their being so close in the middle of nowhere had been a chance occurrence.

As I tried to estimate the distance between the planes, I felt instinctively that the other aircraft "had looked a few centimeters long". Then I reflected and realized this measure does not make any sense until you define just how far from you those centimeters were! That is, a centimeter in a ruler near your nose looks different from a centimeter in the same ruler at arms length, and that in turn looks very different from a centimeter one kilometer away. I concluded on some reflection that when we say, for example, "the man was so far that he looked one centimeter tall" we instinctively tend to refer to the angle subtended at our eye by a centimeter placed somewhere between a half to one arm's length away. For the sake of simplicity, I assumed that our measure is taken one meter from our eyes. This is a non-obvious insight, and tremendously useful.

What does this mean? This means that if a man is 2 meters (6 feet) tall and he shrinks 100 times to appear just 2 cm high, he must be 100 times 1 or 100 meters away. If he looks just 1 cm tall, he must be 200 m away.

In other words,

If an object looks "1 cm high/long", its distance from you is 100 times its height/length!

How convenient!

The plane had looked, say, 2 centimeters long. So how far away was it? Well, that depended on how long it was in reality. It had looked like a commercial Boeing airliner and those often have 30 rows of seats, I thought. A meter per row, so let's say 40 meters, nose and tail included.

So if the plane was 40 meters long and looking like 2 centimeters, the multiple was 2000x. Therefore the plane was probably about 2000 meters away - two kilometers.

Then I wondered if the other plane had been taking off or landing in this wilderness. How could one tell? Well, by seeing its speed. So how fast had it been going? I guessed that it had covered a 90 degree angle in my eyes in about 6 seconds. The distance was the hypotenuse of a right angled triangle (remember school geometry?) with one side of 2 km or 1.414x2 in length or about 3 km. So it appeared to cover 3 km in 6 seconds (say) or 1800 km per hour. Now this was relative speed. Part of this speed was ours - I could find our current speed on the in-flight TV. I don't remember what it was exactly but it was clear that the other plane had been going very fast too, at cruising speed. So it was probably not taking off or landing at that time!

Needless to say, I got immense satisfaction from running these rough numbers.

Something similar had happened one day when I was still at school, watching the Air Force Day celebrations on a TV set. As part of the day's events, air force aircraft carried out mock sorties on some gasoline-filled targets at Hindon air force base. They would fire a missile at an "enemy tank" and it would burst into a fireball with a loud explosion on TV. As I watched, I began to realize that every time a "tank" was blown up on TV, the windows of our apartment would reverberate a half minute later. Highly excited (having been solving Agarwal Classes JEE problems in that year), I calculated 300 m/sec (speed of sound) times 30 seconds = 9 km approximately. Which was approximately the distance between where we lived and Hindon!

Some of you may be thinking this sort of number-checking is just a nerdy engineering trait. I don't think so. I find all intelligent management practitioners asking similar questions. The best venture capitalist wants to probe your business plan numbers in the same way - by quickly running a set of independent calculations in her head. The good CEO does quick back-of-the-envelope numbers to see for himself where, say, working capital can be best shaved. And so on.

In fact, doing this sort of number-checking is like a secret handshake of a secret society - jo ise jaante hain wo iska mahatva pehechaante hain!