The art of estimation

Like the previous entry, this comes from my (now) out-of-print volume, The Speed of Nearly Everything.  I may get around to releasing it as an e-book, if enough people think it's a good idea.

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To a physicist, the notion of an immortal rabbit is quite acceptable. As a boy, my English teacher encouraged me to psychoanalyse Macbeth, even though I objected that we shouldn’t, since Freud hadn’t been invented when Shakespeare was writing. Ever a historically-minded cuss, I argued that it would be more relevant to look at the political situation in London, with a Scot sitting on the throne. Exasperated, he exhorted the class to engage in the willing suspension of disbelief.

And well he might, if he wanted us to accept some of the artifices and conceits of coincidence found in the 19th century novel, but we scientific types were subjected to much hardier fictional nonsense than that.

We routinely solved problems that involve a steel girder of negligible mass, suspended at its centre of gravity by a silken thread, and before we were too far advanced, we heard our first physics joke. It was about the three scientists who were trying to pick the winner of Australia’s premier horse race, the Melbourne Cup, which is held each November.

The mathematician gathers a wealth of data on weather, rainfall, wind, pollen counts and other possible influences, and three years in a row, fails dismally to pick a winner. At the end of those three years, the geneticist has just finished drafting a plan for a breeding program that should, in five generations, produce a winner, but the physicist has got it right, three times in a row.

The others ask him how he did it. He reaches into his pocket and produces an envelope which he turns over. Then he draws a circle on it. “Consider,” he says, “a spherical horse running in a vacuum…”

In fact a spherical cow or spherical horse can be a useful starting point to explore ideas, to get a first approximation that can be extended. Take the yarn about the bumblebee that was shown not to be able to fly: this is usually trotted out as evidence that scientists are thick, but there is a little more to it than that. In 1934, a French entomologist called Antoine Magnan tried to apply an engineer’s equation to bumblebees, and showed that according to that equation, designed for aircraft that did not flap its wings, the bee could not generate enough lift.

A bumblebee, coming in to land (or fall?)

There is a great deal of folklore wrapped around this “event” and who actually was involved, but it appears that the equation was worked out by André Saint-Lagué, and while the incident is often dressed up as “a scientist proving that bumblebees can’t fly”, all that was really shown was that the equation was inadequate to describe the flight of the bumblebee.

Magnan had shown that you can’t apply that particular equation to bumblebees, rather than proving that spherical bumblebees can’t fly, even if real ones, flapping their wings at 130 times a second, move happily along at 3 metres/sec, 11 km/hr or 7 mph. Like Zeno’s paradox (which will be in the next blog entry), Magnan’s calculation merely showed that there was a faulty assumption in there somewhere. The mathematical model was flawed.

When we escaped from the English classroom to the lab, we learned of marvels that could be done with simple apparatus. The muzzle velocity of a bullet could be measured with nothing more than a block of wood, a piece of string, a protractor and a measuring tape.

Our physics teacher, equally as at home with fiction as our English teacher, explained how, in the days of gunpowder and muzzle-loading firearms, slight variations in the ingredients, their amounts and proportions, could make a lot of difference. The most obvious measure was the speed at which a cannon ball or musket ball left the barrel of the gun, or in physics-speak, the muzzle velocity.

The idea was quite simple. You suspend a large block of wood and fire a bullet at it from close range. The bullet lodges in the block, and the energy of the bullet is transferred to the block, which swings like a pendulum. Then one simply has to measure the swing angle and calculate the height the block reaches.

This device even has a name: it is called the ballistic pendulum, and it has been around since the 1742, when it was invented by Benjamin Robins. From the swing, or so we were told, it is a fairly elementary calculation to estimate the energy and hence the velocity of the bullet. Unfortunately, this explanation ignores the 800-pound spherical horse which is rolling around the room.

Some of the energy goes into deforming the bullet and the wood, some is wasted as friction, and to do any calculations, we have to assume that the bullet stops instantaneously (which is as likely as a girder with negligible mass). Of course, if you are trying simply to compare different grades of gunpowder, rather than measuring the muzzle velocities, the losses will be similar in each case, and can be ignored. Whichever powder produces the biggest swing is the best, if everything else is kept constant — and in fairy physics, that always applies.

Robins was born to Quaker parents, but as a mathematician, he tried to make gunnery a science. Along the way, his ballistic pendulum probably showed that Indian saltpetre made the best gunpowder. He died in India in 1751, supervising the construction of forts, and a few years later, the British drove the French out of India, which let them have all that excellent saltpetre for their own use.

Curiously, the pursuit of novel sources for saltpetre during the Napoleonic wars led a French chemist, Bernard Courtois, to discover iodine, but that's another story...the next story, in fact.