The Calculus Diaries - Jennifer Ouellette [45]
SPLASH! The front of the canoe hits the bottom and displaces a large amount of water. Sean is drenched from head to toe, and instantly regrets his chivalrous offer to take the front seat instead of me. Nor does the dousing end there. We soon experience another sudden drop with accompanying splash, and another, and then must endure the shrieking laughter of the animatronic animals reveling in our plight. They have gone from abrasively cheery to vaguely sinister; we even spot Br’er Rabbit on the bank, tied up and struggling, about to be eaten by Br’er Fox. And those mechanical vultures with glowing red eyes look eager to gnaw with abandon on our sodden bones. The animals have found their laughing place, and it is called Das Haus von Schadenfreude.
There is one last lift and one final, fifty-foot drop, accompanied by yet another dousing. This is one of the fastest rates of descent in the entire park. While we know from our exercise with free-fall rides that the collective weight of everyone in our log does not affect the rate at which we fall, it does help determine how wet we are likely to get on this final splash, because the amount of water displaced is proportional to that collective weight.
The good news is that despite being drenched, our log boat floated and didn’t sink, because our average density was less than that of water. Had we sunk, we would have faced a dilemma reminiscent of our old friend Archimedes.
When not drawing countless rectangles under curves, he was having spontaneous epiphanies in his bathtub. Legend has it that Archimedes accepted a challenge from a local tyrant, Hiero of Syracuse. Tyrants are not trusting by nature, and Hiero was no exception. He was convinced a local goldsmith he had hired to make a golden wreath as a gift to the gods had cheated, replacing some of the gold with silver. No self-respecting deity would accept a cheap alloy. But how could he prove dishonesty? Hiero turned to Archimedes for help, who promptly went to the public baths for a good long think. He noticed that the more his body sank into the water, the more water was displaced.
The weight of an object pushes water out of the way, Archimedes reasoned, and the water in turn pushes back. So the buoyant force exerted by a fluid, like water, is equivalent to the weight of the fluid displaced. This gave him an idea for how to test the golden wreath: Gold weighs more than silver, so a crown mixed with silver would need more bulk to achieve the same weight as a crown made of purest gold. He could weigh the crown and submerge it in water to measure its volume, and from that he could calculate the density. Archimedes had stumbled on a way to calculate the volume of irregular objects very precisely. Euphoric over this critical insight, he leaped out of the tub and ran stark naked into the street, shouting “Eureka! Eureka!”30 Once he determined the crown’s volume, then the ratio between its weight and its volume would indicate its density and answer Hiero’s question of purity.
So let’s imagine that, instead of floating, the log boat sank with all its passengers. We can ask everybody to hold their breath while we use Archimedes’ principle to determine the total volume and from that, to calculate their average density. But even had I convinced Disneyland (and my fellow passengers) to let me do that experiment despite the liability issues, I would still lack another crucial piece of the puzzle—I had failed to note the weight of all the other passengers. This is an object lesson in why it’s so important