“Good morning, welcome!” Eusebio was sitting on the floor in the corner, loose pages spread out in front of him. He was bulky for a man, but no less agile for it; Yalda suspected that he’d strived from childhood to match the deftness of smaller peers, much as she had.
She sat facing him, cross-legged, and got straight to the point. She knew exactly what he would have been told in the lecture he’d had the day before; not one word had changed in the introductory physics course since she’d taken it herself, four years previously.
“Conservation of energy and momentum,” she said. “How much did you understand?”
“Maybe half,” he confessed. But Eusebio didn’t claim understanding lightly; Yalda suspected that he’d followed the whole lecture, but longed for a deeper grasp of the subject.
“Let’s start with something simple,” she suggested. “Suppose an object is free to move, without friction. It starts out at rest, and you apply a constant force to it. After some time has passed, tell me how the force, the time, and the object’s velocity are related.”
Eusebio said, “Force equals mass times acceleration; acceleration by time gives velocity. So, the product of force and time equals the product of the object’s mass and its velocity—also known as its ‘momentum’.”
Yalda widened her eyes approvingly. “And in the general situation, where the object need not start from rest? The product of the force and the time for which it’s applied gives…?”
“The change in the object’s momentum.” Eusebio lifted a sheet of calculations. “I confirmed that.”
“Good. So, if two objects interact—if a child throws a stone at an approaching train, and it bounces off the front carriage—what happens to their momenta?”
“The force of the train on the stone is equal and opposite to the force of the stone on the train,” Eusebio replied. “And since both forces act for the same amount of time, they cause equal and opposite changes of momenta: as much as the stone’s momentum rises—measured in the direction of the train’s motion—the train’s will fall.”
Yalda said, “So the total, the sum of the two, is unchanged. What could be simpler?”
“Momentum is simple enough,” Eusebio agreed. “But energy—”
“Energy is almost the same!” Yalda assured him. “It’s just that instead of the product of force and time, you use the product of force and distance traveled. What’s an easy way to turn the first into the second?”
Eusebio thought for a moment. “Multiply it by distance over time, which is the average velocity. For an object that started from rest and accelerated smoothly, that’s half the final velocity it’s reached. So the product of the force and the distance traveled is the product of momentum and half the velocity… or half the mass times the velocity squared. The kinetic energy.”
“Exactly,” Yalda said.
Eusebio understood these calculations well enough, but he was less happy with the bigger picture. “Energy is where the ‘conservation laws’ start to sound more like a long list of exceptions,” he complained.
“Maybe. Tell me about the exceptions.”
“Gravity! Drop a book from my window; its kinetic energy certainly won’t stay the same. And the fact that the book pulls the world up toward it with as much force as the world pulls it down doesn’t help; that keeps momentum balanced, but not kinetic energy.”
“Sure.” Yalda brought one of the diagrams she’d rehearsed onto her chest.
“If you plot the downward force on the book against its height above the ground,” she said, “it’s a constant, a flat, straight line. Now think about the area under that line, up to the point representing the book’s current height. When the book falls, the reduction in the area—the little rectangle that gets chopped off—will equal the force on the book times the distance it travels—which is precisely the amount by which its kinetic energy increases: force times distance.”
Eusebio examined the diagram. “All right.”
“Alternatively, if the book is tossed upward and gravity starts to slow its fall, it will be losing kinetic energy… but the area under the line will increase in a way that precisely balances the loss. So, we call this area ‘potential energy’, and the sum of the two kinds of energy, kinetic and potential, will be conserved.
“This works for other simple forces, too—like the force on an object attached to a stretched spring.”
Eusebio said, “I understand why the mathematics works out as you’ve described it. But isn’t this just a fancy way of saying: kinetic energy isn’t conserved, it changes… and in a few simple cases we understand the forces responsible, well enough to be able to keep track of the changes?”
“Well, yes,” Yalda agreed. “It’s a kind of accounting. But don’t disparage accounting; it can be a powerful tool. Elastic potential energy can tell you how fast a projectile will fly out of a slingshot; gravitational potential energy can tell you how high that projectile will rise.”
Eusebio wasn’t persuaded. He gestured at his marching figurines; two of them had wound down and come to a halt, while the third had ended up on its back, kicking its legs ineffectually. “In the real world, energy isn’t conserved,” he said. “It comes out of food, or burning fuel, and it vanishes as friction.”
“That might sound like the best explanation,” Yalda said, “but those processes are just more complex examples of the very things we’ve been discussing. Friction turns motion into thermal energy, which is the kinetic energy of the constituents of matter. And chemical energy is believed to be a form of potential energy.”
“I understand that heat is a kind of invisible motion,” Eusebio said, “but how does burning fuel fit into this scheme?”
Yalda said, “The way to make sense of a particle of fuel is to imagine a ball of springs all knotted together tightly, then tied up with string. The action of the liberator is like cutting the string: the whole thing flies apart. But instead of a tearing sound and springs flying everywhere, from fuel you get light and hot gas.”
Eusebio was bemused. “That’s a charming image, but I don’t see how it helps in any practical way.”
“Ah, but it does!” Yalda insisted. “By reacting various chemicals together in sealed vessels—which trap all the products, and turn all the light into heat—people have built up tables showing how much potential energy different substances have, relative to each other. Fuel and liberator are like something on the tenth floor of this tower, while the gases they produce are on the ground floor. The difference in chemical energy manifests itself as pressure and heat, just as the difference in gravitational energy, if you dropped a book from that height, would manifest itself in the book’s velocity.”
Eusebio was growing interested now. “And it all works out? Chemical energy is like a kind of accounting, it’s as simple as that?”
Yalda realized that she might have oversold the idea, just slightly. “In principle it should work, but in practice it’s hard to get accurate data. Think of it as a work in progress. But if you ever go out to the chemistry department—”
Eusebio buzzed amusement. “I’m not suicidal!”
“You can always watch their experiments from behind the safety walls.”
“You mean the ‘safety walls’ that need to be rebuilt three or four times a year?”
The truth was, Yalda had only visited Amputation Alley once herself. She said, “All right… be content to reap the benefits from a distance.”
“You say it’s a work in progress,” Eusebio mused. “Fatal explosions aside, what’s the hitch?”
“I’m no expert in their methodology,” Yalda admitted. “I suppose there’s room for errors to creep in when they measure temperature and pressure, and I expect it’s also hard to trap all the light. We can measure the energy in heat, but if there’s light emitted we don’t know how to account for that.”
“So how exactly do you know that they’ve made mistakes?” Eusebio pressed her. “What is it that tells you that their data is wrong?”
“Ah.” Yalda hated to disillusion him, but she had to be honest about the magnitude of the problem. “Someone showed that the values in the last table they published could be used, indirectly, to derive the result that pure, powdered firestone and its liberator contained only slightly more chemical energy than the gases they produce—nowhere near enough to explain the high temperature of the gases. But that extra thermal energy can’t just fall out of the sky; it has to come from a change in chemical energy. And that’s before you even start worrying about the energy carried off by the light.”
“I see,” Eusebio announced cynically. “So ‘chemical energy’ is a beautiful theory… but after all that risk and toil, the results show that it’s actually nonsense?”
Yalda preferred a different interpretation. “Suppose I told you that a friend of a friend of mine had seen a pebble drop from a third floor window, but you knew that the pebble in question had hit the ground with a deafening crash, and made a crater two strides deep. Would you throw out the whole idea of conservation of energy… or would you doubt my third-hand account of the height from which the pebble had fallen?”
Yalda squeezed into the lecture theater just as the guest speaker, Nereo, began ascending to the stage. There were only about four dozen people in the audience, but the venue had been chosen for its facilities, not its capacity, and the optics classes that were given here usually attracted just a couple of dozen students. Her late entry brought some resentful glares, but at least her height gave her the advantage of not needing to jostle for position—and when she realized that she was blocking the view of the young man behind her, she quickly changed places with him.
“My thanks to the scientists of Zeugma for their generous invitation to speak here today,” Nereo began. “I am delighted to have this opportunity to discuss my recent work.” Nereo lived in Red Towers, where his research was supported by a wealthy patron. With no university there, he had no colleagues around him to challenge or encourage him, though perhaps the whims of a rich industrialist were less onerous to deal with than Zeugma’s academic politics.
“I am confident,” Nereo continued, “that this learned audience is intimately familiar with the competing doctrines regarding the nature of light, so I will not spend time recapitulating their strengths and weaknesses. The wave doctrine rose to favor over the particle doctrine more than a year ago, when our colleague Giorgio showed that two narrow slits in an opaque barrier, illuminated with light of a single color, cast a pattern of alternating bright and dark regions—as if waves emerging from the two slits were slipping in and out of agreement with each other. The geometry of this pattern provided a means of estimating the light’s wavelength—and the measurements suggested a wavelength for red light about twice that for violet.”
Yalda looked around for Giorgio, her supervisor; he was standing near the front of the audience. She’d found his experiments persuasive, though many long-time proponents of the particle doctrine were unmoved. Why invoke some fanciful notion of “wavelength”, they argued, when every child who’d ever glanced up at the stars could see that what distinguished one color of light from another was simply its speed of travel?