Work Text:
If Chrome had known how difficult learning the science behind the sorcery that had guided most of his life—if he had known it would include passionate early-morning lectures about how antiderivatives and integrals are completely different things, Chrome, you can’t just use those terms interchangeably—perhaps he would’ve reconsidered his undying quest for knowledge.
He had made one passing remark guessing that an integral, essentially the opposite of a derivative, should therefore just be called an antiderivative, and that had set Senku off. He didn’t even know that “antiderivative” was even a term mathematicians fought over, it was just a seemingly natural conclusion that he pulled out of his butt.
And now Senku’s been going on about it for nearly five hours now. Okay, more like five minutes, but Chrome’s never been a morning person, and the linear progression of time feels anything but linear when all his brain power is spent keeping himself awake. His hand is propping up his head, and he’s shocked out of his bored silence when he notices not only is there drool going down his chin, but it’s also dripping onto the desk.
“This is the fundamental theorem of calculus, so it’s crucial you understand this. It all ties back to the rates of change and initial conditions I taught you for algebra,” Senku continues passionately from across the table, not noticing how Chrome is wiping the table with the end of his shirt.
“So looking back at this problem, we take this integral with respect to what?”
Senku looks at him like he wants Chrome to respond. Crap, what was he even talking about?
“The... uh...”
He’s still getting That Look.
Chrome rubs some sleep from his eyes and finally decides to come clean. “Senku, I know you say this stuff is important, but you’re losing me a little bit with all this calculus. I’m just trying to understand how this applies to real life.”
“Chrome, calculus is the underlying foundation for the laws of physics, which govern our—”
“You keep telling me that,” he interrupts, “but that doesn’t explain how or why. We’ve used plenty of algebra and trigonometry to help build the ship, but I don’t see where taking limits to infinity fits into all that.”
Senku considers this for a moment. “I think I get it. You need a real-life example to prove that all this new stuff isn’t nonsense.”
Chrome nods excitedly, hoping that Senku can give him some calculus-related invention to work on that will help make everything click into place.
“Okay, then. The training wheels are officially coming off. You know that acceleration due to gravity is a constant negative 9.8 meters per second squared. I want you to use that fact and the math I’ve taught you to figure out how long it would take for me to drop this block”—he picks up a wooden block used to help teach the younger children counting—“off the top of the mast of the ship.”
Chrome can only stare at him in utter shock. Not only is this task entirely different from what he was expecting, it brings up more questions than it answers. Until this point, his intuition and lifelong experience as an explorer have been sufficient enough in helping him navigate the world of science, but this seems far more theoretical than the problems he normally tackles. Before he can even come up with a response, Senku’s packing up his papers, writing utensils, and teaching supplies, because he’s got places to be and no time to waste. Chrome can only manage to form words as Senku is heading out the door.
“You’re serious?”
Senku laughs. “Yeah. Remember how acceleration, velocity, and displacement are all related. Remember what I’ve told you about rates of change, derivatives, and integrals. And don’t forget your initial conditions. In a few hours, I’m heading back to Ishigami Village for a week. Have an answer for me by the time I get back.”
And then he leaves, just like that.
Chrome has a bit of free time during the rest of the morning, so he decides to spend that time mulling over the problem he’s been given.
How long would it take to drop a wooden block off the top of the mast of the ship?
It’s a beautiful day outside, but he tells himself that in order to focus he has to stay inside. There’s a blueprint room where Senku, Ryusui, and the other engineers in charge of the ship keep all of their plans and files. It may be stuffy, but maybe being surrounded by all this incredible work will give him motivation to do some of his own.
If only Kaseki weren’t also here, working loudly on some contraption for the ship. He wants nothing more than to be working on some project of his own, before remembering that this assignment is his project for now.
Does Kaseki really have to be chipping away at something, right here of all locations? Won’t the debris get all over the papers and construction plans? Chrome sits there, silently fuming, desperately jealous, and wishing he had a hands-on activity as well. He stares at a blank piece of paper that’s supposed to be flowing with ideas on how to tackle this problem. If only he could come up with something.
After a particularly loud CLUNK!, he crumples up his blank sheet of paper and tosses it in the waste bin. Looking over at Kaseki, he finally gets his lingering curiosity get the best of him. “Why are you doing that here?”
Kaseki looks up from his work and chuckles. “Didn’t see you there, Chrome! Figured I’d give myself a change of scenery while working on this. That reminds me! Senku told me about your little project he’s left you! How’s that coming along?”
“It’s fine, I guess,” he says, unable to meet Kaseki’s eyes and instead staring longingly at the tools he has with him.
“You don’t look fine,” Kaseki replies, honest as ever. “From what Senku told me, he didn’t say anything about not asking for help. You don’t have to do this alone. Everything in the Kingdom of Science has always been a team effort, so why not this problem too?”
The stress clenching every inch of his insides immediately dissipates. Kaseki is right.
“Oh,” is all he can manage to say. His grip on his pencil relaxes. Senku hadn’t said he wasn’t allowed to consult with other people or ask others for help, but of course he also hadn’t told him outright that he was allowed to. Just the kind of move he would pull.
“Well,” he starts, supposing he may as well ask, “what do you know about calculus-based physics?”
“Nothing!” Kaseki tells him, chipping away again. “But I figure my age has taught me a thing or two about problem solving. Start with what you know. Then figure out exactly what it is you need to find. There has to be some way to take what you know and turn it into what you need.”
Five minutes ago, Chrome was questioning the old man for choosing this location as his working spot, but now he realizes how much of a blessing it actually is. He still can’t manage to write out anything, though.
“So what is it that you know?” Kaseki asks.
He answers with a question of his own, tapping the eraser of his pencil rhythmically on the table. “Do you really want a condensed version of all the math that Senku has taught me?”
“No, like I said: start with what you know from the problem Senku gave you. I think he mentioned something about a gravity constant. Not quite sure what that is, but I assume you know what I’m talking about.”
Chrome shrugs and writes down the acceleration due to gravity.
“Well, that’s negative 9.8,” he replies, glad to have something on his paper.
“Negative 9.8 of what? Glass shards? Cups of tea? You’ve got to clarify this stuff for me. I never really learned all this fancy math.”
“Meters per second squared,” he says, quickly adding in his units and chuckling to himself. You have to remember your units, Senku always tells him. The units will tell you everything you need to know.
“Meters per what? Does that have to do with distance? Hmm, maybe it’s best that I stick to arts and crafts...” Kaseki continues on, but Chrome can barely pay attention because he’s just had a bad revelation. The height! That’s probably something he needs to know. He immediately jumps up from his seat, searching around for the longest measuring stick they have.
“W-where are you going?” Kaseki asks as he bolts from the room, desperate to follow his new lead.
“I need to find the height of the mast! I don’t know how, but I think that’ll help me!” he calls over his shoulder before sprinting to the shipyard.
By that afternoon, Chrome has figured out that he needs the height of the mast, but not much else. A piece of paper is tucked in his pocket that simply states the two pieces of information he knows and the thing he’s trying to find:
There are beads of sweat forming on his forehead from his work in the fields, and he takes a moment to wipe it away. Kohaku had teased him earlier when he said he’d be helping Taiju in the fields later that day, saying that he should be working on his problem instead. Loathe as he was to confess it to her, there’s not much else he can do about it. Even though he’s spent months and months learning all this specialized knowledge from Senku, everything seems to fall apart in his brain the moment he tries to piece it together. Chrome knows he does his best work with hands-on projects, so maybe he can trick his brain into making some progress on his very much hands-off project by doing some manual labor in the fields first.
“Hey Chrome,” Taiju says, brilliantly smiling as always. He’s a couple feet away, also harvesting wheat. “Senku told me about the task he gave you before he headed out! How’s it going?”
Chrome doesn’t feel like lying. “Terribly,” he replies, ignoring Taiju’s gaze and crouching back down. “I make one snarky comment about how none of the new stuff he’s teaching me makes sense in the real world, and he gives me this weird theoretical assignment?”
Taiju takes a moment to ponder these words. “I’ve known Senku for a very long time,” he finally says, “and I don’t think he’d give you a problem he knew you couldn’t solve. He will mess with people sometimes, but not when it comes to science.”
Chrome can only sigh in response. Taiju is correct, but that doesn’t make this any less daunting. Remembering his conversation with Kaseki from earlier, he figures he’ll make a shot in the dark and ask Taiju for help. “You know algebra and geometry, right? Did you ever learn calculus?” He doesn’t know when people would normally learn calculus in the old world, so he prays Taiju might know what he’s talking about.
Taiju’s response is slightly dejected. “We were first years in high school when we got petrified, so we never really got to learn this kind of physics and math. Maybe if I had graduated by then I could help you with this.”
“Haven’t you been friends with Senku for forever? Have you picked up anything from him that might help?” Chrome asks, reaching over to pick out some weeds.
“Senku would always complain about how schools teach math in a way that doesn’t properly prepare people for calculus. Kept going on about rates of change and initial conditions and how that all relates to derivatives and integrals, but his explanations always went way too over my head.”
Chrome stops picking at the weeds to think about this, because another idea has sparked in his head. Rates of change. That’s a phrase he’s heard far too often in his recent lessons with Senku. “Well, rates of change at least are pretty straightforward,” he starts, sitting down and beckoning Taiju to join him right there in the hot dirt.
“Think about the car we made,” he says, badly doodling a car on his paper. “If we want to find the speed of the car, we just need to take the distance it travels and divide it by the time it takes to cover that distance. That’s called a rate of change.” He sketches a road for the car to drive on and an arrow to suggest movement.
“I see, like a speedometer on the dashboard! Tells you how many kilometers per hour the car is travelling,” Taiju says, nodding. Chrome isn’t sure what a speedometer or a dashboard are, but he continues his explanation anyway.
“Though when we’re talking about things moving in straight lines, we use the terms velocity and displacement instead of speed and distance. So velocity is the rate of change of displacement, and acceleration is the rate of change of velocity,” he says, jotting this information down below his scratchy drawing:
“Wait a minute,” he says, tapping his pencil on the piece of paper, a new revelation rapidly approaching in his mind.
“Like I said!” Taiju exclaims. “He wouldn’t give you a problem that he knew you couldn’t solve. All this sounds familiar from science class so far!”
“The derivative is the instantaneous rate of change,” he recites automatically before realizing he’s saying what must be complete nonsense to Taiju.
“Huh? You sorta lost me there,” Taiju admits sheepishly, scratching at the back of his head.
“Sorry, that’s just a thing that Senku says is so important, he’s made me verbally repeat it dozens of times.” Chrome adds this revelation to his paper:
“Um... Um...” he says, scrunching his eyes closed and absentmindedly grabbing a handful of weeds right from the ground. “I have the displacement. That’s 22 meters. So do I take the derivative of that to get my velocity? And then take that derivative to get my acceleration?” He asks out loud, fiddling with the weeds in his hands. Taiju just shrugs.
“I suppose you can try that,” he says encouragingly. Without thinking about it, Chrome throws the weeds right back on the ground so he can focus in, and Taiju is kind enough to pick them up and pocket them for him.
Chrome’s heart speeds up, sure he’s reached another breakthrough, before realizing his issue. “I can’t. d is just a number and the derivative of just a number is always zero.”
All this momentum, the supposed weight off his shoulders thinking that he’d finally cracked it, even giving a Senku-style scientific infodump to Taiju, and he’s still left with no real progress. He takes his paper, crumbles it up, and tosses it as far as he can throw. It doesn’t go very far at all, which does little to let out his frustration. This derivative dead-end is disheartening, but since when has he ever backed down from a problem? He broke out of that jail cell with nothing more than a battery and his own sweat! That didn’t require anything more than my intuition, and this stuff is less than intuitive.
“I bet this is frustrating, especially when you have so little to work with.” Taiju un-crumples the piece of paper and hands it back to him. He must’ve retrieved it while Chrome was moping. “But hey, you have more information than you did before! You just need to talk through it out loud.” Taiju stands up and reaches out a hand to help Chrome up, which he accepts.
Taiju continues, picking up his tools again, “So tell me everything you know about derivatives! I bet you’ll remember something that can help set you on the right track.”
“Well,” Chrome begins, getting back to work himself, “it’s only slightly related, but Senku spent like half an hour of our last 8 AM class lecturing me about how antiderivatives and integrals are not the same thing, even though they totally are.” He knows that this is a memory that will haunt him for the rest of his days.
Taiju gives a booming laugh at this. “Sounds like something he’d do!”
“Though now that I think about it,” he says, taking a moment to scratch at his chin, “even if they’re not exactly the same thing, an integral might be the way to go about solving this problem.”
Chrome has the next day off from any manual labor, which is nice because he spent all of the prior evening and most of the night reviewing his notes, thinking of how he could tackle this new integral idea. There’s an outdoor lecture hall used as a classroom for the younger kids these days, and he decides to spend some time there. He doubts the kids would know anything about integrals, but he supposes it wouldn’t hurt to also be surrounded by people working on math problems.
That’s how he finds himself sitting across from Suika, the both of them intensely focused on the math in front of them. Chrome is trying to review all his notes on integrals, and Suika is doing an addition worksheet.
His notes make sense in a formulaic way; he’s been taught many different strategies for how to look at an integral and determine the best way to approach the problem.
The thing that’s missing, and the reason why he was given this task in the first place, is that he doesn’t understand how all these numbers and letters relate to real life. His stack of notes is quite tall, and it’s daunting to review them. He’s extremely tempted to take a break and take a new mineral sample he found earlier that morning to the lab for analyzing—he thinks it could be obsidian, but he knows that if it is, he won’t be able to bring it up to Senku over the phone without being quizzed about the problem he’s been given.
“You look frustrated,” Suika tells him without even looking up from her addition worksheet.
“I guess I sorta am,” Chrome says simply, reaching into his pocket to roll the rock around in his fingers, praying the sample will somehow imbibe within him the knowledge that he needs.
“Why?”
“I can’t figure out this math problem. I figured out I need to use an integral, but I don’t know how to set it up,” Chrome admits. “I’ve never set up an integral myself, I’ve just evaluated other ones given to me.”
“Hmm, we’re not so different, you and I,” Suika says, setting her pencil down and erasing what she just wrote. “You have your stupid integrals, and I have all these stupid addition problems.”
“Do you need help with your addition worksheet?” Chrome asks, eager to feel as though there's some math out here he can do.
“I think I can do it, I just made a mistake,” she says, finishing her erasing.
He is obviously aware that antiderivatives and integrals are not the exact same thing, but since Senku didn’t bother mentioning the difference, he decides they’re basically equivalent for what he needs to do here:
“So what do I do now?” he wonders out loud.
“I don’t know,” Suika says, shrugging. “But you’ve gotta try something.”
Suika is ten years old and she’s doing an addition worksheet, and she definitely doesn’t know anything about integrals, but Chrome has a feeling she’s right anyway.
Velocity is the integral of acceleration.
At least he has something resembling an equation now.
For the next few hours, this equation is scribbled on his paper, a source of truth added to the other bits of information he has, but he knows he’s still missing something else. Going through all his old notes, he noticed how every other integral he did had a dx or a dt or a du at the end. The textbook definition of the letter next to the d, as Senku has so constantly drilled into his mind, is “the variable that you take the integral with respect to.”
With respect to? Chrome wishes he’d had asked for some clarification on that instead of mindlessly jotting it down and taking it as fact. He’d added a dx at the end of his equation because x is a pretty standard variable in the world of math, but got nothing useful after evaluating the integral.
“Velocity is acceleration times x,” he says to Yuzuriha at dinner that night. She’s working on some new clothing and has brought some hand sewing with her, which she expertly works on between bites. “But what is x? That part doesn’t make sense.”
Yuzuriha was also petrified during her first year of high school, and therefore can’t offer much mathematical insight. “You said you took the integral with respect to x?” she asks anyway, engaging him for the sake of hopefully making some progress.
“Yeah, but I never really understood what that meant,” Chrome sighs, taking the mineral sample he had found earlier out of his pocket and once again rolling it between his fingers.
“And x is just some kind of variable? The thing you’re trying to find? I remember that from algebra class,” Yuzuriha says, and Chrome squints over at the fire used to cook the meal, trying to piece together this new insight.
“I have to take the integral with respect to something... with respect to...” he trails off.
“I think you need some sleep. We can talk again in the morning. I think you’re getting close though!” she says cheerfully, cutting a loose thread with some scissors. Chrome decides to take her advice and heads to bed early.
Unfortunately, trying to sleep doesn’t help much. The problem keeps him up almost all night.
The sun has long since set, and Chrome’s getting tired of tossing and turning, his mind completely consumed by the problem he’s been given. This is ridiculous, he thinks to himself, climbing out from under his sheets, tip-toeing past Gen and Taiju towards the entrance of their hut. After slipping outside, he glances up at the sky, where the stars are shining as brightly as always. He’s not the best at telling the time from the stars, but he estimates it’s between two and three in the morning.
One of the first things Senku had taught him and the rest of the village was a more accurate way to tell time: hours, minutes, and even seconds. Before, they had simply relied on the sun to divide up their day. They never had any terms more specific than “sunrise,” “sunset,” or “midday,” because the way they lived their lives, they didn’t need anything more distinct than that. They tracked the months and the moons and the seasons, but then Senku had brought with him a world where time meant so much more. Everything was on a deadline, and there was never enough time. It was a race to finish the medicine for Ruri, a race to finish the telephone over the previous winter, and now, a race to finish the ship so they could set sail before the upcoming winter.
Chrome blinks and everything comes back into focus. While he had been mulling all that over, his feet had taken him to the shipyard, obviously deserted at this time of night. He sits down and curls his knees up to his chest, staring at all the progress they’ve made so far. With the main deck nearing completion, Chrome knows it won’t be much longer until Senku comes back, expecting an answer to the question he had been given.
He can’t figure out this whole “with respect to” thing, and at this rate, he’ll have nothing but a bunch of disorganized ideas to present to Senku upon his return. He’s running out of time. Damn the concept of time! Everything used to happen with respect to mother nature, whatever she had decided to give the village. And now, everything is happening with respect to time.
I know I have to integrate with respect to something... with respect to time!
It’s as if the gears in his mind have clicked into place, just like the gears clicked into place when he and Kaseki were making the water wheel.
“I just had a crazy bad flash of inspiration!” Chrome yells to himself even though it’s the middle of the night. He trips over thin air in a frantic scramble to get up and race down to the classroom where he keeps his notes.
As he pores through his notes, he finds himself in the company of Gen.
Gen often spends his time doing what seems like a lot of lazing around, but Chrome has figured out he’s actually providing some weird kind of moral support. He must’ve heard about Chrome’s newest plight, because he just so happens to sit himself at a bench across from him in the lab, a place he doesn’t frequent often, while absentmindedly playing with a deck of cards.
“So,” Gen starts after a bit of silence. “I’m dying to know. How’s your roblem-pay coming along?”
“To be honest, I simultaneously feel like this is going nowhere and that I’m just about to crack this thing wide open,” Chrome admits right out of the gate.
“Well, you’ve been consulting a whole bunch of people who’ve never taken a day of calculus in their life, correct?” Gen asks, and Chrome thinks that he really must’ve been paying attention to his struggles this whole time to have taken note of that.
“I guess it’s always nice to find someone who’ll listen, even if they don’t totally understand what’s going on,” he says, before realizing what Gen really meant by that statement. “Hang on, didn’t you graduate high school before you got petrified?”
“Yeah, about two months prior,” Gen says, a smile creeping up on his face.
“Did you take calculus at all? Can you help me out then?” Chrome asks, hope building up in his heart.
“I did take calculus, actually!”
“Well then, can you check how I set up this integral? Did I do it right?” Chrome shoves his notebook in Gen’s face with a bit too much force.
Gen glances over the paper for a moment.
“Where v is velocity, a is acceleration, and t is time, of course,” Chrome says with immense excitement.
“I hate to break it to you, but I was never any good at math. I was too busy focusing on the book I had just published to pay much attention in math class,” he says, and Chrome’s excitement immediately deflates.
“Okay, let’s just assume I’m right.” Chrome grabs his notebook back to evaluate the integral.
“That seems a bit too straightforward,” he mumbles to himself. “This can’t be correct.”
“Well, I could still check it for you if you want,” Gen offers, reaching back out for the notebook.
“You just said you didn’t pay attention in math class!” Chrome objects.
Gen stands up so he can read what Chrome wrote upside down. He points at the equation that was just written. “You have to add a +C. I can’t count the number of times my math teacher told me to remember the +C.”
Chrome looks down at his work and sighs, adding a +C.
Well, it seems that Gen’s experience is universal, because he himself can’t count the number of times Senku has told him to remember the +C. He knows the +C means “plus a constant,” but figuring out what the constant is will likely present its own set of issues.
Later that day, a phone call with Senku confirms that he’s on the right track. He rattles off his equation into the microphone with confidence.
“That sounds about right,” Senku says with a laugh. “You even remembered to add your +C!”
“I would’ve forgotten if it weren’t for Gen,” Chrome says without thinking about it. He had never confirmed if he was actually allowed to ask others for help, and he sits there in a slight panic for just a moment wondering if Senku will call him out on it, but that doesn’t happen.
“You have to keep going, though,” Senku says cryptically.
“What do you mean I have to keep going?” Chrome questions.
“Well, you don’t know your velocity, do you? Or what that +C is really supposed to be, right?” Senku asks. “So keep going.”
“Are you going to give me any more hints, or just be vague about this?” Chrome asks, a little snippish.
“I’m not being vague at all. You have all the information you need to solve this problem,” Senku says. “I need to go. It was good checking in with you, and I’ll be back in a few more days.” Senku wastes no more time with goodbyes and hangs up the phone. Chrome almost wishes that the call had gone on for longer, but he knows that he wouldn’t get any more information out of Senku anyway.
Oddly enough, he feels like he needs company to continue his problem. Each revelation he’s had has been with the help of one of his friends, and he feels the need to continue on with that pattern to keep going. Gen’s comment about him only consulting people who hadn’t taken calculus rings through his mind, and he knows exactly who he’s going to bother next.
“Ryusui,” Chrome says, entering the comms room of the Perseus, the newest location Ryusui frequents ever since it was built. “Do you think you could help me with this problem Senku gave me?”
“I’m trying to fix a problem with the ship’s engine,” Ryusui responds. Well, that wasn’t an outright dismissal. He can still work in the general vicinity and maybe weasel some information out of him. Without asking, Chrome sets his things down on a table next to the desk and gets his notes out.
That’s where he had left off. Senku said he needed to keep going, and the only thing he can gather from that suggestion is that he needs to integrate again. His notes confirm his suspicion on the matter:
He has his velocity, he can figure out the +C later, but he needs to connect velocity and displacement, and the way to do that seems to be integrating again.
He knows he already figured out the velocity equation, and substitutes that in.
Plugging in the equation seems simple enough, and the integral is also simple enough to evaluate, leaving him with:
“That can’t be it,” he mutters to himself. Ryusui, who had been paying no attention to him before, appears to be interested all of a sudden.
“What can’t be it?”
“Well, it looks like I have two +Cs,” Chrome says, poking at his notes and not bothering to elaborate.
“Are you integrating something twice?” Ryusui asks, and Chrome could almost cry tears of joy. Finally, finally, after all this time, he’s talking to someone who has more than an ounce of knowledge on the topic. Who may, in fact, know even more than him about it. He leaps up, grabbing his notebook and pencil and runs over to Ryusui’s desk, slamming it down right on top of the engine blueprints.
“Yes I am,” he confirms, somehow out of breath from only moving a few feet. “I am integrating twice, and I have no idea what I’m doing. Do I use the distributive property here? What even is the constant in the +C?”
“Slow down there.” Ryusui now looks as though he regrets opening this can of worms.
“Sorry,” Chrome replies. “You’re just the first person I’ve talked to who really knows what an integral is. I can’t believe I didn’t come talk to you earlier.”
“Just give me a second,” Ryusui says, looking over the notebook. A spark of recognition crosses his face. “Are you deriving the kinematic equations?”
“Uhh... I don’t know. I’m just trying to figure out the problem that Senku gave me, and I guess this is where I ended up,” he responds. Ryusui gives a knowing look.
“I won’t say much, but these are different +Cs,” he says, pointing at the two equations.
“Different? Why are they different? Isn’t +C just +C?” Chrome asks, trying to sap as much information out of Ryusui as possible.
“They just are. I’ve walked by during some of Senku’s lectures he’s given you. Not only are they different, they’re also related to something else.” Ryusui’s vagueness is infuriating, but at least it’s something.
“How do you know all this calculus? I didn’t think sailing required extensive knowledge of math,” Chrome wonders aloud.
“This is a perfectly normal amount of math to know,” Ryusui tells him with confidence, before reconsidering slightly. “Though I will admit my frame of reference might be skewed. My brother is—”
“Forget that, I’ll give you all my Drago if you tell me more about these +Cs,” Chrome says, fishing around in his pockets. Sadly, the only things in there are a spare pencil and various rocks he’s picked up recently.
“Haha!” Ryusui exclaims, accompanied by his trademark snap, “You can’t bribe me!”
“I can’t bribe the greediest man in the world?” Chrome asks sarcastically, trying to recall if he has any spare Drago stashed away in his hut.
“Senku already paid me off! Bribed me before you could, so to say,” Ryusui says, before standing up from his desk to go back to the comms console.
Chrome is left to ponder his +Cs all alone. If they really are different, then fine. He’ll note that in his equations:
Maybe he can still extract some more information out of Ryusui, so he prods him from a different angle. “So what’s wrong with the engine?” he asks.
“Senku went a little too overboard trying to add an additional control system. It started malfunctioning after he left, and I’m trying to disconnect it from the engine.” Ryusui has the control unit open, spilling out mysterious wires and connections.
“Oh, really?” Chrome adds, urging him on.
“It’s like he’s forgotten I’m the world’s best captain! I don’t need an extra control unit to account for initial weather conditions when my sailor’s instinct will work just fine!” Ryusui says with a wink.
Initial conditions.
“Oh my god, they’re initial conditions,” he says, staring at his +Cs. “Did you say that on purpose? You had to have said—”
“—Hey, sorry I’m running late.” It’s Ukyo at the door, showing up with some strange equipment that is likely meant to help with the engine problem. This is bound to occupy Ryusui’s attention completely, but that silly little wink has him convinced he’s finally, finally close to actually cracking this.
Ukyo is quick to get started with the equipment, and of course it’s obnoxiously loud, meaning Chrome will have to go somewhere different if he wants any chance at being able to focus.
“Thanks for the help!” he calls out behind him as he sprints out the door, not bothering to close it behind him.
Ever since the Kingdom of Science merged with the Empire of Might, the population has increased significantly, meaning that there are few places that are totally empty. Chrome has had so much help from everyone over the past few days, but he knows he needs to put these last few pieces together by himself.
The only deserted place is a nearby river, and as he treks there, he overhears various conversations about a phone call from earlier: Senku has departed Ishigami Village early and will be returning in a matter of hours. Chrome’s heart pounds at this information, but he knows he’s so close to getting it.
He perches on a large stone on the side of the river, where a family of ducks are slowly drifting by in the stream. Yeah, he promised himself that he’d figure out this last part on his own, but talking out loud to some wildlife shouldn’t violate that self-imposed rule, right?
“Okay, here’s what I’ve got,” he announces, pulling out his notes again.
There are so many Cs in that second equation that he decides to just focus on the first one.
“So what is really going on here?” he asks himself. “Velocity equals acceleration times time, plus an initial condition. That has to be... the initial velocity?” he wonders aloud, looking up at one yellow baby duck that trails behind its family, swimming slightly closer to where Chrome sits.
The duck does not respond.
“Let’s go with initial velocity. Like, velocity when time equals zero. So let’s call that v0.”
He rewrites his equation.
He’s seen Senku jot down plenty of fancy physics equations over the past year, and if nothing else, this sure does look like one of them.
“So if C1 is really just v0, or the initial velocity, then...” He starts re-writing his second equation. “Does that make C2 the initial displacement?” He looks back up at the duck again, who is now a few feet away from him, making intense eye contact. “Uh...”
The duck is rather intimidating. He decides that C2 is the initial displacement, or the displacement when time equals 0: d0.
“Wait a second,” he says, peering at the two equations he has now. “The initial displacement. That’s when the block is at the top of the mast. That’s 22 meters. And the initial velocity, that’s, um...”
The duck is still making uncomfortable eye contact. It’s completely unmoving, still as the stone statues still scattered around the outskirts of the settlement.
It’s not moving.
“The block’s not moving! So the initial velocity is 0,” he says, taking his equation and adding the information he’s figured out.
“Well, 0 times anything is 0. That’s obvious,” he says, crossing out the middle term.
“Hey Chrome, Senku’s gonna be here soon,” calls a voice from the distance, and he grimaces. Kohaku sometimes does not know when to lay off the teasing, and granted, most of the time the teasing is mutual, but he doesn’t need her snark at this moment. He ignores her call, hoping she’ll get the clue and go away.
“And that d,” he says to the fluffy yellow duck, which has stopped giving him uncomfortable eye contact to swim back towards its family, "the only other displacement I can think of is the one at the end. Which would be 0 meters because the block would be on the ground?”
“Chrome, are you talking to a duck right now?” Kohaku asks, a few feet behind him.
“Kohaku, please. I’m so close to getting this. Please don’t laugh at me,” Chrome begs, setting his pencil down to rub at his tired eyes.
“I’m not laughing at you. I was just asking a reasonable question.” Her voice is slightly trembling as she speaks. Liar.
“Well, if you’re here, you can join this duck in listening to what I’m puzzling out,” he tells her, picking his pencil back up.
“I just don’t get why there are so many letters in your math problem. Math is supposed to be numbers.” Kohaku looks intently at the duck before shrugging and sitting down next to him.
“So if d is 0 meters, and a is negative 9.8 meters per second squared...” he mutters, filling in the new information:
“All that’s left is... to solve for time. Holy shit, little duck, I think I have it. This looks just like an algebra problem now! I can totally do this!” he yells in excitement, rearranging the equation to solve for time.
“Little duck? I thought I was a gorilla to you. Unless you’re still talking to that duck over there?” Kohaku muses, resting her chin on her hand.
“Don’t judge the methods a scientist uses to work through a problem,” Chrome replies, jotting down what he thinks is his final answer.
Off in the distance, he can hear cheers and calls from some people. Turning around, he can see the hot air balloon up in the sky, meaning Senku has almost returned. He adjusts the grip on his pencil and forces himself to slow down, double checking his arithmetic and circling his final answer.
“Well, did you do it?” Kohaku asks. She glances behind her, where a few other people are approaching them, beckoning for Chrome.
“I think so,” he says, standing with determination.
“Chrome, Senku wants to see you!” Taiju roars with excitement, sprinting towards him.
“I suppose we’ll find out,” Kohaku says, standing up to join him.
It’s the moment of truth, and Chrome wasn’t expecting to do this in public. Apparently the whole settlement has heard of his assignment by now, because they’re all eagerly crowded at the shipyard, where Senku is setting up a blackboard. Chrome’s heart rate quickens and his palms sweat. Kohaku tries and fails to hide a grin when he wipes them off on his pants.
“So,” Senku begins, and Chrome’s already panicking because he can’t decipher the look on his face. It has to be the smirk of a mad scientist confident he’s about to see an excellent explanation, because like Taiju said, there’s no way he'd give him a problem he couldn’t solve.
While trying to figure out his expression, Chrome completely misses Senku’s explanation of the problem to the crowd. Next thing he knows, Senku’s looking at him like he’s supposed to say something. He looks down and notices that Senku’s holding a microphone in front of him.
“Wait, what did you say?” he asks, his mind finally returning to reality. Unfortunately, the microphone is close enough to him that this gets picked up and relayed over the speaker, so the whole crowd can see just how badly he was zoning out. There are only a handful of chuckles, which are quickly shushed by others. That's reassuring. They must be rooting for me.
“I said, I can accurately estimate time to the millisecond. So once Suika drops the wooden block, I can tell you how long it takes to reach the ground. But I want you to give me your prediction first. So, Chrome, from 22 meters up, how long should it take to reach the ground?”
The microphone gets shoved back into his face. He looks down to his notebook, at the circled answer he has at the bottom of the page. Chrome swallows before saying, just loud enough for the microphone to pick up. “2.12 seconds.”
Now Senku’s expression is really unreadable. He doesn’t respond, just looks up at Suika and gives a slight nod to her. “Okay!” she cheerfully yells from the top of the mast. “Dropping the block in three... two... one!”
Chrome holds his breath, and theoretically it should be just a bit over two seconds, which is hardly enough time for someone to go light-headed from oxygen deprivation, but he feels like he might pass out anyway. He somehow manages not to faint, and is still standing when the block slams into the ship deck, making a loud CRACK! as it lands and bounces a few times afterwards.
Now he’s staring at the block, heart threatening to beat completely out of his chest. “Don’t lock your knees, dummy,” Kohaku whispers, and when she gently nudges him, it’s almost enough to completely knock him over.
After relaxing his stance, it takes a few more seconds for his vocal cords to kick into gear. “So... was I right?” he asks Senku breathlessly.
“I’ll tell you, once you explain your process. Tell me how you arrived at that answer.” Senku’s face is still unreadable—mainly. It might be him seeing things, but he thinks he sees a bit of that sparkle in his eyes, that look he gets when he sees science at work. Chrome’s stomach lurches as he is handed a piece of chalk for the blackboard.
“Well,” he begins, facing the empty blackboard. He looks down at his notebook and writes down the first equation on the board. “Acceleration due to gravity is negative 9.8 meters per second squared.”
“Speak into the microphone,” Taiju reminds him, setting it up on a stand next to him. He gives a thumbs up and smiles before shuffling off.
“Acceleration due to gravity is negative 9.8 meters per second squared,” he repeats, and he hates how strangled his voice sounds over the speakers. He takes a deep breath before continuing. At least he doesn’t have to see the crowd while facing the board.
“And the height from which the block is dropped is 22 meters,” he continues, writing on the board.
“And Senku said that you can use this, plus the math that he’s taught me, to figure out how long it takes for the block to reach the ground.”
He hesitates for a moment, remembering the crucial conversation he had with Kaseki that set this whole mission on the right track. “Thanks to Kaseki for reminding me to start with what I already know,” he says, turning around to find the old man has made his way to the front of the crowd, ready to cheer him on. A warm feeling swells in his chest, enough to ease his anxieties about the impromptu public speaking.
“I also knew that the rate of change of displacement is velocity and the rate of change of velocity is acceleration. That’s how we can connect the information we’re given.”
Chrome knows he’s correct, he’s basically repeating things Senku has already told him, but he’s terrified of turning around, terrified of seeing what kind of expression Senku might have on his face.
But he knows he wouldn’t have remembered that bit of info without Taiju’s help. So he takes the microphone from the stand with his other hand, and turns around just enough to find Taiju along the edge of the crowd and find his gaze. “And thanks to Taiju for helping me figure that out, even if he wasn’t sure of how to do the rest of this problem.”
He somehow manages to avoid Senku’s gaze before turning around, taking another deep breath before he continues. “To get the rate of change for something, we have to take the derivative.” He swallows, realizing the crowd may not know, or may not remember what a derivative is. He pauses and leans into the mic, “If you want to know what that is, ask Senku, he can explain it better than me.”
That earns some laughs, and he plucks up enough courage to look over his shoulder for a moment. From just that brief glance, he can tell the crowd seems engaged, and he thinks he caught Senku smiling, but he doesn’t look long enough to be sure.
He cringes slightly, before leaning into the mic and quickly explaining his first train of thought he’d had days ago. “And my first attempt at solving this, I tried taking the derivative of the displacement, but that turns out to be zero so I had to try something else,” he rambles out in a single breath.
Everything is silent for a moment.
But he continues, not caring if the audience understood the disaster that just left his mouth. “Even though Senku once gave me a very passionate thirty minute lecture at eight in the morning about the difference between an antiderivative and an integral, from my understanding, and for the purpose of this problem, they’re essentially the same.” It’s at this moment he prays there are no people from the old world who major in math, those who would be likely to be angered by that generalization, but thankfully, he doesn’t hear any heckling, and continues.
“So, I thought I could start with my acceleration and take the integral, then go from there.” He rubs a few things away and re-writes that last line, so it reads:
This time, he turns around to find Suika in the crowd. She’s normally good at hiding, but not this time. She’s right there, in the front row, completely engrossed in what he has to say, though he knows she barely understands any of it. “And thanks to Suika for being there while I figured out the details on that one. You’re a great listener.” She lights up in admiration as he addresses her, and that affirmation gives him a continued boost of confidence.
“It took me a few tries to figure out how to do this and a conversation with Yuzuriha,” he starts, nodding at where she’s standing off to the side of the blackboard. “But to start, we take the integral of the acceleration with respect to time. And that’s because everything in this problem is with respect to time. That’s the thing I needed to find in the first place. And time is what connects the displacement, velocity, and acceleration.”
But Chrome does take a moment to point at the +C, and faces the crowd again. “And, uh, shout out to Gen for reminding me to add the C.” He finds Gen’s eyes and chuckles slightly.
Gen shrugs nonchalantly and calls out, able to project loud enough for the rest of the group to hear him, “That was the thing my math teacher in high school always scolded me about.” Chrome gives a grin in return, feeling more of the tension leave his body. He’s still avoiding eye contact with Senku, though.
“So, we’re about to integrate again to get the displacement.” He adds a 1 next to the C he’s already written for the velocity equation before writing out the displacement equation.
“So it currently looks like adding a bunch of letters, and don’t worry, this also confused me too.” He pauses for a moment, and then in a moment of pettiness, decides to add, “and Kohaku.” A few people in the crowd give a good-natured laugh. This time, he doesn’t turn around, but he can feel her annoyance radiating from wherever she’s standing, her stare boring a hole in the back of his head.
“But then Ryusui helped me remember that these constants are the initial conditions. The initial velocity and the initial displacement. When Suika held that block before she dropped it, it wasn’t moving at all. That means the initial velocity was zero”—he crosses out all the terms with C1—“and she was up 22 meters, so that was the initial displacement. And when the block hit the ground, that’s when the displacement became—”
“Zero!” Suika calls out, and Chrome turns to see her bouncing in excitement, clearly enthralled with what he’s saying. Even if he’s wrong and all of these calculations are completely off, the support from all his friends—he can even see Kohaku with an impressed smirk on her face—is enough to reassure him and give him the confidence he needs to finish.
“So what we’re left with is this:”
“And solving for t in that equation tells us how long the block takes to fall 22 meters.” Chrome quickly scribbles down a few lines below, solving for t, copying what he has in his notebook and praying it’s correct.
“In the end, we have...” he slowly circles his answer. “We have t = 2.12. Meaning it takes 2.12 seconds for the block to drop.” He finally turns around and meets Senku’s eyes, and Chrome’s heart jolts because he doesn’t think he’s ever seen Senku more exhilarated ever in his life. He’s praying this look means he was right, that this week of struggling wasn’t for nothing.
Senku comes up to the blackboard, while both Chrome and the crowd wait with bated breath. “Well?” Chrome asks, pointing the microphone in Senku’s direction.
“Before I tell you, I want you to do one final thing,” Senku says, and Chrome’s simultaneously about to explode from excitement at the prospect of being correct, and faint because if he has to use a single ounce of brain power ever again, he’s going to lose it. Senku takes the piece of chalk and starts writing on the board. “I want you to write a set of general equations, one that will work for motion in any situation, using constant acceleration, time, and the initial and final velocity and displacement,” he says, jotting down:
“These are your ingredients. You already have everything, now you just have to put it all together.”
There’s no way Senku would phrase it like this if I wasn’t right, he thinks, taking the chalk back and finally writing on the board:
He circles the two general equations he’s just written before setting the chalk down. He looks over at Senku, who is still holding the microphone.
“Does everyone see these equations that Chrome just spent a week coming up with, solely from his own intuition and the math I taught him?” Senku beams, pointing to the blackboard.
There are murmurs and nods from the audience. Chrome knows in his heart that he’s right, that he managed to do it, but he still needs that verbal confirmation from Senku—“Those from the old world might recognize these as the kinematic equations, the ones that legendary astronomer Galileo Galilei derived. Those equations didn’t come from nowhere, and Chrome here just proved how they can be intuited by an entry-level calculus student.”
Chrome can’t help but interrupt. “So I was right? Was it really 2.12 seconds?” Senku’s still grinning when he gives a firm nod, and Chrome can’t help but quite literally leap for joy. Next thing he knows, the crowd is cheering and he’s being swarmed by all those who helped or gave him encouragement over the past week.
“You did it!” Taiju yells, lifting him up from the ground.
“I never once doubted you,” Senku says, satisfied.
“Great job!” Suika cheers, “I knew you could do it!”
Congratulations come in from the rest of his friend group, which ends up including Kohaku.
“I have to admit, all that was pretty clever. Maybe you were onto something with talking to that duck,” Kohaku shrugs as Chrome nudges Taiju to let him down.
“That reminds me,” Chrome says, turning to Senku, his heart picking up speed once again. “This is silly, but you know I’ve always wanted to invent something that didn’t exist in the old world? Well, I think I managed to come up with something so ridiculous, there’s no way it used to be a thing.”
“Oh?” Senku asks. “What is it?”
“Well, I had this idea towards the end, when I was trying to figure out the initial conditions. There was a duck nearby, and I just... decided to talk to it. I explained my thought process to it, and it seems to have worked because here I am.”
Senku nods in a way that’s far too knowing for what Chrome wants. “So like rubber ducking? That was something normally done by programmers, but given we have yet to recreate both rubber and computers, I suppose working through a problem while talking to a real duck is the logical equivalent.”
“Aw man!” Chrome says, falling to the ground in defeat as he so often does. “I thought for sure I’d come up with something original this time!”
“Whether or not you came up with something on your own doesn’t matter one millimeter,” Senku says, reaching out a hand to help Chrome up. “What matters is that you understand this physics. When it comes to defeating Why-man, knowledge is power.”
“You mean understanding speed and velocity might help us in the future fight against Why-man?” Chrome asks. Senku’s not even listening, he’s already dragging Chrome away in the direction of the lab.
“Come on, I want to show you how all this relates to projectiles. For all we know, we’ll have to perfectly time throwing revival fluid up in the air to revive ourselves from an oncoming petri-beam some day.”
Senku’s proposition sounds ridiculous, but Chrome goes along with it. Because what are the odds of that happening anyway?