Ponies Learn Calculusby LopsyChaptersChapter 1 - Speed LimitChapter 2 - Camera TrickChapter 3 - Do This In My SleepChapter 4 - Mutually Assured TornadoChapter 1 - Speed Limit Rainbow Dash blasted through the window into Twilight’s library. “Twilight! Twilight! Are you there?” “Mmrmph,” said a muffled voice from below her. Rainbow looked down. Twilight Sparkle was sprawled on the floor under her. “Whoops,” she said sheepishly, removing her hoof from Twilight’s mouth. “Sorry about that. I thought I had a clear landing.” Dash helped Twilight to her hooves. “Don’t worry, I’m not hurt,” Twilight said. “My window, on the other hoof—” “There’s no time for that, Twilight. Take a look at this!” The colorful flier handed her the colorful flyer. “Oh? What about it?” “The Wonderbolts are gonna be watching this like hawks,” Dash said. “I have to snag first place tomorrow.” Twilight flipped over the flyer and started reading the fine print. “I see. It appears that this competition focuses less on speed, and more on maneuverability and complex tricks.” “Exactly. And to be a Wonderbolt, you’ve gotta be the best at both.” “Good luck, Rainbow! I’ll be there to cheer you on. Have you planned out your act yet? Maybe you should rehearse it tonight. I’ll pretend to be a judge, and—” Rainbow raised a hoof to stop her. “Thanks. But what I really need right now is — well — Twilight, do you have any kind of speed-measuring spell?” Twilight stroked her chin. “Speed-measuring? I can’t say I’ve heard of any spells like that. Why?” “Wellllll, I kinda found out today that the stadium’s really small, and they’re judging style over speed, so… the competition has a 20 ponylengths per second speed limit.” “Yes, it says that right here.” Twilight pointed to a sentence written in tiny font on the back of the flyer. “Wait. You mean you didn’t read the rules until today?!” Rainbow turned away, embarrassed. “Look, it’s fine, okay? I threw together a new act today. I just need to know whether it follows the rules. But if you don’t have a speed-measuring spell, I guess I’ll leave…” Rainbow trailed off. She opened the library’s door and sulked out. Twilight galloped outside. “Rainbow! Wait!” she said. “I may not have a spell, but I’ll try my best to help you out. Can I see your act?” Dash grinned. “Not here. This series of awesome tricks is top secret. We have to go somewhere nopony will ever find us.” “Where is that?” “Just follow me.” * * * * * Later, in a clearing deep in the Everfree Forest… “The fastest part of this trick,” Rainbow Dash lectured, “is at the very bottom of this loop-the-loop.” As Twilight watched, Dash spread her wings and took to the sky. She started to flap her wings erratically. Slowly, she turned herself upside-down. This is no simple task. Both bird wings and pegasus wings are built for pushing their owners upward. If anypony tried flying upside down normally, they’d rocket towards the ground. As such, Dash had to jitter her wings up and down in an irregular, alien-looking pattern, just to stay afloat. Most pegasi never bothered learning this trick. And there was nopony else lunatic enough to try what Rainbow did next. She dipped her head downwards, and started to fall. She accelerated towards the ground. “No!” Twilight yelled. Her horn glowed as she prepared to catch Rainbow with her telekinesis. Would her magic have caught Rainbow Dash in time? We’ll never know. A split second before she hit the ground, Dash pumped her wings. She contorted her sleek body, wrenching herself into a hairpin turn. For a split second, Dash was right-side up, her stomach skimming the grass. Then she soared upwards, leaving Twilight’s magic with nothing to grasp but thin air, and her jaw nowhere to go but down. Dash grinned. “So. How was it?” Twilight gaped. “Wow. I wouldn’t call it ‘elegant,’ but… before seeing it, I also wouldn’t have called it biologically possible.” “No, I mean the speed! On the turn, was I going over 20 ponylengths per second or under?” Twilight put her hoof to her chin. “Hmm… I’m not sure. Mind showing me again?” “No problem.” Rainbow darted up into the sky and flipped herself upside-down. * * * * * Seventy minutes later… Twilight shuffled through a sheaf of paper in front of her, filled with numbers and graphs. “Let’s see…” she said. “I’m 90% confident that your top speed is somewhere between 15 and 25 polylengths per second. Mind doing ten or twenty more trials?” Rainbow Dash was slumped on the ground. “Sure… *pant*… thing… *wheeze*… Twilight…” She groaned. “Why is this… *pant*… taking so many tries? I’m totally doing the same thing… *pant*… every time,” she said. “Sorry,” said Twilight. “It’s just so hard to gauge your speed by looking.” Dash sighed. “Rats.” “Maybe you can do the trick without any turning or accelerating? If you were moving at a constant speed in one direction, we could use speed=distance/time to find the answer.” Rainbow rolled her eyes. “Duh. I don’t think the judges will be impressed… *pant*… by your ‘fly in a straight line’ trick, Twi.” Twilight turned away. “Sorry, Rainbow. I wish there was something I could do.” “Why don’t’cha use this?” Twilight did a double-take. Pinkie Pie had suddenly appeared beside her, holding an odd black rectangular device. “Um, Pinkie… what’s that?” Twilight asked. Rainbow let herself flutter to the ground, thankful for the chance to catch her breath. “More importantly, how did you find my hyper-secret hiding place?” Pinkie showed them the front of the device. It had a small rectangular screen. The screen showed an image of Dash hovering upside-down in the air. “It’s a camera, silly! Don’t you see them following us around every Saturday? I saw one heading into the Everfree Forest, and I just knew it had to be following up on something fun!” Rainbow took a sidelong glance at Twilight. They both shook their heads, mentally agreeing not to pry into Pinkie’s mind. “So… how does this ‘camera’ work?” Twilight asked. “It’s easy-peasy-lemon-squeezy.” Pinkie pressed a button on the device, and the screen jumped into motion, showing a video of Dash’s trick. Dash stared at the screen, transfixed. “I knew I looked awesome, but I didn’t know I looked that awesome…” Twilight pushed her aside. “Pinkie Pie, this technology is amazing. I have to study this.” “Gooooood luck! They go away once you start doing boring stuff.” “That doesn’t make any –” Twilight stopped herself, remembering the Pinkie Sense incident. “Ugh. Fine.” “Cameras can do tons of other stuff too! You can watch Rainbow’s trick zoomed super way in, or even go through it frame-by-frame.” “Frame-by-frame?” asked Dash. “Yepper-doodle! Cameras record footage at 30 frames per second. But usually you only watch frame-by-frame if you’re looking for Easter eggs or animation errors.” Twilight started playing with the camera, ignoring Pinkie Pie’s nonsense outburst. You can play with it too, here. You’ll need Flash Player. If you don’t have Flash Player (maybe you’re reading this on a phone), here’s a quick description of what Twilight and Rainbow Dash discovered. The camera let them watch Dash’s trick as many times as they wanted. They could also pause and zoom in. In addition, they could step through the action, frame by frame. The camera records footage at 30 FPS. The footage also shows a small stick on the ground. This is Twilight’s measuring stick. It is one ponylength long, and has markings every 1/10 of a ponylength. Twilight and Dash want to determine whether Dash’s speed is above or below 20 ponylengths per second, when she’s at the very bottom of her loop. Using just the information you have so far, you should be able to figure this out! (If you don’t have Flash Player, you won’t be able to figure out the answer yourself. But maybe, if you think hard, you can figure out a method Twilight and Rainbow can use to find the answer.) Author's Note Next time on Ponies Learn Calculus: Lightbulbs! Onion skins! And maybe even... some actual math? Chapter 2 - Camera TrickTwilight pranced around the field, giddy. “Eureka! Eureka!” “Um, English, Twi?” Rainbow Dash said. “I don’t speak Ancient Geek.” Twilight grabbed Rainbow’s shoulders, shoving her smile right into Dash’s face. “I’ve got it. This camera can tell us your exact speed, Rainbow Dash.” Pinkie Pie poked her head around it. “It can? I didn’t see a ‘Tell Me Rainbow Dash’s Speed’ button. Maybe you need a three-press combo…?” She started mashing buttons. “No, Pinkie.” Rainbow Dash rolled her eyes. “She means she can watch the trick again and again, and it’ll help her guess my speed better.” “Actually,” Twilight said, “it’s neither.” She took the camera back from Pinkie, and paused it at the critical moment of Rainbow’s trick. “The key is to watch the trick frame by frame.” “Wait, what?” Dash stared deep into the screen. Nothing happened. “…Sorry, Twi, I don’t think you can figure out my speed by watching me not move.” “Just watch.” Twilight pressed the frame advance button. On the screen, Rainbow Dash suddenly jumped forward. “Ooooooookay,” said Dash. “I’m watching.” “Now we can figure out how fast you’re moving.” Dash raised an eyebrow. “Bear with me. Pinkie, you said that this device records 30 images every second, right?” “Correctasaurus!” “Then, the time between this moment and this moment,” Twilight said, switching back and forth between the two frames, “is exactly 1/30 seconds.” “Oh. Ohh.” A lightbulb appeared over Rainbow Dash’s head. “Oooohhhhhhhhh! I got it!” Dash switched back and forth between the two frames. “What distance did I travel during those 1/30 seconds? Looks like… half a polylength? A little more?” Twilight pointed at the screen. Specifically, at the measuring stick on the ground. “If you compare your position relative to the measuring stick, it looks like you moved approximately 6/10 ponylengths.” (this is what you'd see if you switched back and forth between the two frames really fast) “Cool. We’ve got the distance, we’ve got the time. Now, the speed is, uh…” “Speed equals distance divided by time,” Twilight reminded her. “I’m not in first grade, Twi. I’ve got this. 6/10 ponylengths, divided by 1/30 seconds, is…” Pinkie jumped in front of her. “Ooooh! Ooooh! I know! It’s 87078952296/4837719572 ponylengths per second!” Dash stared at her. “Uh, what? Is that more or less than 20?” Twilight scribbled some numbers down on a piece of parchment. “It’s 18. Therefore, Rainbow Dash, you were traveling at approximately 18 ponylengths per second during the climax of your loop-the-loop.” Rainbow’s face slowly transformed into a grin. “YES! First-place trophy, here I come!” “Woo-hoo!” Pinkie Pie jumped on top of Dash, tackling her to the ground. “I’ll prepare the trophy-shaped cake. Twilight, aren’t you excited? On a scale from 10 to hot fudge, how excited are you?” Twilight was staring at the camera, transfixed. She examined it from every angle, furiously scribbling down notes. Twilight gazed back and forth between the camera and her equations. A little bit of drool came out of her mouth. “…Let’s just leave her be.” Rainbow said to Pinkie Pie. “She’ll be better in the morning.” Pinkie Pie didn’t respond. She was staring into space, mumbling. “What if the Canterlot mountains were made of rock candy? Ahhh…” Rainbow briefly wondered why so many of her friends occasionally went insane. But the thoughts of tomorrow’s competition soon overtook her. She flew up into the cool evening air, letting the orange sky surround her. She yawned. That rehearsal had taken a lot out of her. She had to wake up early tomorrow morning, too. She settled on top of a nearby cloud and plugged her ears with cloudstuff. On the other hoof… Rainbow unplugged her ears and took to the sky again. Slowly, she flipped herself upside-down. One more run through can’t hurt. After all, I have to be perfect. Chapter 3 - Do This In My Sleep11:58 PM, later that night – basement of Twilight’s library Twilight sat hunched at her workdesk. Disassembled pieces of the camera lay strewn across the surface: panels, wires, and dozens of screws. On the wall behind the desk was a chalkboard, containing Twilight’s detailed diagram showing where each part had been. Twilight’s attention was focused on a heavy black cube. It had been the bulk of the camera, and it still had a few wires sticking out. Her intuition told her that, whatever secrets this technology had to offer, she would find them inside the cube. But it wasn’t going to be easy. When Twilight had disassembled the rest of the camera, most steps had gone as expected. But when she tried to use her magic on anything connected to the black box -- “OWWWWWW!” Twilight jumped back. Her horn throbbed. It felt like somepony had just grabbed it and shook it up and down 60 times in a row. Suddenly, she heard footsteps descending the stairs above. “Twilight?” a voice called down. “What’s with the yelling? Is everything okay?” “Yes, Spike,” she sighed. “Are you sure? Because you’ve yelled ‘OWWWWWWW!’ about five times.” Twilight facehooved. “Sorry, Spike. I’ll try to yell quieter. You can go back to bed.” She was getting tired herself. She decided to get a midnight snack. She trotted upstairs to get a glass of milk. I can’t give up now, she thought. I’m so close! If I could just hold on to the box long enough to give the side one good yank… Twilight re-entered the basement. Her eyes flew wide open. She accidentally released her telekinetic hold on the glass of milk. It hit the floor and shattered. She barely noticed. “What.” The camera had disappeared. The black cube, and every single screw, wire, and panel, were gone. Twilight took a long, deep breath. “You win this round, camera,” she muttered. “But I’ll be back. You’ll rue the day you tried to outwit Twilight Sparkle.” * * * Cloudsdale Stadium, the next day Twilight sat in the stands, surrounded by Pinkie Pie, Rarity, Applejack, and Fluttershy. The aerobatics competition was well underway. So far, each of the pegasi’s dizzying twirls impressed Twilight to no end. But, though she tried her best to hide it, she was starting to get bored. After 14 entrants, Twilight felt like she was eating chocolate cake. Yes, each cake was unique and delicious. But no, that doesn’t mean she wanted to eat 15 more. Fortunately, the announcer had just called a much-needed 10-minute break. She excused herself, saying she had to go the restroom, and then made her way to the judges’ table. She had a question nagging at her head. “Excuse me,” she said, “do you have a minute?” One of the judges, a brown stallion, checked his watch. “We’ve got three. What’s the matter?” “I’ve been wondering, how do you make sure the entrants stay under the 20 p/s speed limit? Is there some kind of spell?” Another judge, a light-green pegasus with tan-colored hair, stepped forward. “That’s easy, girl,” she said. She spat on her hoof and held it out. “My name’s Epsilon, by the way. My friends call me Lonnie.” Twilight ignored the hoof. “I’m Twilight. Nice to meet you… Lonnie.” “Measuring a pegasus’ speed is a piece a’ cake,” Epsilon said. “You’ve just gotta formulize ‘er.” Twilight cocked her head. “Formulize?” “Yeah, girl,” said the green mare. “Check out that pegasus,” she said, pointing at a random stallion. “He’s flying along the path y=x², with constant speed 3 p/s.” Twilight’s eyes widened. “Wh-what! How did you do that?” “Anypony can do it,” she said. “It’s like whistling. Here, look at that grey pegasus.” “The one blowing up a balloon?” “Right. Now, try to formulize that balloon.” Twilight stared at the balloon. Nothing happened. “Er. What am I supposed to do?” “It’s like whistling. It’s hard to explain. Just kind of… meditate. Let some math drift into your head. All zen-like, sha.” Twilight stared at the balloon some more. This is ridiculous, she thought. She let her mind wander for a few seconds. Nothing. Stupid Twilight – thinking anypony could pick up a talent like this just by watching for a few seconds. I bet – “0.5+3√t hooflengths?” came a very quiet voice from behind her. Twilight spun around and saw Fluttershy. “Yup, that’s right, girl! The radius of the balloon was 0.5+3√t hooflengths, at time t seconds after she starting blowing it up.” “Wha!- Fluttershy?! Where did you – What did you – when did you – how did you do that?!” Fluttershy ducked her head into the clouds. “I just did what she told me to…” “But – but –” “Break’s over!” Epsilon trotted back behind the judges’ table. “Get ready. The next competitor’s called ‘Rainbow Dash.’ I hear she’s good.” With no time to get back to their seats, Twilight and Fluttershy decided to watch from the judges’ table. They cheered as Rainbow flew to the center of the stadium. Dash started out with a standard triple aileron roll. Oddly, she was going slower than during the rehearsal. Twilight guessed she was really worried about the speed limit. Dash led the rolls into a twisting vertical loop, to the delight of the crowd. “Did she just yawn?” whispered Fluttershy. “Did she?” Twilight squinted. “She’s probably bored of these tricks and wants to get to the big finish. Yep. That’s probably it.” The big finish came soon enough. Twilight watched Rainbow Dash line herself up. She had pulled it off fifty times late last night, but could she pull it off when it counted? The crowd oohed and aahed as Rainbow slowly rotated upside down. “Wow!” said a nearby stallion. “She even has her eyes closed!” Eyes closed? Twilight thought. I don’t remember that part of the trick. In front of her, Rainbow Dash started her dive. It didn’t look exactly like last night’s rehearsal. For one, she wasn’t rotating to match the angle of her descent. For two, her wings were flapping significantly slower. If Twilight didn’t know better, she would’ve guessed Rainbow were falling. And falling. And… yep, still falling. The stadium fell into a hush. Everypony waited for Dash’s grand recovery. Twilight looked down, in a panic. Rainbow was falling straight towards the weather factory. Twilight charged up her telekinesis, but she was already out of range. By now, not even the Wonderbolts were fast enough to dive and catch Rainbow. Rainbow fell closer and closer to the factory’s wind machine. Twilight grimaced and squinched her eyes. By the way, the Cloudsdale Weather Factory is constructed out of packed clouds. For all the earth ponies and unicorns reading this, picture some hay. Nice and soft when it’s spread out into a haystack. When it’s packed into a bale… not so much. The crowd gasped. Twilight opened her eyes. Rainbow had disappeared. The muttering around her told her what had happened. Rainbow had fallen down a chimney into the factory’s wind machine. To their credit, the factory workers reacted fast. Far below, two of them grabbed a loose cloud and put it directly under the falling pegasus. “Phew,” said Twilight. “At least everypony’s going to be safe.” Rainbow hit the cloud, bounced off of it, and landed belly-first on a large control panel filled with switches and levers. A large grey mass started coming out of the wind machine’s chimney. A large, whirling grey mass. It was moving upwards. Directly towards the stadium. “Tornado!” A Wonderbolt raced through the stadium below them, herding pegasi away. “Evacuate the stadium. Do not panic!” Twilight panicked. “Don’t panic!” said Epsilon. “Seriously, girl. We deal with rogue tornadoes every other week.” “Deal with them? How?” “Well, a tornado is just a bunch a’ air spinning around, right?” “Right…” “So, if you blow a whole bunch a’ air spinning the opposite direction at the exact same speed, it’ll counter it.” Twilight took a few seconds to process this. “Let me get this straight. You’re going to stop this tornado…” “Yeah?” “By launching another tornado at it.” “You’ve got it, girl.” Twilight looked down. Sure enough, the weather team pulled switches and pushed clouds, and a second tornado soon came out of the chimney. It rose, quickly catching up to the first tornado. Twilight gasped as the two whirling twisters collided. Chaos erupted in the air. Dirt flew in every direction. And when the dust settled… A tornado, twice as large as before, was rising towards the stadium. “Uh-oh,” said Epsilon. “That was supposed to happen, right?” said Fluttershy. “Please say that was supposed to happen.” “They got the speed wrong,” said Epsilon. “If two tornadoes aren’t rotating at th’ exact same speed, then instead of cancellin’ out, they merge into one bigger one.” Fluttershy stuffed her head back into the clouds. Epsilon stared down at the tornado, deep in thought. Her hooves twitched. Her eyes unfocused. “…It’s no good,” she concluded. “What?” said Twilight, for the third time that day. “I was trying to formulize the tornado’s spinning speed. But it’s moving too fast to tell. I have to get closer.” Before Twilight could object, Epsilon flapped her wings and glided down towards the spinning mass of chaos. Twilight followed her down. She paused, wings beating furiously, above the whirling grey-brown top of the storm. Twilight strained her ears to hear Epsilon over the storm: “Got it! The tornado will crash into the stadium exactly ten minutes after it formed. And the angle of the tornado, at the time t seconds after it formed, is—” Just then, a plank of wood flew out of the tornado, straight towards Epsilon. She folded her wings and dived down. The plank flew inches over her head. Unfortunately, diving down is a bad idea when you’re hovering above a tornado. Twilight averted her eyes. She heard a scream: “t² degrees! So the airspeed – AAAAAA!” When Twilight opened her eyes again, Lonnie was a green tumbling mass, spinning around like a sock in a washing machine. Twilight riffled through her brain, searching for a spell that would help. Unfortunately, any spell that would destroy the tornado would also hurt Lonnie. Don’t panic, Twilight thought to herself. She closed her eyes and counted to ten. There is one way out. A second tornado. Which means… all I have to do is figure out the tornado’s airspeed. Twilight flew back up to the stadium and retrieved her parchment and quill. She wrote down: Tornado crashes into stadium 10 mins after forming Angle of the tornado, t seconds after forming: t² The “angle of the tornado”? What does that mean? As Twilight was trying to figure this out, a cyan pegasus flapped up behind her and laid face-down on the cloud. “Heeeey,” yawned Rainbow Dash. “’Sup? What’d I miss?” Twilight’s eyes twitched. “…Rainbow?” “Yeah? What’s wrong, you nervous about something?” Twilight gulped. “…Rainbow, are you feeling alright?” “Uhh, yeah? A little nervous about the competition later today, but I’m fine.” “L-later today?” “Yeah. I just had a nightmare about it.” She shivered. “But that’s not gonna stop me! I’m ready to show the Wonderbolts what I’m made of.” Rainbow beamed, looking straight into Twilight’s eyes. Twilight looked away. She didn’t want to be around when Dash found out what happened. If she was going to receive news like this, she’d want it to be in a warm bed with at least three liters of double chocolate ice cream at hoof. Rainbow Dash had to find out eventually, but she vowed to break the news at the least stressful time possible. Or preferably get somepony else to break the news while she waited in a bunker five kilometers away. “…Rainbow! Quick question. Suppose, hypothetically, there was a tornado whose angle was t², where t is the number of seconds after it formed.” “Um, okay.” Rainbow scratched her ear. “Hey, what’s that weird whirring noise?” “Whirring? I don’t hear any whirring. Nope. Definitely no whirring here.” “Never mind. It’s probably just the wind.” Twilight grinned madly. “So,” Dash continued, “angles, huh? So, like, when the tornado first forms, its angle is 0²=0? What does that even mean?” “I’m not sure myself. What about…” Twilight picked a random number. “19 seconds later? After 19 seconds, the tornado’s angle is 19²=361 degrees.” Rainbow scratched her chin. “So in 19 seconds, the tornado spun a little more than one full turn. That’s pretty slow.” “You’re right. It must have sped up since then.” “What? Twilight, you said this was all hypothetical.” “R-right! The hypothetical tornado.” Spitfire yelled from below: “EVACUATE! Seven minutes!” “Whoa. What’s going on?” “Rainbow, wait!” Rainbow poked her head over the edge of the cloud and looked down. “I can explain—” Rainbow’s eyes opened wide. Then her face scrunched up. Then she started screaming. “Well, crap.” Author's Note Again, you have enough information to solve the puzzle before the next chapter. Theoretically, at least. Also: does anyone know a way to inline math formulas more complicated than x²+3? The unicode superscript 2 is already a kludge, and I'm worried that it might not appear on all browsers. Chapter 4 - Mutually Assured Tornado“Rainbow, flying directly into the tornado won't help.” Twilight gripped Rainbow's tail with her telekinesis. “Gotta... be... a hero.” Rainbow struggled against the purple glow. “If you want to be a hero, help me with this math.” Dash did a double-take. “What.” Twilight pulled out her quill and parchment. “If we could determine your speed yesterday with the camera, then we can determine the speed of the tornado.” In fact, Twilight thought, how hard could it be? To find Dash's airspeed, they just had to measure how far she moved in one twentieth of a second. The same trick should work for the tornado. Luckily, they had Epsilon's formula: at time t seconds, the angle of the tornado was t² degrees. “Let's see...” Twilight scribbled. “The tornado will crash into the stadium exactly ten minutes after it formed.” “Don't remind me.” “That's t=600 seconds. So it will have spun... 600²=360000 degrees.” Rainbow Dash kicked the cloud impatiently. “And one frame later?” “600.05², which is... 360060.0025 degrees.” Rainbow whistled. “Nice job, Calculator Sparkle. So it turned 60.0025 degrees during 1/20 of a second. So its speed is... I've got this... 1200.05 degrees per second.” Rainbow blasted away. One millisecond later, her tail tugged her back. She groaned. “Let me go, Twi. I have some tornado canceling to do.” “Wait. If I learned anything from magic kindergarten, it's always check your answers, or you might accidentally turn Luna into a pumpkin.” “Whatever. Check your arithmetic.” “It's not the arithmetic I'm worried about. It's the one-twentieth of a second. What if the tornado sped up or slowed down during that time? Then our answer would be wrong.” Twilight stared at the parchment. She wrote a line of numbers and then crossed them out. She repeated this five times. Her ears drooped. Rainbow peered over the edge of the cloud. The tornado was like that thick tree on her route home: full of angry birds, and a lot closer than she expected. “No pressure Twilight, but...” Twilight put down the quill. “Argh. How am I supposed to figure out whether the tornado speeds up or slows down, if I can't figure out its speed to begin with?” “There's no time! Just use 1/1000 of a second instead of 1/20. There's no way the tornado can speed up in 1/1000 of a second. Even I can't go from 0 to 60 that fast.” “That...” Twilight muttered to herself. “...could actually work.” She tried to pick up her quill. From where she had placed it on the cloud. “Rats.” [If you are reading this to learn calculus, then you should pause here to compute the answer for 1/1000 of a second. Really. If you want a calculator, I recommend WolframAlpha.] [Seriously.] Rainbow pulled a feather out of her wing. Twilight dipped it in ink and wrote: She furrowed her brow. “1200.001. That's not what we got last time.” “Gotcha. One Dash spin cycle special, speed 1200.001 degrees per second, coming up.” Dash looked down. “In T minus... one minute.” “1200.001 isn't what we got last time. I'm going to try 0.0001 seconds, just to make absolutely sure.” Twilight pounded the cloud. “It's still a different answer.” Dash gritted her teeth. “We don’t have all day, Twi. I can see where this is going. You want a time period where the tornado won’t speed up at all. Go down from 0.0001 seconds to zero seconds flat.” Dash flinched. “Ouch. Even I can’t divide by zero.” Twilight grimaced. “That’s too bad. If that had worked, we’d be done. I’m going to try 0.000001 seconds.” “Too late.” Rainbow stuck her head over the edge of the cloud. The roar of the tornado muffled her voice. “I'm going down.” She hopped off the cloud. “If you change your answer in the next thirty seconds... yell loud, OK?” With that, Rainbow disappeared from Twilight's view. Twilight ground her teeth. Calculating machine that she was, thirty seconds was not enough time to square a number like 600.000001 . And even if it was, she doubted she would get the same answer as before. She could keep trying smaller and smaller lengths of time for the rest of the day. For all she knew, she would never reach the true answer -- just a series of better and better approximations. Wait. I don't have enough time to compute the answer for 0.000001 seconds, but I do have enough time to compute it for x seconds. Twilight thanked the Gods of Math for the simple form of the answer. She was about to plug in x=0.000001 when a thought struck her. This entire time, she had been computing a series of better and better approximations by plugging in smaller and smaller values of x. But now that she had the formula 1200+x, it was obvious. Her answers were just getting closer and closer to 1200. No time to check. Twilight prayed to Celestia that she was right. She put on her best Luna impression. “TWELVE HUNDRED DEGREES PER SECOND! EXACTLY!” Rainbow Dash circled beneath the raging grey funnel. The wind started to pick up around her. It carried her along like a straw in a spinning glass. She heard Twilight's scream and adjusted her wings a fraction of a millimeter. (Pegasi are very precise.) She began to feel the whipping wind of a tornado. Her wings shook. Turbulence rocked her body. Imagine riding a vibrating tiger. She hit the right speed. She knew it. All she had to do was push this countertornado up towards the other one. As easy as persuading that tiger to climb Mount Olympus. Dash gritted her teeth. The wind buffeted her from every direction as she spun in a tight circle. If she opened her wings, she'd sail away like a paper bag. Dash closed her eyes. Three, two, one. She lifted her wings and forced them down in a heartbeat. Then the counter-tornado threw her. She tumbled helplessly through the air. Grinning. She got it. She rightened herself and looked back. Two things happened at exactly the same time. One: the twister collided with the stadium's scaffolding. Two: the countertornado hit its mark. Splinters of wood and dirt shot out of the storm. Rainbow crossed her hooves. Both pairs of them. Please slow down, please slow down, please... The raging funnels merged. They began to fade. The winds slowed. Soon, nothing remained but bright blue sky. And one falling green pegasus. Dash flew down, forelegs open. “OOOFFFFF!” Epsilon hit her spine like an Apple family kick. She half-flew, half-fell to the clouds below. Rainbow struggled out from under Epsilon. She waved her hoof in front of her face. “Epsilon. Lonnie. Are you okay? Please be okay.” Epsilon's eyes fluttered open. “Rainbow Dash. You are in so much trouble.”
Chapter 1 - Speed Limit Rainbow Dash blasted through the window into Twilight’s library. “Twilight! Twilight! Are you there?” “Mmrmph,” said a muffled voice from below her. Rainbow looked down. Twilight Sparkle was sprawled on the floor under her. “Whoops,” she said sheepishly, removing her hoof from Twilight’s mouth. “Sorry about that. I thought I had a clear landing.” Dash helped Twilight to her hooves. “Don’t worry, I’m not hurt,” Twilight said. “My window, on the other hoof—” “There’s no time for that, Twilight. Take a look at this!” The colorful flier handed her the colorful flyer. “Oh? What about it?” “The Wonderbolts are gonna be watching this like hawks,” Dash said. “I have to snag first place tomorrow.” Twilight flipped over the flyer and started reading the fine print. “I see. It appears that this competition focuses less on speed, and more on maneuverability and complex tricks.” “Exactly. And to be a Wonderbolt, you’ve gotta be the best at both.” “Good luck, Rainbow! I’ll be there to cheer you on. Have you planned out your act yet? Maybe you should rehearse it tonight. I’ll pretend to be a judge, and—” Rainbow raised a hoof to stop her. “Thanks. But what I really need right now is — well — Twilight, do you have any kind of speed-measuring spell?” Twilight stroked her chin. “Speed-measuring? I can’t say I’ve heard of any spells like that. Why?” “Wellllll, I kinda found out today that the stadium’s really small, and they’re judging style over speed, so… the competition has a 20 ponylengths per second speed limit.” “Yes, it says that right here.” Twilight pointed to a sentence written in tiny font on the back of the flyer. “Wait. You mean you didn’t read the rules until today?!” Rainbow turned away, embarrassed. “Look, it’s fine, okay? I threw together a new act today. I just need to know whether it follows the rules. But if you don’t have a speed-measuring spell, I guess I’ll leave…” Rainbow trailed off. She opened the library’s door and sulked out. Twilight galloped outside. “Rainbow! Wait!” she said. “I may not have a spell, but I’ll try my best to help you out. Can I see your act?” Dash grinned. “Not here. This series of awesome tricks is top secret. We have to go somewhere nopony will ever find us.” “Where is that?” “Just follow me.” * * * * * Later, in a clearing deep in the Everfree Forest… “The fastest part of this trick,” Rainbow Dash lectured, “is at the very bottom of this loop-the-loop.” As Twilight watched, Dash spread her wings and took to the sky. She started to flap her wings erratically. Slowly, she turned herself upside-down. This is no simple task. Both bird wings and pegasus wings are built for pushing their owners upward. If anypony tried flying upside down normally, they’d rocket towards the ground. As such, Dash had to jitter her wings up and down in an irregular, alien-looking pattern, just to stay afloat. Most pegasi never bothered learning this trick. And there was nopony else lunatic enough to try what Rainbow did next. She dipped her head downwards, and started to fall. She accelerated towards the ground. “No!” Twilight yelled. Her horn glowed as she prepared to catch Rainbow with her telekinesis. Would her magic have caught Rainbow Dash in time? We’ll never know. A split second before she hit the ground, Dash pumped her wings. She contorted her sleek body, wrenching herself into a hairpin turn. For a split second, Dash was right-side up, her stomach skimming the grass. Then she soared upwards, leaving Twilight’s magic with nothing to grasp but thin air, and her jaw nowhere to go but down. Dash grinned. “So. How was it?” Twilight gaped. “Wow. I wouldn’t call it ‘elegant,’ but… before seeing it, I also wouldn’t have called it biologically possible.” “No, I mean the speed! On the turn, was I going over 20 ponylengths per second or under?” Twilight put her hoof to her chin. “Hmm… I’m not sure. Mind showing me again?” “No problem.” Rainbow darted up into the sky and flipped herself upside-down. * * * * * Seventy minutes later… Twilight shuffled through a sheaf of paper in front of her, filled with numbers and graphs. “Let’s see…” she said. “I’m 90% confident that your top speed is somewhere between 15 and 25 polylengths per second. Mind doing ten or twenty more trials?” Rainbow Dash was slumped on the ground. “Sure… *pant*… thing… *wheeze*… Twilight…” She groaned. “Why is this… *pant*… taking so many tries? I’m totally doing the same thing… *pant*… every time,” she said. “Sorry,” said Twilight. “It’s just so hard to gauge your speed by looking.” Dash sighed. “Rats.” “Maybe you can do the trick without any turning or accelerating? If you were moving at a constant speed in one direction, we could use speed=distance/time to find the answer.” Rainbow rolled her eyes. “Duh. I don’t think the judges will be impressed… *pant*… by your ‘fly in a straight line’ trick, Twi.” Twilight turned away. “Sorry, Rainbow. I wish there was something I could do.” “Why don’t’cha use this?” Twilight did a double-take. Pinkie Pie had suddenly appeared beside her, holding an odd black rectangular device. “Um, Pinkie… what’s that?” Twilight asked. Rainbow let herself flutter to the ground, thankful for the chance to catch her breath. “More importantly, how did you find my hyper-secret hiding place?” Pinkie showed them the front of the device. It had a small rectangular screen. The screen showed an image of Dash hovering upside-down in the air. “It’s a camera, silly! Don’t you see them following us around every Saturday? I saw one heading into the Everfree Forest, and I just knew it had to be following up on something fun!” Rainbow took a sidelong glance at Twilight. They both shook their heads, mentally agreeing not to pry into Pinkie’s mind. “So… how does this ‘camera’ work?” Twilight asked. “It’s easy-peasy-lemon-squeezy.” Pinkie pressed a button on the device, and the screen jumped into motion, showing a video of Dash’s trick. Dash stared at the screen, transfixed. “I knew I looked awesome, but I didn’t know I looked that awesome…” Twilight pushed her aside. “Pinkie Pie, this technology is amazing. I have to study this.” “Gooooood luck! They go away once you start doing boring stuff.” “That doesn’t make any –” Twilight stopped herself, remembering the Pinkie Sense incident. “Ugh. Fine.” “Cameras can do tons of other stuff too! You can watch Rainbow’s trick zoomed super way in, or even go through it frame-by-frame.” “Frame-by-frame?” asked Dash. “Yepper-doodle! Cameras record footage at 30 frames per second. But usually you only watch frame-by-frame if you’re looking for Easter eggs or animation errors.” Twilight started playing with the camera, ignoring Pinkie Pie’s nonsense outburst. You can play with it too, here. You’ll need Flash Player. If you don’t have Flash Player (maybe you’re reading this on a phone), here’s a quick description of what Twilight and Rainbow Dash discovered. The camera let them watch Dash’s trick as many times as they wanted. They could also pause and zoom in. In addition, they could step through the action, frame by frame. The camera records footage at 30 FPS. The footage also shows a small stick on the ground. This is Twilight’s measuring stick. It is one ponylength long, and has markings every 1/10 of a ponylength. Twilight and Dash want to determine whether Dash’s speed is above or below 20 ponylengths per second, when she’s at the very bottom of her loop. Using just the information you have so far, you should be able to figure this out! (If you don’t have Flash Player, you won’t be able to figure out the answer yourself. But maybe, if you think hard, you can figure out a method Twilight and Rainbow can use to find the answer.) Author's Note Next time on Ponies Learn Calculus: Lightbulbs! Onion skins! And maybe even... some actual math?
Chapter 2 - Camera TrickTwilight pranced around the field, giddy. “Eureka! Eureka!” “Um, English, Twi?” Rainbow Dash said. “I don’t speak Ancient Geek.” Twilight grabbed Rainbow’s shoulders, shoving her smile right into Dash’s face. “I’ve got it. This camera can tell us your exact speed, Rainbow Dash.” Pinkie Pie poked her head around it. “It can? I didn’t see a ‘Tell Me Rainbow Dash’s Speed’ button. Maybe you need a three-press combo…?” She started mashing buttons. “No, Pinkie.” Rainbow Dash rolled her eyes. “She means she can watch the trick again and again, and it’ll help her guess my speed better.” “Actually,” Twilight said, “it’s neither.” She took the camera back from Pinkie, and paused it at the critical moment of Rainbow’s trick. “The key is to watch the trick frame by frame.” “Wait, what?” Dash stared deep into the screen. Nothing happened. “…Sorry, Twi, I don’t think you can figure out my speed by watching me not move.” “Just watch.” Twilight pressed the frame advance button. On the screen, Rainbow Dash suddenly jumped forward. “Ooooooookay,” said Dash. “I’m watching.” “Now we can figure out how fast you’re moving.” Dash raised an eyebrow. “Bear with me. Pinkie, you said that this device records 30 images every second, right?” “Correctasaurus!” “Then, the time between this moment and this moment,” Twilight said, switching back and forth between the two frames, “is exactly 1/30 seconds.” “Oh. Ohh.” A lightbulb appeared over Rainbow Dash’s head. “Oooohhhhhhhhh! I got it!” Dash switched back and forth between the two frames. “What distance did I travel during those 1/30 seconds? Looks like… half a polylength? A little more?” Twilight pointed at the screen. Specifically, at the measuring stick on the ground. “If you compare your position relative to the measuring stick, it looks like you moved approximately 6/10 ponylengths.” (this is what you'd see if you switched back and forth between the two frames really fast) “Cool. We’ve got the distance, we’ve got the time. Now, the speed is, uh…” “Speed equals distance divided by time,” Twilight reminded her. “I’m not in first grade, Twi. I’ve got this. 6/10 ponylengths, divided by 1/30 seconds, is…” Pinkie jumped in front of her. “Ooooh! Ooooh! I know! It’s 87078952296/4837719572 ponylengths per second!” Dash stared at her. “Uh, what? Is that more or less than 20?” Twilight scribbled some numbers down on a piece of parchment. “It’s 18. Therefore, Rainbow Dash, you were traveling at approximately 18 ponylengths per second during the climax of your loop-the-loop.” Rainbow’s face slowly transformed into a grin. “YES! First-place trophy, here I come!” “Woo-hoo!” Pinkie Pie jumped on top of Dash, tackling her to the ground. “I’ll prepare the trophy-shaped cake. Twilight, aren’t you excited? On a scale from 10 to hot fudge, how excited are you?” Twilight was staring at the camera, transfixed. She examined it from every angle, furiously scribbling down notes. Twilight gazed back and forth between the camera and her equations. A little bit of drool came out of her mouth. “…Let’s just leave her be.” Rainbow said to Pinkie Pie. “She’ll be better in the morning.” Pinkie Pie didn’t respond. She was staring into space, mumbling. “What if the Canterlot mountains were made of rock candy? Ahhh…” Rainbow briefly wondered why so many of her friends occasionally went insane. But the thoughts of tomorrow’s competition soon overtook her. She flew up into the cool evening air, letting the orange sky surround her. She yawned. That rehearsal had taken a lot out of her. She had to wake up early tomorrow morning, too. She settled on top of a nearby cloud and plugged her ears with cloudstuff. On the other hoof… Rainbow unplugged her ears and took to the sky again. Slowly, she flipped herself upside-down. One more run through can’t hurt. After all, I have to be perfect.
Chapter 3 - Do This In My Sleep11:58 PM, later that night – basement of Twilight’s library Twilight sat hunched at her workdesk. Disassembled pieces of the camera lay strewn across the surface: panels, wires, and dozens of screws. On the wall behind the desk was a chalkboard, containing Twilight’s detailed diagram showing where each part had been. Twilight’s attention was focused on a heavy black cube. It had been the bulk of the camera, and it still had a few wires sticking out. Her intuition told her that, whatever secrets this technology had to offer, she would find them inside the cube. But it wasn’t going to be easy. When Twilight had disassembled the rest of the camera, most steps had gone as expected. But when she tried to use her magic on anything connected to the black box -- “OWWWWWW!” Twilight jumped back. Her horn throbbed. It felt like somepony had just grabbed it and shook it up and down 60 times in a row. Suddenly, she heard footsteps descending the stairs above. “Twilight?” a voice called down. “What’s with the yelling? Is everything okay?” “Yes, Spike,” she sighed. “Are you sure? Because you’ve yelled ‘OWWWWWWW!’ about five times.” Twilight facehooved. “Sorry, Spike. I’ll try to yell quieter. You can go back to bed.” She was getting tired herself. She decided to get a midnight snack. She trotted upstairs to get a glass of milk. I can’t give up now, she thought. I’m so close! If I could just hold on to the box long enough to give the side one good yank… Twilight re-entered the basement. Her eyes flew wide open. She accidentally released her telekinetic hold on the glass of milk. It hit the floor and shattered. She barely noticed. “What.” The camera had disappeared. The black cube, and every single screw, wire, and panel, were gone. Twilight took a long, deep breath. “You win this round, camera,” she muttered. “But I’ll be back. You’ll rue the day you tried to outwit Twilight Sparkle.” * * * Cloudsdale Stadium, the next day Twilight sat in the stands, surrounded by Pinkie Pie, Rarity, Applejack, and Fluttershy. The aerobatics competition was well underway. So far, each of the pegasi’s dizzying twirls impressed Twilight to no end. But, though she tried her best to hide it, she was starting to get bored. After 14 entrants, Twilight felt like she was eating chocolate cake. Yes, each cake was unique and delicious. But no, that doesn’t mean she wanted to eat 15 more. Fortunately, the announcer had just called a much-needed 10-minute break. She excused herself, saying she had to go the restroom, and then made her way to the judges’ table. She had a question nagging at her head. “Excuse me,” she said, “do you have a minute?” One of the judges, a brown stallion, checked his watch. “We’ve got three. What’s the matter?” “I’ve been wondering, how do you make sure the entrants stay under the 20 p/s speed limit? Is there some kind of spell?” Another judge, a light-green pegasus with tan-colored hair, stepped forward. “That’s easy, girl,” she said. She spat on her hoof and held it out. “My name’s Epsilon, by the way. My friends call me Lonnie.” Twilight ignored the hoof. “I’m Twilight. Nice to meet you… Lonnie.” “Measuring a pegasus’ speed is a piece a’ cake,” Epsilon said. “You’ve just gotta formulize ‘er.” Twilight cocked her head. “Formulize?” “Yeah, girl,” said the green mare. “Check out that pegasus,” she said, pointing at a random stallion. “He’s flying along the path y=x², with constant speed 3 p/s.” Twilight’s eyes widened. “Wh-what! How did you do that?” “Anypony can do it,” she said. “It’s like whistling. Here, look at that grey pegasus.” “The one blowing up a balloon?” “Right. Now, try to formulize that balloon.” Twilight stared at the balloon. Nothing happened. “Er. What am I supposed to do?” “It’s like whistling. It’s hard to explain. Just kind of… meditate. Let some math drift into your head. All zen-like, sha.” Twilight stared at the balloon some more. This is ridiculous, she thought. She let her mind wander for a few seconds. Nothing. Stupid Twilight – thinking anypony could pick up a talent like this just by watching for a few seconds. I bet – “0.5+3√t hooflengths?” came a very quiet voice from behind her. Twilight spun around and saw Fluttershy. “Yup, that’s right, girl! The radius of the balloon was 0.5+3√t hooflengths, at time t seconds after she starting blowing it up.” “Wha!- Fluttershy?! Where did you – What did you – when did you – how did you do that?!” Fluttershy ducked her head into the clouds. “I just did what she told me to…” “But – but –” “Break’s over!” Epsilon trotted back behind the judges’ table. “Get ready. The next competitor’s called ‘Rainbow Dash.’ I hear she’s good.” With no time to get back to their seats, Twilight and Fluttershy decided to watch from the judges’ table. They cheered as Rainbow flew to the center of the stadium. Dash started out with a standard triple aileron roll. Oddly, she was going slower than during the rehearsal. Twilight guessed she was really worried about the speed limit. Dash led the rolls into a twisting vertical loop, to the delight of the crowd. “Did she just yawn?” whispered Fluttershy. “Did she?” Twilight squinted. “She’s probably bored of these tricks and wants to get to the big finish. Yep. That’s probably it.” The big finish came soon enough. Twilight watched Rainbow Dash line herself up. She had pulled it off fifty times late last night, but could she pull it off when it counted? The crowd oohed and aahed as Rainbow slowly rotated upside down. “Wow!” said a nearby stallion. “She even has her eyes closed!” Eyes closed? Twilight thought. I don’t remember that part of the trick. In front of her, Rainbow Dash started her dive. It didn’t look exactly like last night’s rehearsal. For one, she wasn’t rotating to match the angle of her descent. For two, her wings were flapping significantly slower. If Twilight didn’t know better, she would’ve guessed Rainbow were falling. And falling. And… yep, still falling. The stadium fell into a hush. Everypony waited for Dash’s grand recovery. Twilight looked down, in a panic. Rainbow was falling straight towards the weather factory. Twilight charged up her telekinesis, but she was already out of range. By now, not even the Wonderbolts were fast enough to dive and catch Rainbow. Rainbow fell closer and closer to the factory’s wind machine. Twilight grimaced and squinched her eyes. By the way, the Cloudsdale Weather Factory is constructed out of packed clouds. For all the earth ponies and unicorns reading this, picture some hay. Nice and soft when it’s spread out into a haystack. When it’s packed into a bale… not so much. The crowd gasped. Twilight opened her eyes. Rainbow had disappeared. The muttering around her told her what had happened. Rainbow had fallen down a chimney into the factory’s wind machine. To their credit, the factory workers reacted fast. Far below, two of them grabbed a loose cloud and put it directly under the falling pegasus. “Phew,” said Twilight. “At least everypony’s going to be safe.” Rainbow hit the cloud, bounced off of it, and landed belly-first on a large control panel filled with switches and levers. A large grey mass started coming out of the wind machine’s chimney. A large, whirling grey mass. It was moving upwards. Directly towards the stadium. “Tornado!” A Wonderbolt raced through the stadium below them, herding pegasi away. “Evacuate the stadium. Do not panic!” Twilight panicked. “Don’t panic!” said Epsilon. “Seriously, girl. We deal with rogue tornadoes every other week.” “Deal with them? How?” “Well, a tornado is just a bunch a’ air spinning around, right?” “Right…” “So, if you blow a whole bunch a’ air spinning the opposite direction at the exact same speed, it’ll counter it.” Twilight took a few seconds to process this. “Let me get this straight. You’re going to stop this tornado…” “Yeah?” “By launching another tornado at it.” “You’ve got it, girl.” Twilight looked down. Sure enough, the weather team pulled switches and pushed clouds, and a second tornado soon came out of the chimney. It rose, quickly catching up to the first tornado. Twilight gasped as the two whirling twisters collided. Chaos erupted in the air. Dirt flew in every direction. And when the dust settled… A tornado, twice as large as before, was rising towards the stadium. “Uh-oh,” said Epsilon. “That was supposed to happen, right?” said Fluttershy. “Please say that was supposed to happen.” “They got the speed wrong,” said Epsilon. “If two tornadoes aren’t rotating at th’ exact same speed, then instead of cancellin’ out, they merge into one bigger one.” Fluttershy stuffed her head back into the clouds. Epsilon stared down at the tornado, deep in thought. Her hooves twitched. Her eyes unfocused. “…It’s no good,” she concluded. “What?” said Twilight, for the third time that day. “I was trying to formulize the tornado’s spinning speed. But it’s moving too fast to tell. I have to get closer.” Before Twilight could object, Epsilon flapped her wings and glided down towards the spinning mass of chaos. Twilight followed her down. She paused, wings beating furiously, above the whirling grey-brown top of the storm. Twilight strained her ears to hear Epsilon over the storm: “Got it! The tornado will crash into the stadium exactly ten minutes after it formed. And the angle of the tornado, at the time t seconds after it formed, is—” Just then, a plank of wood flew out of the tornado, straight towards Epsilon. She folded her wings and dived down. The plank flew inches over her head. Unfortunately, diving down is a bad idea when you’re hovering above a tornado. Twilight averted her eyes. She heard a scream: “t² degrees! So the airspeed – AAAAAA!” When Twilight opened her eyes again, Lonnie was a green tumbling mass, spinning around like a sock in a washing machine. Twilight riffled through her brain, searching for a spell that would help. Unfortunately, any spell that would destroy the tornado would also hurt Lonnie. Don’t panic, Twilight thought to herself. She closed her eyes and counted to ten. There is one way out. A second tornado. Which means… all I have to do is figure out the tornado’s airspeed. Twilight flew back up to the stadium and retrieved her parchment and quill. She wrote down: Tornado crashes into stadium 10 mins after forming Angle of the tornado, t seconds after forming: t² The “angle of the tornado”? What does that mean? As Twilight was trying to figure this out, a cyan pegasus flapped up behind her and laid face-down on the cloud. “Heeeey,” yawned Rainbow Dash. “’Sup? What’d I miss?” Twilight’s eyes twitched. “…Rainbow?” “Yeah? What’s wrong, you nervous about something?” Twilight gulped. “…Rainbow, are you feeling alright?” “Uhh, yeah? A little nervous about the competition later today, but I’m fine.” “L-later today?” “Yeah. I just had a nightmare about it.” She shivered. “But that’s not gonna stop me! I’m ready to show the Wonderbolts what I’m made of.” Rainbow beamed, looking straight into Twilight’s eyes. Twilight looked away. She didn’t want to be around when Dash found out what happened. If she was going to receive news like this, she’d want it to be in a warm bed with at least three liters of double chocolate ice cream at hoof. Rainbow Dash had to find out eventually, but she vowed to break the news at the least stressful time possible. Or preferably get somepony else to break the news while she waited in a bunker five kilometers away. “…Rainbow! Quick question. Suppose, hypothetically, there was a tornado whose angle was t², where t is the number of seconds after it formed.” “Um, okay.” Rainbow scratched her ear. “Hey, what’s that weird whirring noise?” “Whirring? I don’t hear any whirring. Nope. Definitely no whirring here.” “Never mind. It’s probably just the wind.” Twilight grinned madly. “So,” Dash continued, “angles, huh? So, like, when the tornado first forms, its angle is 0²=0? What does that even mean?” “I’m not sure myself. What about…” Twilight picked a random number. “19 seconds later? After 19 seconds, the tornado’s angle is 19²=361 degrees.” Rainbow scratched her chin. “So in 19 seconds, the tornado spun a little more than one full turn. That’s pretty slow.” “You’re right. It must have sped up since then.” “What? Twilight, you said this was all hypothetical.” “R-right! The hypothetical tornado.” Spitfire yelled from below: “EVACUATE! Seven minutes!” “Whoa. What’s going on?” “Rainbow, wait!” Rainbow poked her head over the edge of the cloud and looked down. “I can explain—” Rainbow’s eyes opened wide. Then her face scrunched up. Then she started screaming. “Well, crap.” Author's Note Again, you have enough information to solve the puzzle before the next chapter. Theoretically, at least. Also: does anyone know a way to inline math formulas more complicated than x²+3? The unicode superscript 2 is already a kludge, and I'm worried that it might not appear on all browsers.
Chapter 4 - Mutually Assured Tornado“Rainbow, flying directly into the tornado won't help.” Twilight gripped Rainbow's tail with her telekinesis. “Gotta... be... a hero.” Rainbow struggled against the purple glow. “If you want to be a hero, help me with this math.” Dash did a double-take. “What.” Twilight pulled out her quill and parchment. “If we could determine your speed yesterday with the camera, then we can determine the speed of the tornado.” In fact, Twilight thought, how hard could it be? To find Dash's airspeed, they just had to measure how far she moved in one twentieth of a second. The same trick should work for the tornado. Luckily, they had Epsilon's formula: at time t seconds, the angle of the tornado was t² degrees. “Let's see...” Twilight scribbled. “The tornado will crash into the stadium exactly ten minutes after it formed.” “Don't remind me.” “That's t=600 seconds. So it will have spun... 600²=360000 degrees.” Rainbow Dash kicked the cloud impatiently. “And one frame later?” “600.05², which is... 360060.0025 degrees.” Rainbow whistled. “Nice job, Calculator Sparkle. So it turned 60.0025 degrees during 1/20 of a second. So its speed is... I've got this... 1200.05 degrees per second.” Rainbow blasted away. One millisecond later, her tail tugged her back. She groaned. “Let me go, Twi. I have some tornado canceling to do.” “Wait. If I learned anything from magic kindergarten, it's always check your answers, or you might accidentally turn Luna into a pumpkin.” “Whatever. Check your arithmetic.” “It's not the arithmetic I'm worried about. It's the one-twentieth of a second. What if the tornado sped up or slowed down during that time? Then our answer would be wrong.” Twilight stared at the parchment. She wrote a line of numbers and then crossed them out. She repeated this five times. Her ears drooped. Rainbow peered over the edge of the cloud. The tornado was like that thick tree on her route home: full of angry birds, and a lot closer than she expected. “No pressure Twilight, but...” Twilight put down the quill. “Argh. How am I supposed to figure out whether the tornado speeds up or slows down, if I can't figure out its speed to begin with?” “There's no time! Just use 1/1000 of a second instead of 1/20. There's no way the tornado can speed up in 1/1000 of a second. Even I can't go from 0 to 60 that fast.” “That...” Twilight muttered to herself. “...could actually work.” She tried to pick up her quill. From where she had placed it on the cloud. “Rats.” [If you are reading this to learn calculus, then you should pause here to compute the answer for 1/1000 of a second. Really. If you want a calculator, I recommend WolframAlpha.] [Seriously.] Rainbow pulled a feather out of her wing. Twilight dipped it in ink and wrote: She furrowed her brow. “1200.001. That's not what we got last time.” “Gotcha. One Dash spin cycle special, speed 1200.001 degrees per second, coming up.” Dash looked down. “In T minus... one minute.” “1200.001 isn't what we got last time. I'm going to try 0.0001 seconds, just to make absolutely sure.” Twilight pounded the cloud. “It's still a different answer.” Dash gritted her teeth. “We don’t have all day, Twi. I can see where this is going. You want a time period where the tornado won’t speed up at all. Go down from 0.0001 seconds to zero seconds flat.” Dash flinched. “Ouch. Even I can’t divide by zero.” Twilight grimaced. “That’s too bad. If that had worked, we’d be done. I’m going to try 0.000001 seconds.” “Too late.” Rainbow stuck her head over the edge of the cloud. The roar of the tornado muffled her voice. “I'm going down.” She hopped off the cloud. “If you change your answer in the next thirty seconds... yell loud, OK?” With that, Rainbow disappeared from Twilight's view. Twilight ground her teeth. Calculating machine that she was, thirty seconds was not enough time to square a number like 600.000001 . And even if it was, she doubted she would get the same answer as before. She could keep trying smaller and smaller lengths of time for the rest of the day. For all she knew, she would never reach the true answer -- just a series of better and better approximations. Wait. I don't have enough time to compute the answer for 0.000001 seconds, but I do have enough time to compute it for x seconds. Twilight thanked the Gods of Math for the simple form of the answer. She was about to plug in x=0.000001 when a thought struck her. This entire time, she had been computing a series of better and better approximations by plugging in smaller and smaller values of x. But now that she had the formula 1200+x, it was obvious. Her answers were just getting closer and closer to 1200. No time to check. Twilight prayed to Celestia that she was right. She put on her best Luna impression. “TWELVE HUNDRED DEGREES PER SECOND! EXACTLY!” Rainbow Dash circled beneath the raging grey funnel. The wind started to pick up around her. It carried her along like a straw in a spinning glass. She heard Twilight's scream and adjusted her wings a fraction of a millimeter. (Pegasi are very precise.) She began to feel the whipping wind of a tornado. Her wings shook. Turbulence rocked her body. Imagine riding a vibrating tiger. She hit the right speed. She knew it. All she had to do was push this countertornado up towards the other one. As easy as persuading that tiger to climb Mount Olympus. Dash gritted her teeth. The wind buffeted her from every direction as she spun in a tight circle. If she opened her wings, she'd sail away like a paper bag. Dash closed her eyes. Three, two, one. She lifted her wings and forced them down in a heartbeat. Then the counter-tornado threw her. She tumbled helplessly through the air. Grinning. She got it. She rightened herself and looked back. Two things happened at exactly the same time. One: the twister collided with the stadium's scaffolding. Two: the countertornado hit its mark. Splinters of wood and dirt shot out of the storm. Rainbow crossed her hooves. Both pairs of them. Please slow down, please slow down, please... The raging funnels merged. They began to fade. The winds slowed. Soon, nothing remained but bright blue sky. And one falling green pegasus. Dash flew down, forelegs open. “OOOFFFFF!” Epsilon hit her spine like an Apple family kick. She half-flew, half-fell to the clouds below. Rainbow struggled out from under Epsilon. She waved her hoof in front of her face. “Epsilon. Lonnie. Are you okay? Please be okay.” Epsilon's eyes fluttered open. “Rainbow Dash. You are in so much trouble.”
