Ponies Learn Calculus

by Lopsy

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.”

Comments

[FiM Fact]

https://www.youtube.com/watch?v=Nrwl6a4-lSg

[pterrorgrine]

Limits! (Essentially.) Now we're really doing calculus!