Chapter 8: The Five Formulas of Motion

Having tossed and turned until midnight, Lia slept until the sky was bright.

She was woken by the sun.

Opening her eyes, she saw the sun already hanging high outside the window.

Warm light streamed through the glass, casting a bright square on the wooden floor.

‘Oh no, I’m late!’

The thought jolted her fully awake.

She sprang out of bed and hastily pulled on her apprentice robe.

Lia burst from her room, taking the spiral staircase two and three steps at a time.

The long sleep had left her full of energy, but the thought of facing Klein’s expressionless face made her heart leap into her throat.

First, she had ruined an experiment, and now she was late on the second day.

Her standing in her mentor’s eyes must have plummeted through the earth’s core.

She stood before the elegant fourth-floor door, took a few deep breaths, and then cautiously pushed it open.

The reprimand she expected never came.

The room was quiet.

Klein wasn’t busy at the workbench; instead, he was seated behind his desk, several parchments covered in writing spread out before him.

He held a quill, intently amending something on the pages, and merely lifted his gaze at the sound of the door opening, showing no sign of displeasure.

Lia quietly sighed in relief and dutifully went to stand by his side, bracing herself for another day of academic interrogation.

“Those descriptions you mentioned yesterday are too convoluted,” Klein began, pointing to a line on the parchment.

“‘The accumulation of swiftness,’ for instance, is too vague for precise description and calculation.”

He tapped another spot with the tip of his quill.

“And this term, ‘swiftness.’

While the meaning is understandable, it lacks conciseness.”

Lia’s mind began to race.

‘Are we establishing academic definitions now?’

“What do you mean, Mentor?”

“Based on my understanding, I’ve devised a few new terms,” Klein said, picking up a blank sheet of parchment.

“The change in an object’s ‘swiftness’ over a unit of time, I will call ‘amplification.’

It represents how quickly ‘swiftness’ increases.”

‘Amplification?

Isn’t that just acceleration?’

Lia retorted silently, ‘Mentor, I think the term ‘acceleration’ would work.’

“That could work.

As for ‘swiftness’ itself,” Klein continued, “I am inclined to replace it with ‘velocity.’

It describes the rate of an object’s movement.”

Velocity.

Lia nodded, thinking the term was a good fit.

She added at the opportune moment, “Th-that book seemed to mention that velocity has direction.

For example, if you throw something into the air, its velocity has a different direction on the way up than on the way down.”

Klein’s quill froze.

He looked at Lia, and after a moment, added a note next to ‘velocity’: has direction.

“Excellent. Then, the distance an object moves in a specific time, we will call ‘displacement.’”

Displacement, velocity, acceleration.

“I organized your points from yesterday,” Klein said, pushing the parchment sheets toward Lia.

“Now, let us continue our discussion.”

His posture was not that of a mentor testing a student, but of an equal scholar initiating a debate.

Lia looked at the familiar logical chains on the parchment and felt as if she were back in her past life, writing research papers.

“The ancient text says that when an object falls from rest, its initial velocity is ‘none,’” Lia said, getting into the zone and beginning to share proactively.

“But from the moment it begins to fall, acceleration comes into play.”

“Therefore, the velocity at any given moment is equal to the initial velocity plus the product of acceleration and time.”

Klein said nothing, merely recording her words swiftly on the parchment.

“Writing out ‘initial velocity’ and ‘acceleration’ every time is too cumbersome,” he said, suddenly pausing his quill.

“Did that ancient text of yours use simpler symbols to represent them?”

Lia’s heart skipped a beat.

She feigned a thoughtful pause, her small brow furrowed in a desperate attempt to look like she was trying to remember.

The sight of her expression almost made Klein laugh.

He lifted the cup from his desk and took a sip of water to hide the curve of his lips.

He thoroughly enjoyed watching her rack her brains to spin this tale.

“I think… I think so!” Lia’s eyes lit up.

“I remember now!

The author of that book was very lazy and liked to use letters instead.”

She picked up a quill and, without hesitation, wrote a line of symbols on Klein’s parchment.

“For initial velocity, it uses a ‘v’ with a small circle below it.

For the final velocity, just a ‘v.’

For ‘acceleration,’ it uses ‘a.’

For the time elapsed, ‘t.’

And for the distance moved, or displacement, ‘s.’”

In one breath, she wrote down v₀, v, a, t, and s.

Only after she had finished did she realize how smoothly she had presented it.

She quickly tried to cover her tracks, “The symbols in the book were a mess.

This is just what I could piece together from memory, so it might not be right.”

Klein gazed at the line of simple, elegant symbols and said nothing.

He simply pulled the parchment toward him and rewrote the formula he had described in words, this time using the symbols Lia had provided.

v = v₀ + at

And just like that, an incredibly simple yet profound formula appeared on the parchment of another world.

Klein’s fingers traced over the formula, and he could feel the harmony and beauty it contained.

“What about displacement?” he asked, looking up.

He had completely ceded the floor to Lia, casting himself as the perfect inquirer and scribe.

He wanted to see just how many more subversive treasures were hidden inside this little girl’s head.

“Displacement…” Lia’s mind spun into high gear again.

“The book said that if the ‘acceleration’ is constant, then the average velocity over a certain distance is equal to the initial and final velocities added together and divided by two.”

She cautiously tossed out the core idea of the second key formula.

The tip of Klein’s quill glided across the parchment.

s = [(v + v₀) / 2] * t

He finished writing the formula but did not stop.

He stared at the two equations before him, the light in his deep blue eyes growing ever brighter.

He began to derive new equations on his own.

He rearranged the first formula to get t = (v – v₀) / a, then substituted it into the second.

The quill flew across the parchment, leaving behind lines of rigorous deduction.

s = [(v + v₀) / 2] * [(v – v₀) / a]

s = (v² – v₀²) / 2a

2as = v² – v₀²

A brand-new formula was born.

This formula perfectly connected displacement, velocity, and acceleration, yet it completely eliminated the variable of time, ‘t’!

Klein’s breath caught for a moment.

This was true theory!

Interlocking, mutually confirming, concise, and irrefutable!

He did not stop.

He then substituted the first formula, v = v₀ + at, into the second, this time eliminating the final velocity, ‘v.’

s = [(v₀ + at + v₀) / 2] * t

s = [(2v₀ + at) / 2] * t

s = v₀t + (1/2)at²

The fourth formula!

Lia stood to the side, hardly daring to breathe as she watched Klein enter a state of feverish deduction.

She had merely provided a starting point, two fundamental components, and Klein, with his terrifying mathematical prowess, had single-handedly constructed the skeleton of the entire theoretical system.

Finally, after studying the four formulas on the parchment, Klein isolated v₀ from the second equation, getting v₀ = (2s/t) – v, and substituted it back into the first.

A fifth formula leaped onto the page.

s = vt – (1/2)at²

Five formulas.

They lay in neat order on the parchment, each one a model of simplicity, yet together they formed a perfect theoretical framework describing uniformly accelerated linear motion.

They were like an intricate magic circle—interlinked, mutually reinforcing, and shimmering with the light of truth.

The room fell into a dead silence.

Klein stood up, gazing at the parchment as if it were a flawless work of art.

He was now one hundred percent certain that this was not something that could have been recorded in some so-called ancient text.

Because when Lia had explained these concepts, her awkwardness—the way she struggled for analogies, trying to break down a complete theory into scattered fragments of knowledge—could not have been faked.

That ancient text simply did not exist.

All of this knowledge originated from the little girl before him, who barely reached his chest.