The Language God Talks | SPARO ARCHIVE VOL.1 — Transcript

You see, nature is always moving.

Things change, planets orbit, atoms vibrate, and rockets fly.

And for a long time, people were puzzled by a very simple question: How do we describe motion?

Sounds easy, doesn't it?

You say, well, the rocket moved from the launchpad to space.

But that's not enough, we want to know exactly how it moved, was it smooth, did it jerk, how fast was it going at the exact moment the engines cut off?

To answer this, we need a special kind of mathematics, a mathematics of change, we call it calculus.

But don't let the name scare you.

It is just a way of looking at small changes.

Now, if I ask you how fast is the rocket going, you might say 10,000 miles per hour.

But what does that mean? If the rocket travels 10,000 miles in one hour, the average speed is 10,000 miles per hour.

That's simple arithmetic, distance divided by time.

But here is the trouble, the rocket is speeding up.

If the speed is changing every second, what do we mean by the speed right now, at this exact instant?

There is a subtle point here.

If we take right now literally, we mean a moment with no time duration.

If no time passes, the rocket moves no distance, so if we try to calculate speed, zero divided by zero, this is meaningless, you cannot divide by zero.

So how can something have a speed if it hasn't moved any distance in zero time?

This paradox stopped the ancients cold, they didn't know how to handle it.

To solve this, we have to play a trick, a very clever trick.

We say, let's not look at zero time, let's look at a very short time.

Suppose we want the speed exactly at five seconds.

We wait a tiny bit longer, say until 5.1 seconds, we measure the change.

Now, here is the calculus part, we make that time interval smaller, and smaller, and smaller.

We find that the ratio stops changing wildly, it approaches a specific number.

We call that limit the derivative.

It is the speed at that exact moment, when you see DS over DT in a book, it looks scary.

But all it means is, how much did the distance change for a tiny, tiny change in time?

Let's try a real example.

Imagine a spent rocket stage falling back to Earth, the distance it falls follows a rule, 16t squared.

How fast is it falling at exactly one second?

Using our trick, we look at one second plus a tiny bit of time, we'll call it delta t.

We do the algebra, 16 times 1 plus delta t squared, we subtract the original distance, and we divide by that tiny time.

Now, watch this, as we make that tiny time vanish to zero, the extra bits disappear.

What is left?

32 feet per second.

That is the magic, by looking at the change, we found the velocity.

But we aren't done, in a rocket, you don't feel speed, you feel the push.

You feel the change in speed, if the velocity is changing, we do the trick again.

We ask, how much does the velocity change in a tiny amount of time?

That is acceleration, the derivative of the derivative.

In our falling rocket, the acceleration is constant, 32, gravity pulls at 32 feet per second per second.

That is why you feel a constant weight.

Finally, suppose we are in mission control, we have the data, we know the acceleration at every second.

Can we figure out how far we went?

Yes.

We just go backward.

This is called integration.

If the derivative breaks the motion down into tiny snapshots to find the speed, the integral glues all those tiny snapshots back together to find the total distance.

So, you see, calculus is not just symbols.

Differentiation is just looking at the speedometer of the rocket, integration is looking at the odometer to see how far you went.

It is the language we use to talk about things that change, and since in this universe, everything is changing, it is the language of nature.

That's all calculus is.

Citizens of the world, this is Aaron, head of space missions. I am speaking tonight to address the unfortunate wave of rumors and misinformation currently spreading regarding the recent cosmic anomaly.

To those who mistake our silence for a lack of action, understand that precision takes time.

SPARO is not standing still, we are currently engaged in a massive coordinated effort in direct collaboration with NASA. Every resource we possess is being utilized.

We are not ignoring the event, we are solving it.

Our CEO has been directing these operations personally.

Sir, I believe the people need to hear from you.

I will just say that, stay curious, stay bold, stay with SPARO.