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[music] Maybe I don't know my position and my velocity, but I know my acceleration.

Maybe my acceleration at sometime t is just a constant 8 units per second

squared, right? So, I'm going to accelerate at a constant

rate. Is that enough information to determine my

velocity? We have to think back to how velocity and

acceleration are related. My definition and acceleration is the

change to the derivative in velocity, right.

Acceleration is rate of change in velocity.

Or in other words, right, velocity is an anti-derivative of acceleration.

So, I know that a of t is 8. Let's now solve for v.

Okay, so a of t, my acceleration, is just constant function 8, and v of t is the

anti-derivative of my acceleration, right. If I anti-differentiate acceleration, I'm

going to get velocity. It means I'm anti-differentiating 8.

Well what anti-differentiates to 8 or 8t, plus some constant C?

It's that constant again. Right?

Knowing my acceleration doesn't determine my velocity, it only determines my

velocity up to some constant. I could be going really fast or really

slow, but still accelerating at the same rate.

Well, in any case, I've at least got a formula for my velocity with that

constant. Now, can I use that formula to determine

my position? Same kind of game.

My velocity is the rate in change in my position, right?

My velocity is the derivative of position. And that means position is an

anti-derivative of velocity. So, let's anti-differentiate.

So, p of t is an anti-derivative of my velocity, and I figured out my velocity a

minute ago. My velocity is 8t plus c.

So, I want to anti-differentiate 8t plus c.

Well, it's an anti-derivative of a sum, so it's the sum of the anti-derivatives.

And what's an anti-derivative for 8 times something?

Well, that's a constant multiple, so it's 8 times the anti-derivative of t, plus an

anti-derivative for C. Now, 8 times, what's an anti-derivative

for t? Well one of them is t squared over 2.

And what's an anti-derivative for C? Well, C times t is an anti-derivative for

C. Then, I should add some constant here.

I'm going to call that constant big D. Right.

So, here's my position. I guess I could write this a little bit

more nicely, coz the 8 and the dividing by 2 simplify that I could write it as 4t

squared, plus Ct, plus D. Well, that's kind of weird, right?

Why are there two constants in my answer? Just knowing that my acceleration is 8

units per second squared, doesn't tell me my initial velocity, right?

This quantity C here, is really v of 0. Right?

It's my initial velocity. And I could be accelerating at a rate of 8

units per second squared. But starting with any of a range of

possible initial velocities, right? I could be going really fast at first or

really slow at first. But still, always accelerating at 8 units

per second squared. So, knowing this doesn't nail down C.

Similarly, just knowing my velocity doesn't nail down my initial position.

Right? This D here is really p of 0.

Right? If I plug in t equals 0, I just get D.

And that means that D is really providing my initial position.

So, if I know my initial position and my initial velocity and my acceleration, then

I can nail down an exact formula for my position.

I can get rid of these mystery constants. The important lesson to take away here is

that anti-differentiating can actually be useful.

Anti-differentiating let's us take, say, velocity information and produce the

position data. Or even better, in this case, it let us

take acceleration information, figure out the velocity, and then, take that velocity

information to figure out my position.