[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.