I'd like to now show you two mathematical facts about how we
integrate quantities of motion.
Suppose we have an object, let's call this the i hat direction,
and it's moving, and it has an x component of velocity.
And suppose it starts at some initial position
and goes at some final position.
And in the initial position, it was at time ti,
and the final position was at t final.
Now, we have two very important integrals
that we want to look at which govern how we describe,
how we integrate acceleration with respect to time,
and also how we'll integrate acceleration
with respect to space.
Now, recall that our acceleration, which
may be a function of time, is the derivative
of the x component of velocity with respect to time.
So, let's first look at a simple integration
of acceleration with respect to time and see what we get.
So, if we integrate acceleration,
now I'm going to introduce an integration variable, dt prime.
And our integration variable goes from our initial time
to some final time.
And if we use our fact that acceleration
is the derivative of the velocity,
then we can write this is dvx, dt prime, dt prime again.
After a while I'll drop the dummy variables
and the endpoints of the integral.
And this simply becomes the integral
from t prime, t initial.
t prime equals t final of dvx prime.
Now notice we've done a change of variables
in the integration.
So instead of now talking about the endpoints
of the integral from t prime, t initial
to t final, now what we're doing is we changed our integration
variable.
And so what we have is, we have the velocity integration
variable is going from some initial value.
And that integration variable is going to some final value.
So again, our initial conditions may
have some initial velocity and some final velocity--
three different ways of describing
the initial and final states.
This integral is a very straightforward integral,
the x final minus vx initial, which
is the change in the x component of the velocity.
And that's our classic result that we've
done that the integration of acceleration
with respect to time is a change in velocity.
Now, let's see what happens as a comparison
when we integrate our acceleration not with respect
to time, but suppose that it's a function of space.
And so now we're integrating again.
We have a dummy variable.
We have to be careful, because this
is the x component of acceleration,
but x prime is our integration variable,
and that's going from some initial position
to some final position.
Now, we can write this as again, make the substitution,
dx dt, dx prime, going from the initial to the final.
But now notice we're going to rewrite this integrand as dvx
times dx prime, dt prime, dt.
And when we do that, we have this result that dx prime,
dt prime is precisely what we mean by vx.
Now, I'll introduce some dummy variables there as well.
So what our integrand becomes is dvx prime, vx prime.
That's not a function, it's a product
of a differential times vx prime.
And now our new integration variable
is going from some initial value to some final value.
Once again, this is a straightforward integral.
For those who haven't seen integrals like this,
it's just something like x dx is x squared over 2,
but our integration variable is vx prime.
So what we get is 1/2 vx final squared
minus vx initial squared.
And so what we see here is two fundamentally different facts
that if you integrate acceleration
with respect to time, you get the change in velocity.
But if you integrate acceleration
with respect to space, you get 1/2 times the change
not of velocity, but of the x component
of the velocity squared.
And both of these facts are central to how
we'll analyze the concept of work
and how we applied Newton's second law.