Let's now consider two dimensional motion,
and let's try to analyze how to describe
the change in velocity.
So again, let's choose a coordinate system.
We have an origin plus y plus x.
And let's draw the trajectory of our object.
And now let's draw the object at two different times.
So for instance, if I call this the location at time t1,
and a little bit later here, this
is the location of the object at time t2.
We'll call our unit vectors i hat and j hat.
We know that the direction of the velocity
is tangent to this curve.
So if we draw v at time t1-- and over here,
notice the direction has changed v at time t2.
And what we'd like to do now is describe, just as before,
that our acceleration a of t is the derivative of the velocity
as a function of time.
What that means is the limit as delta t goes
to 0 of delta v over delta t.
Now, it's much harder to visualize the delta v
in this drawing.
And partly, the reason for that is these velocity vectors
are located at two different points.
And right now, the backs of these vectors
have different places in space.
But remember that delta v is just v, in this case,
at time t2 minus v at time t1.
And our principle for subtracting two vectors
at different locations in space is
to draw the vectors where we put the tails at the same location.
So here's a tail at this vector.
We're just going to translate that vector in space.
That is still v at time t1.
These vectors are equal.
They have the same length, and they have the same direction.
And so delta v is just the vector
that connects here to there.
That's what we mean by delta v.
And so you can see in this particular case
that it's not obvious from looking at the orbit what
the delta v is.
So what we need to do is just trust our calculus.
And so when we write the velocity as dx dt i hat plus dy
dy j hat, and we're now treating each direction independently.
We call this vx i hat plus vy j hat.
So that's our velocity vector.
Then our acceleration is just the derivative of the velocity.
We take each direction separately,
so we have dv x dt i hat plus dv y dt j hat.
Now, again, notice that velocity v of x
is already the first derivative of the position
of the exponent function.
So what we really have here is the second derivative
of the position function in the i hat
direction and the second derivative of the component
function in the y direction.
And that is what we call the instantaneous acceleration.
Now, again, this is sometimes awkward to draw,
but you always must remember that this x component
of the acceleration by definition
is the second derivative of the component function
or the first derivative of the component
function for the velocity.
And likewise, the y component of the acceleration ay
is the second derivative of the component
function for position.
And that's also equal, by definition,
to the first derivative of the component of the velocity
vector.
And that's how we describe the acceleration.
As before, we can talk about the magnitude of a vector.
And the magnitude of a we'll just write as a.
It's the components squared, added together, taken
square root.
And that's our magnitude.
And so now we've described all of our kinematic quantities
in two dimensions-- the position,
the velocity as the derivative of the position,
and the acceleration as the derivative of the velocity
where each direction is treated independently.