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Let's now consider two
dimensional motion,

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and let's try to
analyze how to describe

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the change in velocity.

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So again, let's choose
a coordinate system.

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We have an origin plus y plus x.

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And let's draw the
trajectory of our object.

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And now let's draw the object
at two different times.

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So for instance, if I call
this the location at time t1,

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and a little bit
later here, this

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is the location of
the object at time t2.

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We'll call our unit
vectors i hat and j hat.

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We know that the
direction of the velocity

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is tangent to this curve.

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So if we draw v at time
t1-- and over here,

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notice the direction has
changed v at time t2.

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And what we'd like to do now
is describe, just as before,

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that our acceleration a of t is
the derivative of the velocity

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as a function of time.

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What that means is the
limit as delta t goes

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to 0 of delta v over delta t.

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Now, it's much harder
to visualize the delta v

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in this drawing.

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And partly, the reason for
that is these velocity vectors

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are located at two
different points.

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And right now, the
backs of these vectors

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have different places in space.

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But remember that delta v
is just v, in this case,

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at time t2 minus v at time t1.

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And our principle for
subtracting two vectors

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at different
locations in space is

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to draw the vectors where we put
the tails at the same location.

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So here's a tail at this vector.

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We're just going to translate
that vector in space.

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That is still v at time t1.

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These vectors are equal.

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They have the same length, and
they have the same direction.

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And so delta v is
just the vector

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that connects here to there.

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That's what we mean by delta v.

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And so you can see in
this particular case

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that it's not obvious from
looking at the orbit what

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the delta v is.

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So what we need to do is
just trust our calculus.

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And so when we write the
velocity as dx dt i hat plus dy

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dy j hat, and we're now treating
each direction independently.

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We call this vx i
hat plus vy j hat.

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So that's our velocity vector.

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Then our acceleration is just
the derivative of the velocity.

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We take each
direction separately,

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so we have dv x dt i
hat plus dv y dt j hat.

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Now, again, notice
that velocity v of x

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is already the first
derivative of the position

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of the exponent function.

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So what we really have here
is the second derivative

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of the position
function in the i hat

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direction and the second
derivative of the component

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function in the y direction.

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And that is what we call the
instantaneous acceleration.

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Now, again, this is
sometimes awkward to draw,

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but you always must remember
that this x component

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of the acceleration
by definition

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is the second derivative
of the component function

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or the first derivative
of the component

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function for the velocity.

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And likewise, the y component
of the acceleration ay

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is the second derivative
of the component

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function for position.

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And that's also
equal, by definition,

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to the first derivative of
the component of the velocity

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vector.

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And that's how we
describe the acceleration.

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As before, we can talk about
the magnitude of a vector.

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And the magnitude of a
we'll just write as a.

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It's the components squared,
added together, taken

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square root.

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And that's our magnitude.

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And so now we've described all
of our kinematic quantities

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in two dimensions--
the position,

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the velocity as the
derivative of the position,

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and the acceleration as the
derivative of the velocity

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where each direction is
treated independently.