WEBVTT
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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.