WEBVTT
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Recall, when we were examining
the motion an object,
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in two dimensions, we
introduced Cartesian coordinates
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and a position vector.
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Now let's suppose
the object has moved
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to a new point, along the orbit.
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Well, we'll write
another vector r of t.
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And let's say this took a
time delta t to the new point.
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And what we want to
define is the displacement
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of that object.
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So that's a vector delta r.
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And recall that a vector
of time of t plus delta t
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is equal to the old vector r of
t plus this displacement vector
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delta r.
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Now what we want to consider is
a limit as delta t goes to 0.
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And let's just look
graphically at what that means.
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As we move this delta,
delta t-- as delta t
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gets smaller and smaller and
our object is getting closer
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and closer to its
position at time t,
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and the position vector
r of t plus delta t
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is getting closer and
closer to r of t delta t,
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the key fact is that if we
do a tangent to the orbit,
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then the limit of delta
r is approaching tangent
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to the curve.
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So in the limit,
delta r, the direction
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is tangent to the orbit.
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So that's our first key
property of delta r.
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Now the second thing
we want to express
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is, if we write delta
r, as a displacement
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in the i-hat direction
and a displacement
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in the j-hat
direction, now again,
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maybe we can just clean
this up a little bit,
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and see what we mean by that.
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So here's our delta r.
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And we have a little
delta x in this direction,
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delta y in that direction.
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Remember delta x or delta y
can be positive or negative.
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That's all right.
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Now if we want to define
our velocity as the limit,
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as delta t goes to 0 of
delta r over delta t,
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then what we see is we have two
pieces, the limit as delta t
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goes to 0, of delta x over
delta t i-hat, plus the limit
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as delta t goes to 0 of
delta y delta t j-hat.
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And the definition
of these limits,
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we'll write that as
the derivative dr, dt.
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So the velocity is dr, dt.
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And that's equal to dx, dt,
how that coordinate function is
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changing in time, i-hat
plus dy, dt j-hat.
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Now as far as notation
goes, we write
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this philosophy as an x
component of the velocity
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plus a y component of
the velocity, where
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the x component,
the x, is dx, dt.
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And the y component is dy, dt.
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Now recall that the direction
was tangent to the curve,
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but the magnitude
of the velocity,
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what we call the
speed, is just the sum
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of the squares of the
components, the square root.
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And so now we've
describe what we
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refer to as the
instantaneous velocity.
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So far we've looked that a
trajectory in two dimensions.
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Let's again consider
some type of motion
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where we choose a positive
y-axis, a positive x-axis,
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an origin, e at vectors,
i-hat and j-hat.
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And I'll have some
type of trajectory,
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where our object is
moving like that.
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We know that at this
particular time,
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the velocity is tangent to
this trajectory, at that point.
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And now, what we'd
like to do, is
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try to describe--
we've described
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it's two components
ex and vy as a vector.
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So if you did vector
decomposition,
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you would write a vector like
this and a vector like that.
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This is the x component.
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That's the y component.
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And now if I define
this angle theta,
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we know that a vector has a
direction and a magnitude.
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We've seen what we call
the magnitude the speed.
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So that's just the sum of
these components squared,
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square root.
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Speed is always positive.
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So we always take the
positive square root.
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And now what about the direction
of this vector in the xy plane?
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Well, we can see
from our geometry
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that the tangent theta
is given by the y
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component over the x component.
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Or one could say that the angle
theta, at this given time,
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is the inverse
function of vy over vx.
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And so now we've described not
only the direction of velocity,
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but the angle that it's making
with the horizontal axis.
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And so we have now completely
described the velocity,
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instantaneous velocity,
vector at time t
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in terms of its two component
functions, its speed
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and the angle that makes
at the positive x-axis.