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Math 18.02 (Spring 2009): Lecture 6
February 13
Reading Material:
From
Simmons
: 17.4 and
Lecture Notes
K.
Last time:
Parametric equations for lines and curves
Today:
·
,
×
. Kepler’s second law.
2
Velocity and Acceleration
We recall that a curve
γ
in 3D is described by three
parametric equations
x
=
x
(
t
)
y
=
y
(
t
)
z
=
z
(
t
)
where
t
belongs to an interval [
a, b
]. We can associate to the point
P
(
t
) = (
x
(
t
)
, y
(
t
)
, z
(
t
)) that
determines the curve the vector position
→
OP
(
t
) =
~
r
(
t
) =
x
(
t
)
ˆ
i
+
y
(
t
)
ˆ
j
+
z
(
t
)
ˆ
k,
that is the vector that starts at the origin
O
and ends on the point
P
(
t
) of the curve:
It is now clear that in order to describe a curve it is completely equivalent to either give parametric
equations or give the vector position
~
r
(
t
).
Deﬁnition 1.
Given a curve
γ
described by a vector
~
r
(
t
) =
x
(
t
)
ˆ
i
+
y
(
t
)
ˆ
j
+
z
(
t
)
ˆ
k
for
t
in
[
a, b
]
, we
deﬁne the
velocity
of
~
r
(
t
)
to be the vector
~v
(
t
)
deﬁned as
~v
(
t
) =
dx
dt
(
t
)
ˆ
i
+
dy
dt
(
t
)
ˆ
j
+
dz
dt
(
t
)
ˆ
k
=
x
0
(
t
)
ˆ
i
+
y
0
(
t
)
ˆ
j
+
z
0
(
t
)
ˆ
k.
1
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View Full DocumentWe deﬁne the
speed
of
~
r
(
t
)
to be
s
(
t
) =

~v
(
t
)

=
p
(
x
0
(
t
))
2
+ (
y
0
(
t
))
2
+ (
z
0
(
t
))
2
.
We usually also write
~v
(
t
) =
d~
r
dt
(
t
)
,
or even simpler
~v
=
d~
r
dt
.
The vector
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 Spring '08
 Auroux

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