Section 14.4
Motion in Space: Velocity and Acceleration
“Applications of vector functions and vector derivatives”
In Calc 1 and 2, we saw that derivatives and integrals were closely
related to the concept of speed, distance traveled and acceleration. In
this section, we shall generalize these ideas to vectors, tracking the
motion of a body in 3dimensional space (rather than the rather fake
2d space developed in Calc 2).
1.
Velocity, Acceleration and Force Vectors
Suppose that
vector
r
(
t
) is the position function for a particle
P
traveling
through space.
Then we define its velocity vector and acceleration
vector as follows:
Definition 1.1.
The velocity (the rate of change of position with re
spect to time) of
P
at time
t
is defined to be
vectorv
(
t
) = lim
h
→
0
vector
r
(
t
+
h
)

vector
r
(
t
)
h
=
vector
r
′
(
t
)
and the acceleration (the rate of change of velocity with respect to
time) of
P
at time
t
is defined to be
vectora
(
t
) = lim
h
→
0
vectorv
(
t
+
h
)

vectorv
(
t
)
h
= lim
h
→
0
vector
r
′
(
t
+
h
)

vector
r
′
(
t
)
h
=
vector
r
′′
(
t
)
.
We define the speed of
P
at time
t
to be

vector
r
′
(
t
)

(this is the physical
rate of change of distance with respect to time).
Calculation of these values are straight forward.
Example 1.2.
Find the velocity, acceleration and speed of a particle
whose position in space is given by
vector
r
(
t
) = sin (
t
)
vector
i
+
t
vector
j
+ cos (
t
)
vector
k.
We have
vector
r
′
(
t
) = cos (
t
)
vector
i
+
vector
j

sin (
t
)
vector
k
,
vector
r
′′
(
t
) =

sin (
t
)
vector
i

cos (
t
)
vector
k
, and

vector
r
′
(
t
)

=
radicalbig
(cos
2
(
t
) + 1 + sin
2
(
t
)) =
√
2.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Stefanov
 Calculus, Derivative, Integrals, Acceleration, Force, Mass, Velocity

Click to edit the document details