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Discussion Questions
A. Is it possible for an airplane to maintain a constant velocity vector but not a
constant |v|. How about the opposite -- a constant |v| but not a constant
velocity vector. Explain.
B. New York and Rome are at about the same latitude, so the earth’s rotation
carries them both around nearly the same circle. Do the two cities have the
same velocity vector (relative to the center of the earth). If not, is there any
way for two cities to have the same velocity vector.
8.2The Acceleration Vector
When all three acceleration components are constant, i.e. when the v
x
-t,
v
y
-t, and v
z
-t graphs are all linear, we can define the acceleration vector as
a=
.
v/
.
t[ only for constant acceleration] ,
which can be written in terms of initial and final velocities as
a=
(
v
f
-v
i
)/
.
t[ only for constant acceleration] .
If the acceleration is not constant, we define it as the vector made out of the
a
x
, a
y
, and a
z
components found by applying the slope-of-the-tangent-line
technique to the v
x
-t, v
y
-t, and v
z
-t graphs.
Now there are two ways in which we could have a nonzero acceleration.
Either the magnitude or the direction of the velocity vector could change.
This can be visualized with arrow diagrams as shown in the figure. Both the
magnitude and direction can change simultaneously, as when a car acceler-
ates while turning. Only when the magnitude of the velocity changes while
its direction stays constant do we have a
.
v vector and an acceleration
vector along the same line as the motion.
Self-Check
(1) In figure (a), is the object speeding up or slowing down. (2) What would the
diagram look like if v
i
was the same as v
f
. (3) Describe how the
.
v vector is
different depending on whether an object is speeding up or slowing down.
If this all seems a little strange and abstract to you, you’re not alone. It
doesn’t mean much to most physics students the first time someone tells
them that acceleration is a vector, and that the acceleration vector does not
have to be in the same direction as the velocity vector. One way to under-
stand those statements better is to imagine an object such as an air freshener
or a pair of fuzzy dice hanging from the rear-view mirror of a car. Such a
hanging object, called a bob, constitutes an accelerometer. If you watch the
bob as you accelerate from a stop light, you’ll see it swing backward. The
horizontal direction in which the bob tilts is opposite to the direction of the
acceleration. If you apply the brakes and the car’s acceleration vector points
backward, the bob tilts forward.
After accelerating and slowing down a few times, you think you’ve put
your accelerometer through its paces, but then you make a right turn.
Surprise! Acceleration is a vector, and needn’t point in the same direction as
the velocity vector. As you make a right turn, the bob swings outward, to
your left. That means the car’s acceleration vector is to your right, perpen-
(a) A change in the magnitude of the
velocity vector implies an acceleration.
(b) A change in the direction of the
velocity vector also produces a non-
zero
.
v vector, and thus a nonzero
acceleration vector,
.
v/
.
t.
v
i
v
f
-v
i
v
f
.
v
v
i
v
f
-v
i
v
f
.
v
(1) It is speeding up, because the final velocity vector has the greater magnitude. (2) The result would be zero,
which would make sense. (3) Speeding up produced a
.
v vector in the same direction as the motion. Slowing
down would have given a
.
v that bointed backward.
Section 8.2The Acceleration Vector