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acceleration vector therefore changes back and forth between the up and
down directions, but is never in the same direction as the horse’s motion. In
this chapter, we will examine more carefully the properties of the velocity,
acceleration, and force vectors. No new principles are introduced, but an
attempt is made to tie things together and show examples of the power of
the vector formulation of Newton’s laws.
8.1The Velocity Vector
For motion with constant velocity, the velocity vector is
v=
.
r/
.
t[ only for constant velocity ] .
The
.
r vector points in the direction of the motion, and dividing it by the
scalar
.
t only changes its length, not its direction, so the velocity vector
points in the same direction as the motion. When the velocity is not
constant, i.e. when the x-t, y-t, and z-t graphs are not all linear, we use the
slope-of-the-tangent-line approach to define the components v
x
, v
y
, and v
z
,
from which we assemble the velocity vector. Even when the velocity vector
is not constant, it still points along the direction of motion.
Vector addition is the correct way to generalize the one-dimensional
concept of adding velocities in relative motion, as shown in the following
example:
Example: velocity vectors in relative motion
Question: You wish to cross a river and arrive at a dock that is
directly across from you, but the river’s current will tend to carry
you downstream. To compensate, you must steer the boat at an
angle. Find the angle
.
, given the magnitude, |v
WL
|, of the water’s
velocity relative to the land, and the maximum speed, |v
BW
|, of
which the boat is capable relative to the water.
Solution: The boat’s velocity relative to the land equals the
vector sum of its velocity with respect to the water and the
water’s velocity with respect to the land,
v
BL
= v
BW
+ v
WL
.
If the boat is to travel straight across the river, i.e. along the y
axis, then we need to have v
BL,x
=0. This x component equals the
sum of the x components of the other two vectors,
v
BL,x
= v
BW,x
+ v
WL,x
,
or
0 = -|v
BW
| sin
.
+ |v
WL
| .
Solving for
.
, we find
sin
.
= |v
WL
|/|v
BW
| ,
.
=sin
–1
v
WL
v
BW
.
v
BW
v
WL
.
x
y
Chapter 8Vectors and Motion