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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