Independence of Perpendicular
Components of Motion

A force vector which is directed
upward and rightward has two parts - an upward part and a
rightward part. That is to say, if you pull upon an object
in an upward and rightward direction, then you are exerting
an influence upon the object in two separate directions - an
upward direction and a rightward direction. These two parts
of the two-dimensional vector are referred to as
components.
A
component
describes the affect of a single vector in a given
direction. Any force vector which is exerted at an angle to
the horizontal can be considered as having two parts or
components. The vector sum of these two components is always
equal to the force at the given angle. This is depicted in
the diagram below.

Any vector - whether it be a force
vector, displacement vector, velocity vector, etc. -
directed at an angle can be thought of as being composed of
two perpendicular components. These two components can be
represented as legs of a right triangle formed by projecting
the vector onto the x- and y-axis.

The two perpendicular parts or
components of a vector are independent of each other.
Consider the pull upon Fido as an example. If the horizontal
pull upon Fido increases, then Fido would be accelerated at
a greater rate to the right; yet this greater horizontal
pull would not exert any vertical influence upon Fido.
Pulling horizontally with more force does not lift Fido
vertically off the ground. A change in the horizontal
component does not affect the vertical component. This is
what is meant by the phrase "perpendicular components of
vectors are independent of each other." A change in one
component does not affect the other component. Changing a
component will affect the motion in that specific direction.
While the change in one of the components will alter the
magnitude of the resulting force, it does not alter the
magnitude of the other component.

The resulting motion of a plane
flying in the presence of a crosswind is the combination (or
sum) of two simultaneous velocity vectors which are
perpendicular to each other. Suppose that a plane is
attempting to fly northward from Chicago to the Canada
border by simply directing the plane due northward. If the
plane encounters a crosswind directed towards the west, then
the resulting velocity of the plane would be northwest. The
northwest velocity vector consists of two components - a
north component resulting from the plane's motor (the
plane velocity) and a westward component resulting
from the crosswind (the wind velocity). These two
components are independent of each other. An alteration in
one of the components will not affect the other component.
For instance, if the wind velocity increased, then the plane
would still be covering ground in the northerly direction at
the same rate. It is true that the alteration of the wind
velocity would cause the plane to travel more westward;
however, the plane still flies northward at the same speed.
Perpendicular components of motion do not affect each
other.

Now consider an air balloon
descending through the air toward the ground in the presence
of a wind which blows eastward. Suppose that the downward
velocity of the balloon is 3 m/s and that the wind is
blowing east with a velocity of 4 m/s. The resulting
velocity of the air balloon would be the combination (i.e.,
the vector sum) of these two simultaneous and independent
velocity vectors. The air balloon would be moving downward
and eastward.

If the wind velocity increased, the
air balloon would begin moving faster in the eastward
direction, but its downward velocity would not be altered.
If the balloon were located 60 meters above the ground and
was moving downward at 3 m/s, then it would take a time of
20 seconds to travel this vertical distance.

d = v • t
So
t = d /
v = (60 m) / (3 m/s) =
20
seconds

During the 20 seconds taken by the
air balloon to fall the 60 meters to the ground, the wind
would be carrying the balloon in the eastward direction.
With a wind speed of 4 m/s, the distance traveled eastward
in 20 seconds would be 80 meters. If the wind speed
increased from the value of 4 m/s to a value of 6 m/s, then
it would still take 20 seconds for the balloon to fall the
60 meters of downward distance. A motion in the downward
direction is affected only by downward components of motion.
An alteration in a horizontal component of motion (such as
the eastward wind velocity) will have no affect upon
vertical motion. Perpendicular components of motion are
independent of each other. A variation of the eastward wind
speed from a value of 4 m/s to a value of 6 m/s would only
cause the balloon to be blown eastward a distance of 120
meters instead of the original 80 meters.

In the most recent section of Lesson
1, the topic of relative velocity and riverboat motion
was discussed. A boat on a river often heads straight across
the river, perpendicular to its banks. Yet because of the
flow of water (i.e., the current) moving parallel to the
river banks, the boat
does not land on the bank directly across from the starting
location. The resulting motion of the boat is the
combination (i.e., the vector sum) of these two simultaneous
and independent velocity vectors - the boat velocity plus
the river velocity. In the diagram at the right, the boat is
depicted as moving eastward across the river while the river
flows southward. The boat starts at Point A and heads itself
towards Point B. But because of the flow of the river
southward, the boat reaches the opposite bank of the river
at Point C. The time required for the boat to cross the
river from one side to the other side is dependent upon the
boat velocity and the width of the river. Only an eastward
component of motion could affect the time to move eastward
across a river.

Suppose that the boat velocity is 4 m/s and the river
velocity is 3 m/s. The magnitude of the resultant velocity
could be determined to be 5 m/s using the Pythagorean
Theorem. The time required for the boat to cross a 60-meter
wide river would be dependent upon the boat velocity of 4
m/s. It would require 15 seconds to cross the 60-meter wide
river.

d = v • t
So
t = d /
v = (60 m) / (4 m/s) =
15
seconds

The southward river velocity will not affect the time
required for the boat to travel in the eastward direction.
If the current increased such that the river velocity became
5 m/s, then it would still require 15 seconds to cross the
river. Perpendicular components of motion are independent of
each other. An increase in the river velocity would simply
cause the boat to travel further in the southward direction
during these 15 seconds of motion. An alteration in a
southward component of motion only affects the southward
motion.

All vectors can be thought of as having perpendicular
components. In fact, any motion that is at an angle to the
horizontal or the vertical can be thought of as having two
perpendicular motions occurring simultaneously. These
perpendicular components of motion occur independently of
each other. Any component of motion occurring strictly in
the horizontal direction will have no affect upon the motion
in the vertical direction. Any alteration in one set of
these components will have no affect on the other set. In
Lesson 2, we will apply this
principle to the motion of projectiles which typically
encounter both horizontal and vertical motion.

Check Your
Understanding

1.
A plane flies northwest out of O'Hare Airport in Chicago at
a speed of 400 km/hr in a direction of 150 degrees (i.e., 30
degrees north of west). The Canadian border is located a
distance of 1500 km due north of Chicago. The plane will
cross into Canada after approximately ____ hours.

a. 0.13

b. 0.23

c. 0.27

d. 3.75

e. 4.33

f. 6.49

g. 7.50

h. None of these are
even close.

2. Suppose that the plane in
question 1 was flying with a velocity of 358 km/hr in a
direction of 146 degrees (i.e., 34 degrees north of west).
If the Canadian border is still located a distance of 1500
km north of Chicago, then how much time would it take to
cross the border?

3.
TRUE or
FALSE:

A boat heads straight across a
river. The river flows north at a speed of 3 m/s. If the
river current was greater, then the time required for the
boat to reach the opposite shore would not
change.

a. True

b. False

4.
A boat begins at point A and heads straight across a
60-meter wide river with a speed of 4 m/s (relative to the
water). The river water flows north at a speed of 3 m/s
(relative to the shore). The boat reaches the opposite shore
at point C. Which of the following would cause the boat to
reach the opposite shore at a location SOUTH of C?

a. The boat heads across the river at 5 m/s.

b. The boat heads across the river at 3 m/s.

c. The river flows north at 4 m/s.

d. The river flows north at 2 m/s.

e. Nonsense! None of these affect the location where
the boat lands.