Independence of Perpendicular Components of Motion
A force vector that 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
effect of a single vector in a given direction. Any force vector that 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 is 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 that
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 that 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 effect 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.
the topic of
relative velocity and riverboat motion was discussed previously. 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 effect upon the motion in the vertical direction. Any
alteration in one set of these components will have no effect on the other set. InLesson 2, we will apply this principle to the motion of projectiles that
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. |
||
Answer: G
The plane has both a northward and a westward motion. The
northward motion towards the Canadian border is dependent upon the component of
velocity in the northern direction. To solve this problem, the northern
component of the plane's velocity must first be determined.
vNorth = 400 • sine
(150 degrees) = 200 km/hr
Once determined, the d = v • t equation can be used to determine
the time that it takes the plane to reach the Canadian border.
dNorth = vNorth • t t = dNorth / vNorth = (1500 km) / (200 km/hr) = 7.5 hr
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?
Answer: 7.5 hr
The same strategy used to answer #1 would have to be used in
question #2. First find the northward component of the plane's velocity.
vNorth = 358 • sine
(146 degrees) = 200 km/hr
Once determined, the d = v • t equation can be used to determine
the time that it takes the plane to reach the Canadian border.
dNorth = vNorth • t t = dNorth / vNorth = (1500 km) / (200 km/hr) = 7.5 hr
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 were greater, then the time required
for the boat to reach the opposite shore would not change.
a.
True
b. False
Answer: A - True
The flow of the
water down the river and parallel to its banks has no affect on
the across-the-river motion of the boat. Perpendicular
components of motion are independent of each other.
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.
Answer: A and D
The
only two variables which would effect the
northward motion of the boat would be the northward velocity of the river and
the time required to cross the river. The northward distance is simply given by
the equation
dNorth = vNorth • t
A
decrease in the northward river velocity would decrease the distance and a
decrease in time would decrease the distance. If the boat traveled faster across the river, the time would be
decreased.