Chapter
Forces in Two Dimensions
5
In this chapter you will:
•
Represent vector
quantities both graphically
and algebraically.
•
Use Newton’s laws to
analyze motion when
friction is involved.
•
Use Newton’s laws and
your knowledge of vectors
to analyze motion in two
dimensions.
Section
Vectors
5.1
In this section you will:
•
Evaluate the sum of two or more vectors in two dimensions
graphically.
•
Determine the components of vectors.
•
Solve for the sum of two or more vectors algebraically by
adding the components of the vectors.
What do the arrows represent?
Vectors and Scalars
A vector has magnitude as
well as direction.
Some vector quantities:
displacement, velocity, force,
momentum
A scalar has only a magnitude.
Some scalar quantities: mass,
time, temperature
SCALAR
A SCALAR quantity
is any quantity in
physics that has
MAGNITUDE ONLY
Number value
with units
Scalar
Example
Magnitude
Speed
35 m/s
Distance
25 meters
Age
16 years
VECTOR
A VECTOR quantity
is any quantity in
physics that has
BOTH MAGNITUDE
and DIRECTION
Vector
Example
Magnitude and
Direction
Velocity
35 m/s, North
Acceleration
10 m/s2, South
Force
20 N, East
An arrow above the symbol
illustrates a vector quantity.
It indicates MAGNITUDE and
DIRECTION
Vectors are represented by
arrows.
• The length of the arrow reflects the
magnitude of the measurement
The head of the vector must point
in the direction of the quantity
Addition of Vectors – Graphical Methods
For vectors in one
dimension, simple
addition and subtraction
are all that is needed.
You do need to be careful
about the signs, as the
figure indicates.
VECTOR APPLICATION
ADDITION: When two (2) vectors point in the SAME direction, simply
add them together.
EXAMPLE: A man walks 46.5 m east, then another 20 m east.
Calculate his displacement relative to where he started.
46.5 m, E
+
66.5 m, E
20 m, E
MAGNITUDE relates to the
size of the arrow and
DIRECTION relates to the
way the arrow is drawn
What is the resultant vector?
What do the arrows represent?
When an object is
tossed, it travels in
a “parabolic” path.
There is a
velocity
component
vertically
(vy)
Gravity
always
points
downward
There is a velocity
component horizontally (vx)
The velocity of the toss is actually the sum of the two vectors
Steps for Graphical Addition
1. Choose an appropriate scale (e.g.. 1cm = _____ m/s)
2. Draw all vectors with tail starting at origin
3. Redraw vector from “head to tail” while maintaining
original direction of vector.
4. From tail of first vector to head of last connect lines
(this is resultant) direction is towards head of last
original vector
5. Measure length and convert back using scale.
6. Measure resultant from 0 degrees.
What happens now?
Graphical:
Scale: 1 box = 50 km/h
Head to Tail Method
Tail to Tail Method
Finding the Resultant Vector
• Pythagorean Theorem
c 2 = a2 + b 2
Head to Tail Method
Direction can be measured in
degrees
1. 3.0 m/s, 45 deg + 5.0
m/s, 135 deg
5.83 m/s, 104 deg
5.0 m/s, 45 deg + 2.0
m/s, 180 deg
3.85m/s, 66.5 deg
Elaboration Vector Activity
1.
You take a walk in the park for 15 steps using a compass that points 25º North of East.
•How would you use the simulation to represent your path?
•Explain why the same representation works for illustrating this different scenario: You drive at 15 miles/hour using a compass that points 25º North of East.
•Write another scenario using different units that could also be represented the same.
2.
In the simulation, a vector is described by four measurements: R, Ө, Rx, and Ry. Put a vector in the work area, and then investigate to make sense of what these
four things represent. In your investigation, use a wide variety of vector measurements and all three styles of Component Displays. Then, describe in your own words
what the measurements represent and what “component” means.
3.
Suppose you are driving 14 miles/hour with a compass reading of 35°north of east.
•Represent the vector using the simulation. How fast is your car traveling in the north direction? How fast in the east direction?
•Figure out how the components could be calculated using geometry if you couldn’t use the simulation.
•Check your ideas by testing with other vectors and then write a plan for finding the components of any vector.
4. To get to the sandwich shop, you left home and drove 6 miles south and then 10 miles west.
•If a bird flew from your house to the sandwich shop in a straight line, how far do you think the bird would fly? Use the simulation to check your reasoning.
•What direction should it fly from your house to get to the shop?
•Explain how you could use the simulation to answer these questions.
•Explain how you could use geometry equations to answer these questions.
5.
Suppose you and a friend are test driving a new car. You drive out of the car dealership and go 10 miles east, and then 8 miles south. Then, your friend drives 8
miles west, and 6 miles north.
•If you had the dealer’s homing pigeon in the car, how far do you think it would have to fly to get back to the dealership? Use the simulation to test ideas.
•The distance that the bird has to fly represents the sum of the 4 displacement vectors. Use the simulation to test ideas you have about vector addition. After your tests,
describe how you can use the simulation to add vectors.
6.
A paper airplane is given a push so that it could fly 7m/s 35° North of East, but there is wind that also pushes it 8 m/s 15° North of East.
•Use the simulation to solve the problem. How fast could it go and in what direction would it travel?
•Think about your math tools and design a way to add vectors without the simulation.
•Check your design by adding several vectors mathematically and then checking your answers using the simulation.
Addition of Vectors – Graphical Methods
If the motion is in two dimensions, the situation is
somewhat more complicated.
Here, the actual travel paths are at right angles to
one another; we can find the displacement by
using the Pythagorean Theorem.
Addition of Vectors – Graphical Methods
Adding the vectors in the opposite order gives the
same result:
Exploration
Commutative Property
Addition of Vectors – Graphical Methods
Even if the vectors are not at right
angles, they can be added graphically by
using the “tail-to-tip” method.
Addition of Vectors – Graphical Methods
The parallelogram method may also be used;
here again the vectors must be “tail-to-tip.”
Concept Development 3-2 Front only
at this point.
NON-COLLINEAR VECTORS
When two (2) vectors are PERPENDICULAR to each other, you must
use the PYTHAGOREAN THEOREM
FINISH
Example: A man travels 120 km east
then 160 km north. Calculate his
resultant displacement.
the hypotenuse is
called the RESULTANT
160 km, N
VERTICAL
COMPONENT
S
R
T
T
A
120 km, E
HORIZONTAL COMPONENT
Pythagorean Theorem
30.814 m/s
14.369
m/s
25
34 m/s
Since components always form a right triangle, the
Pythagorean theorem holds: (14.369)2 + (30.814)2 =
(34)2.
Note that a component can be as long, but no longer,
than the vector itself. This is because the sides of a
right triangle can’t be longer than the hypotenuse.
c
Law of Cosines
Θ
A
a
Law of Cosines:
B
R
b
R2 = A2 + B2 - 2 AB cos Θ
These two sides are repeated.
This side is always opposite this angle.
It matters not which side is called A, B, and R, so long as the two
rules above are followed. This law is like the Pythagorean theorem
with a built in correction term of -2 AB cos Θ . This term allows us
to work with non-right triangles. Note if Θ = 90, this term drops
out (cos 90 = 0), and we have the normal Pythagorean theorem.
Example Vector Problem

A motorboat heads due east at 16 m/s
across a river that flows due north at 9.0
m/s.
 What is the resultant velocity of the
boat?
 If the river is 136 m wide, how long
does it take the motorboat to reach the
other side?
Graphical Solution
Draw vectors, tip to tail
 Using your scale, measure length of R

R
9 m/s
16 m/s
Solution : Algebraic Method
• What is the resultant velocity of the
boat? A2 + B2 = C2
•
(9 m/s)2 + (16 m/s)2 = R2
Solution: Calculation of time
• If the river is 136 m wide, how
long does it take the motorboat to
reach the other side?
• V = Δd/Δt
• 16 m/s = 136 m/t
• 136 m /16 m/s =Δt
• t = 8.5 s
Example Vector Problem
An airplane is flying 200
mph at 50°. Wind velocity
is 50 mph at 270°. What
is the velocity of the plane?
90
180
o
o
0
270
o
o
90
180
o
o
0
270
o
o
90
180
o
o
0
270
o
o
90
180
o
o
0
270
o
o
Subtraction of Vectors, and
Multiplication of a Vector by a Scalar
In order to subtract vectors, we
define the negative of a vector, which
has the same magnitude but points
in the opposite direction.
Then we add the negative vector:
Opposite of a Vector
v
-v
If v is 17 m/s up
and to the right,
then -v is 17 m/s
down and to the
left. The directions
are opposite; the
magnitudes are the
same.
Scalar Multiplication
x
3x
-2x
½x
Scalar multiplication means
multiplying a vector by a real
number, such as 8.6. The
result is a parallel vector of a
different length. If the scalar
is positive, the direction
doesn’t change. If it’s
negative, the direction is
exactly opposite.
Blue is 3 times longer than red in the
same direction. Black is half as long
as red. Green is twice as long as
red in the opposite direction.
VECTOR APPLICATION
SUBTRACTION: When two (2) vectors point in the OPPOSITE direction,
simply subtract them.
EXAMPLE: A man walks 46.5 m east, then another 20 m west.
Calculate his displacement relative to where he started.
46.5 m, E
20 m, W
26.5 m, E
Elaboration
Worksheets 1 and 2
Homework
Page 121 1-4
Closure
Vectors
Kahoot Principles &
Problems Vectors 1
Objectives
The students will be able to:
1. Use the trigonometric component method to resolve a vector
components in the x and y directions.
2. Use the trigonometric component method to determine the vector
resultant in problems involving vector addition or subtraction of two
or more vector quantities.
150 N
Horizontal
component
Vertical
component
Vector Components
A 150 N force is exerted up and to
the right. This force can be
thought of as two separate forces
working together, one to the right,
and the other up. These
components are perpendicular to
each other. Note that the vector
sum of the components is the
original vector (green + red =
black). The components can also
be drawn like this:
Adding Vectors by Components
Any vector can be expressed as the sum
of two other vectors, which are called its
components. Usually the other vectors are
chosen so that they are perpendicular to
each other.
Finding Components with Trig
v sin 
Multiply the magnitude of the original vector by
the sine & cosine of the angle made with the
given. The units of the components are the
same as the units for the original vector.
v

v cos 
Here’s the
correspondence:
cosine  adjacent side
sine  opposite side
Component Example
14.369 m/s
30.814 m/s
25
34 m/s
A helicopter is flying at 34 m/s at 25 S of W (south of west).
The magnitude of the horizontal component is 34 cos 25 
30.814 m/s. This is how fast the copter is traveling to the
west. The magnitude of the vertical component is 34 sin 25
 14.369 m/s. This is how fast it’s moving to the south.
Note that 30.814 + 14.369 > 34. Adding up vector components
gives the original vector (green + red = black), but adding up
the magnitudes of the components is meaningless.
Pythagorean Theorem
14.369 m/s
30.814 m/s
25
34 m/s
Since components always form a right triangle,
the Pythagorean theorem holds: (14.369)2 +
(30.814)2 = (34)2.
Note that a component can be as long, but no
longer than, the vector itself. This is because the
sides of a right triangle can’t be longer than the
hypotenuse.
3-4 Adding Vectors by Components
If the components are
perpendicular, they can be found
using trigonometric functions.
Other component pairs
v cos 
v
v sin 


v

v
There are an infinite number of component pairs into which a
vector can be split. Note that green + red = black in all 3
diagrams, and that green and red are always perpendicular.
The angle is different in each diagram, as well as the lengths
of the components, but the Pythagorean theorem holds for
each. The pair of components used depends on the geometry
of the problem.
WHAT ABOUT DIRECTION?
In the example, DISPLACEMENT asked for and since it is a VECTOR quantity,
we need to report its direction.
N
W of N
E of N
N of E
N of E
N of W
E
W
S of W
NOTE: When drawing a right triangle that
conveys some type of motion, you MUST
draw your components HEAD TO TOE.
S of E
W of S
E of S
S
NEED A VALUE – ANGLE!
Just putting N of E is not good enough (how far north of east ?).
We need to find a numeric value for the direction.
To find the value of the
angle we use a Trig
function called TANGENT.
200 km
160 km, N
 N of E
120 km, E
So the COMPLETE final answer is : 200 km, 53.1 degrees North of East
What are your missing
components?
Suppose a person walked 65 m, 25 degrees East of North. What
were his horizontal and vertical components?
H.C. = ?
V.C = ?
25
65 m
The goal: ALWAYS MAKE A RIGHT
TRIANGLE!
To solve for components, we often use
the trig functions since and cosine.
adjacent side
hypotenuse
adj  hyp cos 
cosine 
opposite side
hypotenuse
opp  hyp sin 
sine 
adj  V .C.  65 cos 25  58.91m, N
opp  H .C.  65 sin 25  27.47m, E
Example - Solve
A bear, searching for food wanders 35 meters east then 20 meters north.
Frustrated, he wanders another 12 meters west then 6 meters south. Calculate
the bear's displacement.
-
12 m, W
-
=
6 m, S
20 m, N
35 m, E

=
14 m, N
R  14 2  232  26.93m
14 m, N
R
23 m, E
14
 .6087
23
  Tan 1 (0.6087)  31.3
Tan 
23 m, E
The Final Answer: 26.93 m, 31.3 degrees NORTH of EAST
Example - Solve
A boat moves with a velocity of 15 m/s, N in a river which
flows with a velocity of 8.0 m/s, west. Calculate the
boat's resultant velocity with respect to due north.
Rv  82  152  17 m / s
8.0 m/s, W
15 m/s, N
Rv

8
Tan   0.5333
15
  Tan 1 (0.5333)  28.1
The Final Answer : 17 m/s, @ 28.1 degrees West of North
Example - South
A plane moves with a velocity of 63.5 m/s at 32 degrees South of East. Calculate
the plane's horizontal and vertical velocity components.
adjacent side
cosine 
hypotenuse
adj  hyp cos 
H.C. =?
32
63.5 m/s
opposite side
sine 
hypotenuse
opp  hyp sin 
V.C. = ?
adj  H .C.  63.5 cos 32  53.85 m / s, E
opp  V .C.  63.5 sin 32  33.64 m / s, S
Example - Solve
A storm system moves 5000 km due east, then shifts course at 40
degrees North of East for 1500 km. Calculate the storm's
resultant displacement.
1500 km
adjacent side
hypotenuse
V.C.
adj  hyp cos 
cosine 
opposite side
hypotenuse
opp  hyp sin 
sine 
40
5000 km, E
H.C.
adj  H .C.  1500 cos 40  1149.1 km, E
opp  V .C.  1500 sin 40  964.2 km, N
5000 km + 1149.1 km = 6149.1 km
R
964.2 km

6149.1 km
The Final Answer: 6224.2 km @ 8.92
degrees, North of East
Adding Vectors by Components
Adding vectors:
1. Draw a diagram; add the vectors graphically.
2. Choose x and y axes.
3. Resolve each vector into x and y components.
4. Calculate each component using sines and cosines.
5. Add the components in each direction.
6. To find the length and direction of the vector, use:
Elaboration
• Transparency 5-1 Vector Components
• Worksheets 3-4
• Section 5-1 Additional Problems
Closure
Vectors 2
Kahoot
Homework for Chapter 5 p125
• Problems on pages 125 # 5-10
Closure
Vectors 3
Kahoot