Two-Dimensional Motion and Vectors
Preview
Section 1 Introduction to Vectors
Section 2 Vector Operations
Section 3 Extra Questions
Section 1
Two-Dimensional Motion and Vectors
Section 1
What do you think?
• How are measurements such as mass and
volume different from measurements such as
velocity and acceleration?
• How can you add two velocities that are in
different directions?
Two-Dimensional Motion and Vectors
Section 1
Introduction to Vectors
• Scalar - a quantity that has magnitude but no
direction
– Examples: volume, mass, temperature, speed
• Vector - a quantity that has both magnitude and
direction
– Examples: acceleration, velocity, displacement, force
Two-Dimensional Motion and Vectors
Section 1
Vector Properties
• Vectors are generally drawn as arrows.
– Length represents the magnitude
– Arrow shows the direction
• Resultant - the sum of two or more vectors
Two-Dimensional Motion and Vectors
Section 1
Finding the Resultant Graphically
• Method
– Draw each vector in the proper
direction.
– Establish a scale (i.e. 1 cm = 2 m)
and draw the vector the
appropriate length.
– Draw the resultant from the tip of
the first vector to the tail of the last
vector.
– Measure the resultant.
• The resultant for the addition of
a + b is shown to the left as c.
Two-Dimensional Motion and Vectors
Vector Addition
• Vectors can be moved
parallel to themselves
without changing the
resultant.
– the red arrow represents
the resultant of the two
vectors
Section 1
Two-Dimensional Motion and Vectors
Section 1
Vector Addition
• Vectors can be added in
any order.
– The resultant (d) is the
same in each case
• Subtraction is simply the
addition of the opposite
vector.
Two-Dimensional Motion and Vectors
Properties of Vectors
Click below to watch the Visual Concept.
Visual Concept
Section 1
Two-Dimensional Motion and Vectors
Section 1
Sample Resultant Calculation
• A toy car moves with a
velocity of .80 m/s across
a moving walkway that
travels at 1.5 m/s. Find
the resultant speed of the
car.
Two-Dimensional Motion and Vectors
Section 1
Now what do you think?
• How are measurements such as mass and
volume different from measurements such as
velocity and acceleration?
• How can you add two velocities that are in
different directions?
Two-Dimensional Motion and Vectors
Section 2
What do you think?
• What is one disadvantage of adding vectors by
the graphical method?
• Is there an easier way to add vectors?
Two-Dimensional Motion and Vectors
Section 2
Vector Operations
• Use a traditional x-y coordinate system as shown below
on the right.
• The Pythagorean theorem and tangent function can be
used to add vectors.
– More accurate and less time-consuming than the graphical
method
Two-Dimensional Motion and Vectors
Section 2
Pythagorean Theorem and Tangent Function
Two-Dimensional Motion and Vectors
Vector Addition - Sample Problems
• 12 km east + 9 km east = ?
– Resultant: 21 km east
• 12 km east + 9 km west = ?
– Resultant: 3 km east
• 12 km east + 9 km south = ?
– Resultant: 15 km at 37° south of east
• 12 km east + 8 km north = ?
– Resultant: 14 km at 34° north of east
Section 2
Two-Dimensional Motion and Vectors
Section 2
Resolving Vectors Into Components
Two-Dimensional Motion and Vectors
Section 2
Resolving Vectors into Components
• Opposite of vector addition
• Vectors are resolved into x and y components
• For the vector shown at right,
find the vector components vx
(velocity in the x direction) and
vy (velocity in the y direction).
Assume that that the angle is
20.0˚.
• Answers:
– vx = 89 km/h
– vy = 32 km/h
Two-Dimensional Motion and Vectors
Section 2
Adding Non-Perpendicular Vectors
• Four steps
–
–
–
–
Resolve each vector into x and y components
Add the x components (xtotal = x1 + x2)
Add the y components (ytotal = y1 + y2)
Combine the x and y totals as perpendicular vectors
Two-Dimensional Motion and Vectors
Adding Vectors Algebraically
Click below to watch the Visual Concept.
Visual Concept
Section 2
Two-Dimensional Motion and Vectors
Section 2
Classroom Practice
• A camper walks 4.5 km at 45° north of east and
then walks 4.5 km due south. Find the camper’s
total displacement.
• Answer
– 3.4 km at 22° S of E
Two-Dimensional Motion and Vectors
Section 2
Now what do you think?
• Compare the two methods of adding vectors.
• What is one advantage of adding vectors with
trigonometry?
• Are there some situations in which the graphical
method is advantageous?
Two-Dimensional Motion and Vectors
Section 4
Classroom Practice Problem
• A plane flies northeast at an airspeed of 563
km/h. (Airspeed is the speed of the aircraft
relative to the air.) A 48.0 km/h wind is blowing
to the southeast. What is the plane’s velocity
relative to the ground?
• Answer: 565.0 km/h at 40.1° north of east
• How would this pilot need to adjust the direction
in order to to maintain a heading of northeast?
Two-Dimensional Motion and Vectors
Section 4
Now what do you think?
• Suppose you are traveling at a constant 80 km/h when a
car passes you. This car is traveling at a constant 90
km/h.
– How fast is it going, relative to your frame of
reference?
– How fast is it moving, relative to Earth as a frame of
reference?
• Does velocity always depend on the frame of reference?
• Does acceleration depend on the frame of reference?
Two-Dimensional Motion and Vectors
Preview
• Multiple Choice
• Short Response
• Extended Response
Section 4
Two-Dimensional Motion and Vectors
Section 4
Multiple Choice
1. Vector A has a magnitude of 30 units. Vector B
is perpendicular to vector A and has a
magnitude of 40 units. What would the
magnitude of the resultant vector A + B be?
A. 10 units
B. 50 units
C. 70 units
D. zero
Two-Dimensional Motion and Vectors
Section 4
Multiple Choice, continued
Use the diagram to answer questions 3–4.
3. What is the direction of the
resultant vector A + B?
A. 15º above the x-axis
B. 75º above the x-axis
C. 15º below the x-axis
D. 75º below the x-axis
Two-Dimensional Motion and Vectors
Section 4
Multiple Choice, continued
Use the diagram to answer questions 3–4.
4. What is the direction of the
resultant vector A – B?
F. 15º above the x-axis
G. 75º above the x-axis
H. 15º below the x-axis
J. 75º below the x-axis
Two-Dimensional Motion and Vectors
Section 4
Multiple Choice, continued
Use the passage below to answer questions 5–6.
A motorboat heads due east at 5.0 m/s across a river that flows
toward the south at a speed of 5.0 m/s.
5. What is the resultant velocity relative to an observer on the shore ?
A. 3.2 m/s to the southeast
B. 5.0 m/s to the southeast
C. 7.1 m/s to the southeast
D. 10.0 m/s to the southeast
Two-Dimensional Motion and Vectors
Section 4
Multiple Choice, continued
Use the passage below to answer questions 5–6.
A motorboat heads due east at 5.0 m/s across a river that flows
toward the south at a speed of 5.0 m/s.
6. If the river is 125 m wide, how long does the boat take to cross the
river?
F. 39 s
G. 25 s
H. 17 s
J. 12 s
Two-Dimensional Motion and Vectors
Section 4
Multiple Choice, continued
7. The pilot of a plane measures an air velocity of 165 km/h south
relative to the plane. An observer on the ground sees the plane pass
overhead at a velocity of 145 km/h toward the north.What is the
velocity of the wind that is affecting the plane relative to the
observer?
A. 20 km/h to the north
B. 20 km/h to the south
C. 165 km/h to the north
D. 310 km/h to the south
Two-Dimensional Motion and Vectors
Section 4
Multiple Choice, continued
8. A golfer takes two putts to sink his ball in the hole once he is on the
green. The first putt displaces the ball 6.00 m east, and the second
putt displaces the ball 5.40 m south. What displacement would put
the ball in the hole in one putt?
F. 11.40 m southeast
G. 8.07 m at 48.0º south of east
H. 3.32 m at 42.0º south of east
J. 8.07 m at 42.0º south of east
Two-Dimensional Motion and Vectors
Section 4
Short Response
13. If one of the components of one vector along
the direction of another vector is zero, what can
you conclude about these two vectors?
Two-Dimensional Motion and Vectors
Section 4
Short Response, continued
14. A roller coaster travels 41.1 m at an angle of
40.0° above the horizontal. How far does it move
horizontally and vertically?
Two-Dimensional Motion and Vectors
Section 4
Short Response, continued
14. A roller coaster travels 41.1 m at an angle of
40.0° above the horizontal. How far does it move
horizontally and vertically?
Answer: 31.5 m horizontally, 26.4 m vertically