Download Chapter 3 Two-Dimensional Motion and Vectors

Chapter 3 Two-Dimensional Motion and Vectors
Section 1 Introduction to Vectors
Preview
 Objectives, scalars and vectors, graphical addition of vectors, triangle method of addition,
properties of vectors
Objectives
 Distinguish between a scalar and a vector
 Add and subtract vectors using the graphical method
 Multiply and divide vectors by scalars
Scalars and Vectors
 A scalar is a physical quantity that has ______________________ but no
_______________________
o Examples: __________________, volume, the number of pages in your textbook
 A vector is a physical quantity that has ____________ magnitude and direction
o Examples: ________________________, velocity, acceleration
Graphical Addition of Vectors
 A ________________________ vector represents the sum of two of more vectors.
 Vectors can be added ___________________________
 A student walks from his house to his friend’s house (a), then from his friend’s house to the
school (b). The student’s resultant displacement (c) can be found by using a _______________
and a ____________________________.
Triangle Method of Addition
 Vectors can be moved _________________________ to themselves in a diagram.
 Thus, you can draw one vector with its ______________ starting at the _____________ of the
other as long as the size and direction of each vector do not change.
 The resultant vector can then be drawn from the _____________ of the first vector to the
___________ of the last vector.
Section 2 Vector Operations
Preview
 Objectives, coordinate systems in two dimensions, determining resultant magnitude and
direction, ample problem, resolving vectors into components, adding vectors that are not
perpendicular.
Objectives
 Identify appropriate coordinate systems for solving problems with vectors
 Apply the Pythagorean Theorem and tangent functions to calculate the magnitude and direction
of a resultant vector.
 Resolve vectors into components using sine and cosine functions
 Add vectors that are not perpendicular.
Coordinate Systems in Two Dimensions
 One method for diagramming the motion of an object employs vectors and the use of the ____and _____- axes
 Axes are often designated using ________________ directions
 In the figure shown here, the ___________________ y-axis points north and the positive x-axis
points _____________________
Determining Resultant Magnitude and Direction
 In section 1, the magnitude and direction of a resultant were found _______________________
 With this approach, the accuracy of the answer depends on how carefully the diagram is
__________________ and _________________________
 A simpler method uses the ____________________________ ______________________ and
the tangent function
 The Pythagorean Theorem
o Use the Pythagorean theorem to find the _________________________ of the resultant
vector
o The Pythagorean theorem states that for any _________________ triangle, the square
of the hypotenuse – the side opposite the right angle – equals the sum of the squares of
the other two sides, or legs.

The Tangent Function
o Use the tangent function to find the _______________________ of the resultant vector
o For any right triangle, the tangent of the angle is defined as the ratio of the
_________________ and ____________________ legs with respect to ta specified acute
angle of a right triangle.
Sample Problem
 An archaeologist climbs the Great Pyramid in Giza, Egypt. The pyramid’s height is 136 m and its
width is 2.30 x 102 m. What is the magnitude and the direction of the displacement of the
archaeologist after she has climbed from the bottom of the pyramid to the top?
1. Define:
 Given:

Unknown:

Diagram:
2. Plan:


Choose an equation or situation: the Pythagorean Theorem can be used to find
the magnitude of the archaeologist’s displacement. The direction of the
displacement can be found by using the inverse tangent function.
Rearrange the equations to isolate the unknowns:
3. Calculate:
4. Evaluate: Because d is the hypotenuse, the archaeologist’s displacement should be less
than the sum of the height and half of the width. The angle is expected to be more than
45° because the height is greater than half the width.
Example Problems
1. A truck driver is attempting to deliver some furniture. First, he travels 8 km east, and then he
turns around and travels 3 km west. Finally, he turns again and travels 12 km east to his
destination.
a. What distance has the driver traveled?
b. What is the driver’s total displacement?
2. While following the directions on a treasure map, a pirate walks 45.0 m north and then turns
and walks 7.5 km east. What single straight-line displacement could the pirate have taken to
reach the treasure?
3. Emily passes a soccer ball 6.0 m directly across the field to Kara. Kara then kicks the ball 14.5 m
directly down the field to Luisa. What is the total displacement of the ball as it travels between
Emily and Luisa?
4. A hummingbird, 3.4 m above the ground, flies 1.2 m along a straight path. Upon spotting a
flower below, the hummingbird drops directly downward 1.4 m to hover in front of the flower.
What is the hummingbird’s total displacement?
Resolving Vectors into Components
 You can often describe an object’s motion more conveniently by breaking a single vector into
two _______________________________, or resolving the vector
 The ____________________________ of a vector are the projections of the vector along the
axes of a coordinate system.
 Resolving a vector allows you to analyze the motion in each ___________________________.
 Consider an airplane flying at 95 km/h
1. The hypotenuse (_____________) is the resultant vector that describes the airplane’s
total velocity
2. The _____________________ leg represents the x component (______), which
describes the airplane’s horizontal speed.
3. The opposite leg represents the y component (______), which describes the airplane’s
______________________ speed.


The ____________ and _______________________ functions can be used to find the
components of a vector.
The sine and cosine functions are defined in terms of the lengths of the sides of
________________ triangles.
Example Problems:
1. How fast must a truck travel to stay beneath an airplane that is moving 105 km/h at an angle of
25° to the ground?
2. What is the magnitude of the vertical component of the velocity of the plane in item 1?
3. A truck drives up a hill with a 15° incline. If the truck has a constant speed of 22 m/s, what are
the horizontal and vertical components of the truck’s velocity?
4. What are the horizontal and vertical components of a cat’s displacement when the cat has
climbed 5 m directly up a tree?
Adding Vectors that are not Perpendicular
 Suppose that a plane travels first 5 km at an angle of 35°, then climbs at 10° for 22 m, as shown
below (you must draw). How can you find the total displacement?
 Because the original displacement vectors do not form a ________________ triangle, you can
not directly apply the ____________________ function or the Pythagorean Theorem.



You can find the magnitude and the direction of the resultant by ____________________ each
of the plane’s displacement vectors into its x and y components.
Then the components along each axis can be __________________________ together.
As shown in the figure, these sums will be the two _________________________ components
of the resultant, d. the resultant’s magnitude can then be found by using the Pythagorean
Theorem, and its direction can be found by using the ____________________ tangent function.
Sample Problem:
 Adding vectors algebraically: a hiker walks 27.0 km from her base camp at 35° south of east. The
next day, she walks 41.0 km in a direction of 65° north of east and discovers a forest ranger’s
tower. Find the magnitude and direction of her resultant displacement.
1. Select the coordinate system. Then sketch and label each vector
 Given:


Tip: Ɵ1 is negative, because clockwise movement from the positive x-axis is
negative by convention
Unknown:
2. Find the x and y components of all vectors: Make a separate sketch of the displacements
for each day. Use the cosine and sine functions to find the components
3. Find the x and y components of the total displacement
4. Use the Pythagorean Theorem to find the magnitude of the resultant vector.
5. Use a suitable trigonometric function to find the angle.
Example Problems:
1. A football player runs directly down the field for 35 m before turning to the right at an angle of
25° from his original direction and running and additional 15 m before getting tackled. What is
the magnitude and direction of the runner’s total displacement?
2. A plane travels 2.5 km at an angle of 35° to the ground and then changes direction and travels
5.2 km at an angle of 22° to the ground. What is the magnitude and direction of the runner’s of
the plane’s total displacement?
3. During a rodeo, a clown runs 8.0 m north, turns 55° north of east, and runs 3.5 m. Then, after
waiting for the bull to come near, the clown turns due east and runs 5.0 m to exit the arena.
What is the clown’s total displacement?
4. An airplane flying parallel to the ground undergoes two consecutive displacements. The first is
75 km 30.0° west of north, and the second is 155 km 60.0° east of north. What is the total
displacement of the airplane?