Chapter 4. Kinematics in Two Dimensions
A car turning a corner, a
basketball sailing toward the
hoop, a planet orbiting the
sun, and the diver in the
photograph are examples of
two-dimensional motion or,
equivalently, motion in a
plane.
Chapter Goal: To learn to
solve problems about motion
in a plane.
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Chapter 4. Kinematics in Two Dimensions
Topics:
• Acceleration
• Kinematics in Two Dimensions
• Projectile Motion
• Relative Motion
• Uniform Circular Motion
• Velocity and Acceleration in Uniform
Circular Motion
• Nonuniform Circular Motion and Angular
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Acceleration
Chapter 4. Reading Quizzes
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
A ball is thrown upward at a 45° angle. In
the absence of air resistance, the ball
follows a
A. tangential curve.
B. sine curve.
C. parabolic curve.
D. linear curve.
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
A ball is thrown upward at a 45° angle. In
the absence of air resistance, the ball
follows a
A. tangential curve.
B. sine curve.
C. parabolic curve.
D. linear curve.
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
A hunter points his rifle directly at a
coconut that he wishes to shoot off a tree.
It so happens that the coconut falls from
the tree at the exact instant the hunter
pulls the trigger. Consequently,
A. the bullet passes above the coconut.
B. the bullet hits the coconut.
C. the bullet passes beneath the coconut.
D. This wasn’t discussed in Chapter 4.
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
A hunter points his rifle directly at a
coconut that he wishes to shoot off a tree.
It so happens that the coconut falls from
the tree at the exact instant the hunter
pulls the trigger. Consequently,
A. the bullet passes above the coconut.
B. the bullet hits the coconut.
C. the bullet passes beneath the coconut.
D. This wasn’t discussed in Chapter 4.
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
The quantity with the symbol ω is called
A. the circular weight.
B. the circular velocity.
C. the angular velocity.
D. the centripetal acceleration.
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
The quantity with the symbol ω is called
A. the circular weight.
B. the circular velocity.
C. the angular velocity.
D. the centripetal acceleration.
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
For uniform circular motion, the
acceleration
A. points toward the center of the circle.
B. points away from the circle.
C. is tangent to the circle.
D. is zero.
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
For uniform circular motion, the
acceleration
A. points toward the center of the circle.
B. points away from the circle.
C. is tangent to the circle.
D. is zero.
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Chapter 4. Basic Content and Examples
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Acceleration
The average acceleration of a moving object is defined as
the vector
As an object moves, its velocity vector can change in two
possible ways.
1.The magnitude of the velocity can change, indicating a
change in speed, or
2. The direction of the velocity can change, indicating that
the object has changed direction.
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Tactics: Finding the acceleration vector
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Tactics: Finding the acceleration vector
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
EXAMPLE 4.1 Through the valley
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
EXAMPLE 4.1 Through the valley
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
EXAMPLE 4.1 Through the valley
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
EXAMPLE 4.1 Through the valley
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Projectile Motion
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Projectile Motion
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Projectile Motion
Projectile motion is made up of two independent motions:
uniform motion at constant velocity in the horizontal
direction and free-fall motion in the vertical direction. The
kinematic equations that describe these two motions are
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
EXAMPLE 4.4 Don’t try this at home!
QUESTION:
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
EXAMPLE 4.4 Don’t try this at home!
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
EXAMPLE 4.4 Don’t try this at home!
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
EXAMPLE 4.4 Don’t try this at home!
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
EXAMPLE 4.4 Don’t try this at home!
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
EXAMPLE 4.4 Don’t try this at home!
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
EXAMPLE 4.4 Don’t try this at home!
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Problem-Solving Strategy: Projectile Motion
Problems
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Problem-Solving Strategy: Projectile Motion
Problems
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Problem-Solving Strategy: Projectile Motion
Problems
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Problem-Solving Strategy: Projectile Motion
Problems
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Relative Motion
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Relative Motion
If we know an object’s velocity measured in one reference
frame, S, we can transform it into the velocity that
would be measured by an experimenter in a different
reference frame, S´, using the Galilean transformation of
velocity.
Or, in terms of components,
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
EXAMPLE 4.8 A speeding bullet
QUESTION:
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
EXAMPLE 4.8 A speeding bullet
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
EXAMPLE 4.8 A speeding bullet
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Uniform Circular Motion
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Uniform Circular Motion
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Uniform Circular Motion
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
EXAMPLE 4.13 At the roulette wheel
QUESTIONS:
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
EXAMPLE 4.13 At the roulette wheel
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
EXAMPLE 4.13 At the roulette wheel
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
EXAMPLE 4.13 At the roulette wheel
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
EXAMPLE 4.13 At the roulette wheel
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Centripetal Acceleration
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
EXAMPLE 4.14 The acceleration of a Ferris
wheel
QUESTION:
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
EXAMPLE 4.14 The acceleration of a Ferris
wheel
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
EXAMPLE 4.14 The acceleration of a Ferris
wheel
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
EXAMPLE 4.14 The acceleration of a Ferris
wheel
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Chapter 4. Summary Slides
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
General Principles
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
General Principles
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Important Concepts
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Important Concepts
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Applications
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Applications
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Applications
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Applications
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Chapter 4. Clicker Questions
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
This acceleration will cause the particle to
A. slow down and curve downward.
B. slow down and curve upward.
C. speed up and curve downward.
D. speed up and curve upward.
E. move to the right and down.
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
This acceleration will cause the particle to
A. slow down and curve downward.
B. slow down and curve upward.
C. speed up and curve downward.
D. speed up and curve upward.
E. move to the right and down.
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
During which time interval is the particle
described by these position graphs at
rest?
A. 0–1 s
B. 1–2 s
C. 2–3 s
D. 3–4 s
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
During which time interval is the particle
described by these position graphs at
rest?
A. 0–1 s
B. 1–2 s
C. 2–3 s
D. 3–4 s
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
A 50 g ball rolls off a table and lands 2 m
from the base of the table. A 100 g ball
rolls off the same table with the same
speed. It lands at a distance
A. less than 2 m from the base.
B. 2 m from the base.
C. greater than 2 m from the base.
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
A 50 g ball rolls off a table and lands 2 m
from the base of the table. A 100 g ball
rolls off the same table with the same
speed. It lands at a distance
A. less than 2 m from the base.
B. 2 m from the base.
C. greater than 2 m from the base.
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
A plane traveling horizontally to the right at
100 m/s flies past a helicopter that is going
straight up at 20 m/s. From the helicopter’s
perspective, the plane’s direction and speed
are
A.
B.
C.
D.
E.
right and up, more than 100 m/s.
right and up, less than 100 m/s.
right and down, more than 100 m/s.
right and down, less than 100 m/s.
right and down, 100 m/s.
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
A plane traveling horizontally to the right at
100 m/s flies past a helicopter that is going
straight up at 20 m/s. From the helicopter’s
perspective, the plane’s direction and speed
are
A.
B.
C.
D.
E.
right and up, more than 100 m/s.
right and up, less than 100 m/s.
right and down, more than 100 m/s.
right and down, less than 100 m/s.
right and down, 100 m/s.
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
A particle moves cw around a circle at
constant speed for 2.0 s. It then reverses
direction and moves ccw at half the
original speed until it has traveled
through the same angle. Which is the
particle’s angle-versus-time graph?
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
A particle moves cw around a circle at
constant speed for 2.0 s. It then reverses
direction and moves ccw at half the
original speed until it has traveled
through the same angle. Which is the
particle’s angle-versus-time graph?
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Rank in order, from largest to smallest,
the centripetal accelerations (ar)a to (ar)e of
particles a to e.
A. (ar)b > (ar)e > (ar)a > (ar)d > (ar)c
B. (ar)b > (ar)e > (ar)a = (ar)c > (ar)d
C. (ar)b = (ar)e > (ar)a = (ar)c > (ar)d
D. (ar)b > (ar)a = (ar)c = (ar)e > (ar)d
E. (ar)b > (ar)a = (ar)a > (ar)e > (ar)d
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Rank in order, from largest to smallest,
the centripetal accelerations (ar)a to (ar)e of
particles a to e.
A. (ar)b > (ar)e > (ar)a > (ar)d > (ar)c
B. (ar)b > (ar)e > (ar)a = (ar)c > (ar)d
C. (ar)b = (ar)e > (ar)a = (ar)c > (ar)d
D. (ar)b > (ar)a = (ar)c = (ar)e > (ar)d
E. (ar)b > (ar)a = (ar)a > (ar)e > (ar)d
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
The fan blade is slowing down. What are
the signs of ω and α?
Α. ω is positive and α is positive.
B. ω is negative and α is positive.
C. ω is positive and α is negative.
D. ω is negative and α is negative.
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
The fan blade is slowing down. What are
the signs of ω and α?
Α. ω is positive and α is positive.
B. ω is negative and α is positive.
C. ω is positive and α is negative.
D. ω is negative and α is negative.
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.