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.

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 Acceleration

Chapter 4. Reading Quizzes

A ball is thrown upward at a 45° angle A ball is thrown upward at a 45° angle. In the absence of air resistance, the ball follows a tangential curve. sine curve. parabolic curve. linear curve. Answer: C

A ball is thrown upward at a 45° angle A ball is thrown upward at a 45° angle. In the absence of air resistance, the ball follows a tangential curve. sine curve. parabolic curve. linear curve.

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, the bullet passes above the coconut. the bullet hits the coconut. the bullet passes beneath the coconut. This wasn’t discussed in Chapter 4. Answer: B

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, the bullet passes above the coconut. the bullet hits the coconut. the bullet passes beneath the coconut. This wasn’t discussed in Chapter 4.

The quantity with the symbol ω is called the circular weight. the circular velocity. the angular velocity. the centripetal acceleration. Answer: C

The quantity with the symbol ω is called the circular weight. the circular velocity. the angular velocity. the centripetal acceleration.

For uniform circular motion, the acceleration points toward the center of the circle. points away from the circle. is tangent to the circle. is zero. Answer: A

For uniform circular motion, the acceleration points toward the center of the circle. points away from the circle. is tangent to the circle. is zero.

Chapter 4. Basic Content and Examples

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. 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.

Tactics: Finding the acceleration vector

Tactics: Finding the acceleration vector

EXAMPLE 4.1 Through the valley

EXAMPLE 4.1 Through the valley

EXAMPLE 4.1 Through the valley

EXAMPLE 4.1 Through the valley

Projectile Motion

Projectile Motion

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

EXAMPLE 4.4 Don’t try this at home! QUESTION:

EXAMPLE 4.4 Don’t try this at home!

EXAMPLE 4.4 Don’t try this at home!

EXAMPLE 4.4 Don’t try this at home!

EXAMPLE 4.4 Don’t try this at home!

EXAMPLE 4.4 Don’t try this at home!

EXAMPLE 4.4 Don’t try this at home!

Problem-Solving Strategy: Projectile Motion Problems

Problem-Solving Strategy: Projectile Motion Problems

Problem-Solving Strategy: Projectile Motion Problems

Problem-Solving Strategy: Projectile Motion Problems

Relative Motion

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,

EXAMPLE 4.8 A speeding bullet QUESTION:

EXAMPLE 4.8 A speeding bullet

EXAMPLE 4.8 A speeding bullet

Uniform Circular Motion

Uniform Circular Motion

Uniform Circular Motion

EXAMPLE 4.13 At the roulette wheel QUESTIONS:

EXAMPLE 4.13 At the roulette wheel

EXAMPLE 4.13 At the roulette wheel

EXAMPLE 4.13 At the roulette wheel

EXAMPLE 4.13 At the roulette wheel

Centripetal Acceleration

EXAMPLE 4.14 The acceleration of a Ferris wheel QUESTION:

EXAMPLE 4.14 The acceleration of a Ferris wheel

EXAMPLE 4.14 The acceleration of a Ferris wheel

EXAMPLE 4.14 The acceleration of a Ferris wheel

Chapter 4. Summary Slides

General Principles

General Principles

Important Concepts

Important Concepts

Applications

Applications

Applications

Applications

Chapter 4. Clicker Questions

This acceleration will cause the particle to slow down and curve downward. slow down and curve upward. speed up and curve downward. speed up and curve upward. move to the right and down. Answer: A

This acceleration will cause the particle to slow down and curve downward. slow down and curve upward. speed up and curve downward. speed up and curve upward. move to the right and down. STT6.1

During which time interval is the particle described by these position graphs at rest? Answer: C

During which time interval is the particle described by these position graphs at rest? STT33.1

A 50 g ball rolls off a table and lands 2 m from the base of the table 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 less than 2 m from the base. 2 m from the base. greater than 2 m from the base. Answer: B

A 50 g ball rolls off a table and lands 2 m from the base of the table 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 less than 2 m from the base. 2 m from the base. greater than 2 m from the base. STT6.3

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 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. Answer: C

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 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. STT6.4

A particle moves cw around a circle at constant speed for 2. 0 s 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? Answer: B

A particle moves cw around a circle at constant speed for 2. 0 s 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? STT7.1

Rank in order, from largest to smallest, the centripetal accelerations (ar)a to (ar)e of particles a to e. (ar)b > (ar)e > (ar)a > (ar)d > (ar)c (ar)b > (ar)e > (ar)a = (ar)c > (ar)d (ar)b = (ar)e > (ar)a = (ar)c > (ar)d (ar)b > (ar)a = (ar)c = (ar)e > (ar)d (ar)b > (ar)a = (ar)a > (ar)e > (ar)d Answer: B

Rank in order, from largest to smallest, the centripetal accelerations (ar)a to (ar)e of particles a to e. (ar)b > (ar)e > (ar)a > (ar)d > (ar)c (ar)b > (ar)e > (ar)a = (ar)c > (ar)d (ar)b = (ar)e > (ar)a = (ar)c > (ar)d (ar)b > (ar)a = (ar)c = (ar)e > (ar)d (ar)b > (ar)a = (ar)a > (ar)e > (ar)d STT7.2

The fan blade is slowing down. What are the signs of ω and α?  is positive and  is positive. B.  is negative and  is positive. C.  is positive and  is negative. D.  is negative and  is negative. Answer: B

The fan blade is slowing down. What are the signs of ω and α?   is positive and  is positive. B.  is negative and  is positive. C.  is positive and  is negative. D.  is negative and  is negative. STT13.1