This chapter discusses motion in two dimensions, including vectors, projectile motion, and uniform circular motion. Key concepts are distinguishing scalars from vectors, adding and subtracting vectors analytically and graphically, and describing position, velocity, and acceleration as vectors. Projectile motion equations give the horizontal and vertical motion of a projectile. Uniform circular motion describes constant speed but changing direction, with equations for centripetal acceleration. Example problems apply these concepts.

1. Essential College Physics
Chapter 3: Motion in
Two Dimensions

2. Chapter 3 Objectives
• Distinguish scalars and vectors.
• Understand vectors in component form and
as a magnitude and direction.
• Add and subtract vectors both analytically
and graphically.
• Describe position, velocity, and acceleration
vectors.

3. Chapter 3 Objectives
• Understand and analyze projectile motion.
• Understand and analyze uniform circular
motion.

4. The Paperboy Problem
• A paperboy’s route happens to also take him
to school. He leaves home and walks 8
blocks east. He then walks 4 blocks north
and, finally, walks 5 blocks west to get to
school.
• Through what distance (in blocks) did he
walk?
• What is the displacement between his home
and school?

5. The Paperboy Problem Figure

6. Trigonometry Review

7. Trigonometry Review
SOHCAHTOA (41)
opposite
sin( )
hypotenuse
 
adjacent
cos( )
hypotenuse
 
opposite
tan( )
adjacent
 

8. Example 3.1 (42)

9. Scalars vs. Vectors
Scalars
• Only magnitude
• Normal variable symbols
• Examples: time, volume,
mass
• Distance and speed
Vectors
• Magnitude and direction
• Symbolized by
superscripted arrow or
bold symbols
• Examples: force,
momentum
• Displacement and velocity

10. Unit Vectors (43)
ˆi 1,0
ˆj 0,1

11. A Position Vector
ˆ ˆv 4i 3j 

12. Displacement Vector (44)

13. Position Vector (45)

14. Vector Magnitude and Direction (45)
• Magnitude
• Direction
• Vector Components
2 2
r x y 
y1
x
tan ( )
 
x r cos( )   y r sin( )  

15. Vector Magnitude and Direction (45)

16. Graphical Interpretation of Vector Arithmetic

17. Reviewing New Concepts (47)
• To add vectors, add the components
• To subtract vectors, subtract the components
• Graphically form the vector sum by
placing the tail of at the head of . The sum
is the vector drawn from the tail of to the
head of .
• Graphically subtract vectors by adding the
opposite.
• To multiply a vector by a scalar, multiply each
component of the vector by that scalar.
1 2r r
2r 1r
1r
2r

18. Vector Definition of
Average Velocity (47)
r
v
t




19. Velocity Magnitude and Direction (48)
• Magnitude
• Direction
• Velocity Components
2 2
x yv v v 
y
x
v1
vtan ( )
 
xv v cos( )   yv v sin( )  

20. Problems
• Example 3.3 (49)
• Conceptual Example 3.4 (50)

21. Vector Definition of
Average Acceleration
v
a
t




22. Projectile Motion
• Independence thought experiment
• Independence demonstration

23. Projectile Motion
Monkey and Hunter Video
http://www.youtube.com/watch?v=MkKEwrshYWs&feature=related

24. Projectile Motion Equations (52)
Horizontal
xx v t 
Vertical
21
0y 2y v t g t    
y 0yv v g t  
2 2
y 0yv v 2 g y   

25. Projectile Motion Problems
• Example 3.5 (53)
• Conceptual Example 3.6 (53)
• Example 3.7 (54)
• Example 3.9 (55)

26. Projectile Motion Lab
• The purpose is to predict the range of a
projectile
• One will be fired horizontally
• One will be fired at an angle

27. Uniform Circular Motion (56)
• Constant speed, but changing direction
• Speed equation
• Changing velocity vector means there is an
acceleration
2 R
v
T

Speed, m/s
Radius, m
Period, s

28. Centripetal Acceleration (57)

29. Centripetal Acceleration
Formulas (57)
2
r
v
a
R

2
r 2
4 R
a
T


Centripetal
acceleration, m/s2
Speed, m/s
Radius, m
Period, s

30. Example 3.10 (58)

31. Chapter 3 Problem Sets
• Honors: P. 61, Ex. 34, 46, 50, 54, 66, 70, 72,
82, 88, 94
• UConn: P. 61, Ex. 34, 46, 50, 54, 66, 70, 72, 74,
76, 82, 88, 94