This chapter discusses motion in a plane, including vector addition and subtraction using graphical and component methods. It also covers velocity, acceleration, and projectile motion, where an object experiences no acceleration horizontally but constant downward acceleration due to gravity vertically. Examples demonstrate calculating the position, velocity, and displacement of projectiles over time, as well as determining where a projectile will land.
3. MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 3
Graphical Addition and Subtraction
of Vectors
A vector is a quantity that has both a
magnitude and a direction. Position is an
example of a vector quantity.
A scalar is a quantity with no direction. The
mass of an object is an example of a scalar
quantity.
4. MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 4
Notation
Vector: F
F
or
The magnitude of a vector: .
or
or F
F
F
Scalar: m (not bold face; no arrow)
The direction of vector might be “35 south of east”;
“20 above the +x-axis”; or….
5. MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 5
To add vectors graphically they must be placed “tip to
tail”. The result (F1 + F2) points from the tail of the first
vector to the tip of the second vector.
This is sometimes called the resultant vector R
F1
F2
R
Graphical Addition of Vectors
9. MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 9
Relative Displacement Vectors
C = A + B
C- A = B
Vector Addition
Vector Subtraction
B is a relative displacement vector of point P3
relative to P2
12. MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 12
Think of vector subtraction A B as A+(B), where the
vector B has the same magnitude as B but points in the
opposite direction.
Graphical Subtraction of Vectors
Vectors may be moved any way you please (to place them
tip to tail) provided that you do not change their length nor
rotate them.
18. MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 18
x y z
A = A i + A j+ A k
ˆ ˆ ˆ
x y z
B = B i +B j + B k
ˆ ˆ ˆ
Vectors with Rectangular Unit Vectors
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Dot Product - Scalar
The dot product multiplies the portion of A that is parallel to B with B
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Dot Product - Scalar
The dot product multiplies the portion of A that is parallel to B with B
In 2 dimensions
In any number of dimensions
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Cross Product - Vector
The cross product multpilies the portion of A that is perpendicular to
B with B
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x y z
x y z
i j k
A A A
B B B
ˆ ˆ ˆ
y z z y
x z z x
x y x y
= (A B - A B )i
+ (A B - A B ) j
+ (A B - A B ) k
ˆ
ˆ
ˆ
A B = A Bsin( )
In 2 dimensions
In any number of dimensions
Cross Product - Vector
24. MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 24
y
x
ri rf
t
r
vav Points in the direction of r
r
vi
The instantaneous
velocity points
tangent to the path.
vf
A particle moves along the curved path as shown. At time t1
its position is ri and at time t2 its position is rf.
25. MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 25
t
t
r
v lim
0
velocity
ous
Instantane
The instantaneous velocity is represented by the slope
of a line tangent to the curve on the graph of an
object’s position versus time.
t
r
vav
velocity
Average
t
x
v x
,
av
:
be
would
component
-
x
The
A displacement over an interval of
time is a velocity
27. MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 27
y
x
vi
ri rf
vf
A particle moves along the curved path as shown. At time t1
its position is r0 and at time t2 its position is rf.
v
Points in the
direction of v.
t
v
aav
28. MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 28
t
v
aav
on
accelerati
Average
A nonzero acceleration changes an
object’s state of motion
Δt 0
Δv
Instantaneous acceleration = a = lim
Δt
These have interpretations similar to vav and v.
29. MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 29
Motion in a Plane with Constant
Acceleration - Projectile
What is the motion of a struck baseball? Once it leaves the
bat (if air resistance is negligible) only the force of gravity
acts on the baseball.
Acceleration due to gravity has a constant value near the
surface of the earth. We call it g = 9.8 m/s2
Only the vertical motion is affected by gravity
30. MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 30
The baseball has ax = 0 and ay = g, it moves with constant
velocity along the x-axis and with a changing velocity along the y-
axis.
Projectile Motion
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Example:
An object is projected from the origin. The initial velocity
components are vix = 7.07 m/s, and viy = 7.07 m/s.
Determine the x and y position of the object at 0.2 second
intervals for 1.4 seconds. Also plot the results.
2
f i iy y
f i ix
1
Δy = y - y = v Δt + a Δt
2
Δx = x - x = v Δt
Since the object starts from the origin, y and x
will represent the location of the object at time t.
33. MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 33
0
2
4
6
8
10
12
0 0.5 1 1.5
t (sec)
x,y
(m)
This is a plot of the x position (black points) and y position
(red points) of the object as a function of time.
Example continued:
34. MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 34
Example continued:
0
0.5
1
1.5
2
2.5
3
0 2 4 6 8 10
x (m)
y
(m)
This is a plot of the y position versus x position for the
object (its trajectory). The object’s path is a parabola.
35. MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 35
Example (text problem 3.50): An arrow is shot into the air
with = 60° and vi = 20.0 m/s.
(a) What are vx and vy of the arrow when t = 3 sec?
The components of the initial
velocity are:
m/s
3
.
17
sin
m/s
0
.
10
cos
i
iy
i
ix
v
v
v
v
At t = 3 sec:
m/s
1
.
12
m/s
0
.
10
t
g
v
t
a
v
v
v
t
a
v
v
iy
y
iy
fy
ix
x
ix
fx
x
y
60°
vi
CONSTANT
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(b) What are the x and y components of the displacement
of the arrow during the 3.0 sec interval?
y
x
r
2
x f i ix x ix
2 2
y f i iy y iy
1
Δr = Δx = x - x = v Δt + a Δt = v Δt +0 = 30.0 m
2
1 1
Δr = Δy = y - y = v Δt + a Δt = v Δt - gΔt = 7.80 m
2 2
Example continued:
37. MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 37
Example: How far does the arrow in the previous example
land from where it is released?
The arrow lands when y = 0. 0
2
1 2
t
g
t
v
y iy
Solving for t: sec
53
.
3
2
g
v
t
iy
The distance traveled is: ix
Δx = v Δt = 35.3 m
iy
1
Δy = (v - gΔt) t = 0
2
39. MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 39
Projectiles Examples
• Problem solving strategy
• Symmetry of the motion
• Dropped from a plane
• The home run