PHYS 1441 – Section 002
Lecture #6
Monday, Feb. 4, 2013
Dr. Jaehoon Yu
•
•
•
•
•
Properties and operations of vectors
Components of the 2D Vector
Understanding the 2 Dimensional Motion
2D Kinematic Equations of Motion
Projectile Motion
• 1st term exam
Announcements
– In class, Wednesday, Feb. 13
– Coverage: CH1.1 – what we finish Monday, Feb. 11, plus Appendix
A1 – A8
– Mixture of free response problems and multiple choice problems
– Please do not miss the exam! You will get an F if you miss any
exams!
• Quiz #2
– Early in class this Wednesday, Feb. 6
– Covers CH1.6 – what we finish today (CH3.4?)
• Homework
– I see that Quest now is accepting payments through Feb.25.
Please take an action quickly so that your homework is
Monday,undisrupted..
Feb. 4, 2013
PHYS 1441-002, Spring 2013
Dr. Jaehoon Yu
2
Vector and Scalar
Vector quantities have both magnitudes (sizes)
and directions Velocity, acceleration, force, momentum, etc
Normally denoted in BOLD letters, F, or a letter with arrow on top
Their sizes or magnitudes are denoted with normal letters, F, or
absolute values:
Scalar quantities have magnitudes only
Can be completely specified with a value
and its unit Normally denoted in normal letters, E
Speed, energy,
heat, mass,
time, etc
Both have units!!!
Monday, Feb. 4, 2013
PHYS 1441-002, Spring 2013
Dr. Jaehoon Yu
3
Properties of Vectors
• Two vectors are the same if their sizes and the directions
are the same, no matter where they are on a coordinate
system!!  You can move them around as you wish as
long as their directions and sizes are kept the same.
Which ones are the
same vectors?
y
D
A=B=E=D
F
A
Why aren’t the others?
B
x
E
Monday, Feb. 4, 2013
C
PHYS 1441-002, Spring 2013
Dr. Jaehoon Yu
C: The same magnitude
but opposite direction:
C=-A:A negative vector
F: The same direction
but different magnitude
4
•
Vector Operations
Addition:
– Triangular Method: One can add vectors by connecting the head of one vector to
the tail of the other (head-to-tail)
– Parallelogram method: Connect the tails of the two vectors and extend
– Addition is commutative: Changing order of operation does not affect the results
A+B=B+A, A+B+C+D+E=E+C+A+B+D
A+B
B
A
•
A
=
B
A+B
OR
A+B
B
A
Subtraction:
– The same as adding a negative vector:A - B = A + (-B)
A
A-B
•
-B
Since subtraction is the equivalent to adding a
negative vector, subtraction is also commutative!!!
Multiplication by a scalar is
increasing the magnitude A, B=2A
Monday, Feb. 4, 2013
A
PHYS 1441-002, Spring 2013
Dr. Jaehoon Yu
B=2A
5
Example for Vector Addition
A car travels 20.0km due north followed by 35.0km in a direction 60.0o West
of North. Find the magnitude and direction of resultant displacement.
Bsin60o N
r
2
2
= A2 + B2 ( cos2 q + sin 2 q ) + 2AB cosq
B 60o Bcos60o

( A + Bcosq ) + ( Bsinq )
r=
= A2 + B2 + 2ABcosq
=
20
A
( 20.0 )2 + ( 35.0 )2 + 2 ´ 20.0 ´ 35.0 cos60
= 2325 = 48.2(km)
E
q = tan
B sin 60
-1
A + B cos 60
Do this using
35.0 sin 60
= tan
components!!
20.0 + 35.0 cos 60
30.3
= tan -1
= 38.9 to W wrt N
37.5
-1
Monday, Feb. 4, 2013
PHYS 1441-002, Spring 2013
Dr. Jaehoon Yu
6
Components and Unit Vectors
Coordinate systems are useful in expressing vectors in their components
y
Ay
(-,+)
(+,+)
A
(Ax,Ay)
Monday, Feb. 4, 2013
(+,-)
}
Ay 

(-,-)
Ax 
Ax
x
PHYS 1441-002, Spring 2013
Dr. Jaehoon Yu
A  Ax  Ay
2
2
Components
} Magnitude
7
Unit Vectors
• Unit vectors are the ones that tells us the
directions of the components
– Very powerful and makes vector notation and operations
much easier!
• Dimensionless
• Magnitudes these vectors are exactly 1
• Unit vectors are usually expressed in i, j, k or
So a vector A can be
expressed as
Monday, Feb. 4, 2013
Ax  Ay
PHYS 1441-002, Spring 2013
Dr. Jaehoon Yu
8
Examples of Vector Operations
Find the resultant vector which is the sum of A=(2.0i+2.0j) and B =(2.0i-4.0j)
( 4.0) + ( -2.0)
2
2
= 16 + 4.0 = 20 = 4.5(m)
q=
tan
-1
Cy
Cx
=
Find the resultant displacement of three consecutive displacements:
d1=(15i+30j +12k)cm, d2=(23i+14j -5.0k)cm, and d3=(-13i+15j)cm
Magnitude
Monday, Feb. 4, 2013
( 25) + (59) + (7.0)
2
2
PHYS 1441-002, Spring 2013
Dr. Jaehoon Yu
2
= 65(cm)
9
2D Displacement
Displacement:
Monday, Feb. 4, 2013
PHYS 1441-002, Spring 2013
Dr. Jaehoon Yu
10
2D Average Velocity
Average velocity is the
displacement divided by
the elapsed time.
t - to
Monday, Feb. 4, 2013
=
PHYS 1441-002, Spring 2013
Dr. Jaehoon Yu
11
2D Instantaneous Velocity
The instantaneous velocity indicates how fast the car
moves and the direction of motion at each instant of time.
Monday, Feb. 4, 2013
PHYS 1441-002, Spring 2013
Dr. Jaehoon Yu
12
2D Average Acceleration
t - to
Monday, Feb. 4, 2013
PHYS 1441-002, Spring 2013
Dr. Jaehoon Yu
=
13
Displacement, Velocity, and Acceleration in 2-dim
• Displacement:
• Average Velocity:
How is each of
these quantities
defined in 1-D?
• Instantaneous
Velocity:
• Average
Acceleration
• Instantaneous
Acceleration:
Monday, Feb. 4, 2013
PHYS 1441-002, Spring 2013
Dr. Jaehoon Yu
14
Kinematic Quantities in 1D and 2D
Quantities
1 Dimension
Displacement
Dx = x f - xi
Average Velocity
Inst. Velocity
Average Acc.
Inst. Acc.
Monday, Feb. 4,What
2013
2 Dimension
Dx x f - xi
vx =
=
Dt
t f - ti
Dx
vx º lim
Dt ®0 Dt
Dvx vxf - vxi
ax º
=
Dt
t f - ti
ax º lim
Dt®0
Dvx
Dt
PHYS between
1441-002, Spring
is the difference
1D2013
and 2D quantities?
Dr. Jaehoon Yu
15
A Motion in 2 Dimension
This is a motion that could be viewed as two motions
combined into one. (superposition…)
Monday, Feb. 4, 2013
PHYS 1441-002, Spring 2013
Dr. Jaehoon Yu
16
Motion in horizontal direction (x)
(v
)
vx = vxo + axt
x=
v = v + 2ax x
x = vxot + ax t
2
x
2
xo
Monday, Feb. 4, 2013
PHYS 1441-002, Spring 2013
Dr. Jaehoon Yu
1
2
+
v
t
xo
x
1
2
2
17
Motion in vertical direction (y)
v y = v yo + a y t
y=
1
2
(v
)
+
v
t
yo
y
v = v + 2a y y
2
y
2
yo
y = v yot + a y t
1
2
Monday, Feb. 4, 2013
PHYS 1441-002, Spring 2013
Dr. Jaehoon Yu
2
18
A Motion in 2 Dimension
Imagine you add the two 1 dimensional motions on the left.
It would make up a one 2 dimensional motion on the right.
Monday, Feb. 4, 2013
PHYS 1441-002, Spring 2013
Dr. Jaehoon Yu
19
Kinematic Equations in 2-Dim
x-component
y-component
vx = vxo + axt
v y = v yo + a y t
x=
1
2
(v
)
+
v
t
xo
x
2
y
2
xo
Dx = vxot + ax t
1
2
Monday, Feb. 4, 2013
(v
)
+
v
t
yo
y
v = v + 2a y y
v = v + 2ax x
2
x
y=
1
2
2
2
yo
Dy = v yot + ayt
PHYS 1441-002, Spring 2013
Dr. Jaehoon Yu
1
2
20
2
Ex. A Moving Spacecraft
In the x direction, the spacecraft in zero-gravity zone has an initial velocity
component of +22 m/s and an acceleration of +24 m/s2. In the y direction, the
analogous quantities are +14 m/s and an acceleration of +12 m/s2. Find (a) x
and vx, (b) y and vy, and (c) the final velocity of the spacecraft at time 7.0 s.
Monday, Feb. 4, 2013
PHYS 1441-002, Spring 2013
Dr. Jaehoon Yu
21
How do we solve this problem?
1. Visualize the problem  Draw a picture!
2. Decide which directions are to be called positive (+) and
negative (-).
3. Write down the values that are given for any of the five
kinematic variables associated with each direction.
4. Verify that the information contains values for at least
three of the kinematic variables. Do this for x and y
separately. Select the appropriate equation.
5. When the motion is divided into segments, remember
that the final velocity of one segment is the initial velocity
for the next.
6. Keep in mind that there may be two possible answers to
a kinematics problem.
Monday, Feb. 4, 2013
PHYS 1441-002, Spring 2013
Dr. Jaehoon Yu
22
Ex. continued
In the x direction, the spacecraft in a zero gravity zone has an initial velocity
component of +22 m/s and an acceleration of +24 m/s2. In the y direction, the
analogous quantities are +14 m/s and an acceleration of +12 m/s2. Find (a) x
and vx, (b) y and vy, and (c) the final velocity of the spacecraft at time 7.0 s.
x
ax
vx
vox
t
?
+24.0 m/s2
?
+22.0 m/s
7.0 s
y
ay
vy
voy
t
?
+12.0 m/s2
?
+14.0 m/s
7.0 s
Monday, Feb. 4, 2013
PHYS 1441-002, Spring 2013
Dr. Jaehoon Yu
23
First, the motion in x-direciton…
x
ax
vx
vox
t
?
+24.0 m/s2
?
+22 m/s
7.0 s
Dx = voxt + axt
(
1
2
)(
2
) ( 24m s )(7.0 s)
= 22m s 7.0 s +
2
1
2
vx = vox + axt
(
) (
= 22m s + 24m s
Monday, Feb. 4, 2013
2
2
= +740 m
)(7.0 s) = +190m s
PHYS 1441-002, Spring 2013
Dr. Jaehoon Yu
24
Now, the motion in y-direction…
y
ay
vy
voy
t
?
+12.0 m/s2
?
+14 m/s
7.0 s
Dy = voy t + a y t
(
)(
1
2
2
) (
= 14m s 7.0 s + 12m s
1
2
v y  voy + a y t
(
) (
= 14m s + 12m s
Monday, Feb. 4, 2013
2
2
)(7.0 s)
2
= +390 m
)(7.0 s) = +98m s
PHYS 1441-002, Spring 2013
Dr. Jaehoon Yu
25
The final velocity…
v
v y = 98m s
q
vx = 190 m s
(
) + ( 98m s)
(98 190) = 27
v  190 m s
q = tan
-1
2
= 210 m s
A vector can be fully described when
the magnitude and the direction are
given. Any other way to describe it?
Yes, you are right! Using
components and unit vectors!!
Monday, Feb. 4, 2013
2
m s
PHYS 1441-002, Spring 2013
Dr. Jaehoon Yu
26
If we visualize the motion…
Monday, Feb. 4, 2013
PHYS 1441-002, Spring 2013
Dr. Jaehoon Yu
27