Vectors
2D & 3D Motion
Lecturer:
Professor Stephen T. Thornton
Reading QuizI
Given that A + B = C,
and that lAl + lBl =
lCl , how are vectors
A and B oriented with
respect to each
other?
A) they are perpendicular to each other
B) they are parallel and in the same direction
C) they are parallel but in the opposite
direction
D) they are at 45° to each other
E) they can be at any angle to each other
Reading Quiz
Given that A + B = C,
and that lAl + lBl =
lCl , how are vectors
A and B oriented with
respect to each
other?
A) they are perpendicular to each other
B) they are parallel and in the same direction
C) they are parallel but in the opposite
direction
D) they are at 45° to each other
E) they can be at any angle to each other
The only time vector magnitudes will simply add together is when the
direction does not have to be taken into account (i.e., the direction is
the same for both vectors). In that case, there is no angle between
them to worry about, so vectors A and B must be pointing in the
same direction.
Last Time
Problem solving
Motion with constant acceleration
Free fall
Today
More conceptual quizzes
Vectors
Two and three dimensional motion
How to Succeed in Physics
1) Don’t get behind. Don’t wait until the last
evening to start homework!
2) When you can’t work problems, it usually
means you don’t understand the concepts or
have difficulty with the mathematics.
3) Work lots of problems that have answers at
the back of the book.
4) Try the questions at the end of the chapter.
5) Don’t get behind!
Falling Water Balloons. Roger sees
water balloons fall past his window. He
notices that each balloon strikes the
sidewalk 0.83 s after passing his window.
Roger’s room is on the third floor, 15 m
above the sidewalk.
(a) How fast are the balloons traveling
when they pass Roger’s window?
(b) Assuming the balloons are being
released from rest, from what floor are
they being released? Each floor of the
dorm is 5.0 m high.
Conceptual Quiz
A ball is thrown straight upward with
some initial speed. When it reaches
the top of its flight (at a height h), a
second ball is thrown straight
upward with the same initial speed.
Where will the balls cross paths?
A) at height h
B) above height h/2
C) at height h/2
D) below height h/2 but above 0
E) at height 0
Conceptual Quiz
A ball is thrown straight upward with
some initial speed. When it reaches
the top of its flight (at a height h), a
second ball is thrown straight
upward with the same initial speed.
Where will the balls cross paths?
A) at height h
B) above height h/2
C) at height h/2
D) below height h/2 but above 0
E) at height 0
The first ball starts at the top with no initial speed. The second ball
starts at the bottom with a large initial speed. Because the balls travel
the same time until they meet, the second ball will cover more distance
in that time, which will carry it over the halfway point before the first
ball can reach it.
Conceptual Quiz
A) it speeds up all the time
The graph of position vs.
B) it slows down all the time
time for a car is given below.
C) it moves at constant velocity
What can you say about the
D) sometimes it speeds up and
velocity of the car over time?
sometimes it slows down
E) not really sure
x
t
Conceptual Quiz
A) it speeds up all the time
The graph of position vs.
B) it slows down all the time
time for a car is given below.
C) it moves at constant velocity
What can you say about the
D) sometimes it speeds up and
velocity of the car over time?
sometimes it slows down
E) not really sure
The car slows down all the time
because the slope of the x vs. t graph is
diminishing as time goes on.
Remember that the slope of x vs. t is
the velocity! At large t, the value of the
position x does not change, indicating
that the car must be at rest.
x
t
Conceptual Quiz
A) decreases
Consider the line labeled B in
B) increases
the v vs. t plot. How does the
C) stays constant
speed change with time for
D) increases, then decreases
line B?
E) decreases, then increases
v
A
t
B
Conceptual Quiz
A) decreases
Consider the line labeled B in
B) increases
the v vs. t plot. How does the
C) stays constant
speed change with time for
D) increases, then decreases
line B?
E) decreases, then increases
v
A
t
B
In case B, the initial velocity is positive
but the magnitude of the velocity
decreases toward zero. After this, the
magnitude increases again, but
becomes negative, indicating that the
object has changed direction.
v
Conceptual Quiz
A
C
t
v
B
the floor and bounces right
back up to you. Which
represents the velocity v vs. t
graph for this motion?
t
v
t
You drop a very bouncy rubber
ball. It falls, and then it hits
v
D
t
vConceptual
A
Quiz
v
C
t
v
B
You drop a very bouncy
t
v
D
t
t
the velocity v vs. t graph for
Initially, the ball is falling down, so its
velocity must be negative (if UP is
positive). Its velocity is also
increasing in magnitude as it falls.
Once it bounces, it changes direction
and then has a positive velocity,
which is also decreasing as the ball
this motion?
moves upward.
rubber ball. It falls, and then
it hits the floor and bounces
right back up to you. Which
of the following represents
Falling Ball. A ball is dropped from
the top of a 50.0-m-high cliff. At the
same time, a carefully aimed stone is
thrown straight up from the bottom of
the cliff with a speed of 24.0 m/s. The
stone and ball collide part way up.
How far above the base of the cliff
does this happen?
Review of Vectors
A scalar is a number with units. It
can be positive, negative, or zero.
A vector has both a magnitude and
direction.
When writing a vector we will put an
arrow over it. Sometimes a vector is
in boldface.
Directions to the library
3 blocks west, 3
blocks north.
Start
Ax  A cos
Ay  A sin 
A A  A
2
x
2
y
 Ay 
  tan  
 Ax 
1
Conceptual Quiz
The components of a vector A satisfy Ax  0
and Ay  0. What is the range of possible
direction angles for A?
A

A) Between 00 and 900.
B)
C)
D)
E)
Between 900 and 1800.
Between 1800 and 2700.
Between 2700 and 3600.
Cannot be determined with information given.
Answer: C
y

x
A
C) Between 1800 and 2700
The Sum of Two Vectors
We can add
vectors.
C=A+B
A+B = C = B+A??
Can we add
(commutative)
vectors either way?
Doesn’t matter: A+B = B+A = C
Component Addition of Vectors
C A B
Vector Subtraction
We can subtract
vectors. The
minus sign just
changes the
direction.
Vector Subtraction
D=A-B
Head
Tail
We can add
negative
vectors.
Conceptual
Quiz
Which of the following are equal vectors?
A) H and -B
B) A and - G
C) F and - I
D) A and C
E) A, J and H
Answer:
C) F and - I
If we look closely, we see they have the
same magnitude and with the minus sign,
they have the same direction.
Unit Vectors
More common to use
ˆi and ˆj or just i, j or i, j
ˆ y.
ˆ All of these
than x,
mean the same thing.
They are unit vectors.
Multiplying a Vector by a Scalar
We can
multiply a
vector by a
scalar.
Vector Component Use
A  3iˆ + 4jˆ
B  2iˆ -2jˆ
ˆ ˆ  (4jˆ  2j)
ˆ = 5iˆ  2ˆj
A  B  (3i+2i)
ˆ ˆ  (2iˆ  2j)
ˆ  ˆi  6ˆj
A - B  (3i+4j)
ˆ ˆ  2(2iˆ  2j)
ˆ  ˆi  8jˆ
A - 2B  (3i+4j)
Unit vectors make vector addition and
subtraction reasonably easy.
Vector Components
Consider
ˆ ˆ  2(2iˆ  2j)
ˆ  ˆi  8jˆ
A - 2B  (3i+4j)
Draw this new vector. Find magnitude
and angle of this vector. Special Extra
Problem 3 on Vectors.
Good review of vector use:
http://www.physics.uoguelph.ca/tutorials/
vectors/vectors.html
Vector Addition & Subtraction.
For the vectors given in the
figure, determine ABC .
Conceptual Quiz
A) it doubles
If each component of a
vector is doubled, what
happens to the angle of
that vector?
B) it increases, but by less than double
C) it does not change
D) it is reduced by half
E) it decreases, but not as much as half
Conceptual Quiz
A) it doubles
If each component of a
vector is doubled, what
happens to the angle of
that vector?
B) it increases, but by less than double
C) it does not change
D) it is reduced by half
E) it decreases, but not as much as half
The magnitude of the vector clearly doubles if each of its
components is doubled. But the angle of the vector is given by tan
 = 2y/2x, which is the same as tan  = y/x (the original angle).
Conceptual Quiz
If two vectors are given
A) same magnitude, but can be in any
direction
such that A + B = 0, what B) same magnitude, but must be in the same
direction
can you say about the
magnitude and direction
of vectors A and B?
C) same magnitude, but must be in opposite
directions
D) different magnitudes, but must be in the
same direction
E) different magnitudes, but must be in
opposite directions
Conceptual Quiz
If two vectors are given
such that A + B = 0, what
can you say about the
magnitude and direction
of vectors A and B?
A) same magnitude, but can be in any
direction
B) same magnitude, but must be in the same
direction
C) same magnitude, but must be in opposite
directions
D) different magnitudes, but must be in the
same direction
E) different magnitudes, but must be in
opposite directions
The magnitudes must be the same, but one vector must be pointing in
the opposite direction of the other in order for the sum to come out to
zero. You can prove this with the tip-to-tail method.
Conceptual Quiz
A certain vector has x and y components
that are equal in magnitude. Which of the
following is a possible angle for this vector
in a standard x-y coordinate system?
A) 30°
B) 180°
C) 90°
D) 60°
E) 45°
Conceptual Quiz
A certain vector has x and y components
that are equal in magnitude. Which of the
following is a possible angle for this vector
in a standard x-y coordinate system?
A) 30°
B) 180°
C) 90°
D) 60°
E) 45°
The angle of the vector is given by tan  = y/x. Thus, tan  = 1
in this case if x and y are equal, which means that the angle
must be 45°.