Vectors 2D & 3D Motion Lecturer: Professor Stephen T. Thornt on.
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Transcript of Vectors 2D & 3D Motion Lecturer: Professor Stephen T. Thornt on.
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 QuizReading QuizII
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.
Reading QuizReading Quiz
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.
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 QuizConceptual 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 QuizConceptual Quiz
t
x
A) it speeds up all the time
B) it slows down all the time
C) it moves at constant velocity
D) sometimes it speeds up and
sometimes it slows down
E) not really sure
The graph of position vs. The graph of position vs.
time for a car is given below. time for a car is given below.
What can you say about the What can you say about the
velocity of the car over time?velocity of the car over time?
Conceptual QuizConceptual Quiz
A) it speeds up all the time
B) it slows down all the time
C) it moves at constant velocity
D) sometimes it speeds up and
sometimes it slows down
E) not really sure
t
x
The graph of position vs. The graph of position vs.
time for a car is given below. time for a car is given below.
What can you say about the What can you say about the
velocity of the car over time?velocity of the car over time?
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.
Conceptual QuizConceptual Quiz
Consider the line labeled B in Consider the line labeled B in
the the vv vs. vs. tt plot. How does the plot. How does the
speedspeed change with time for change with time for
line B?line B?
v
t
A
B
A) decreasesA) decreases
B) increasesB) increases
C) stays constantC) stays constant
D) increases, then decreasesD) increases, then decreases
E) decreases, then increasesE) decreases, then increases
Conceptual QuizConceptual Quiz
Consider the line labeled B in Consider the line labeled B in
the the vv vs. vs. tt plot. How does the plot. How does the
speed change with time for speed change with time for
line B?line B?
v
t
A
B
A) decreasesA) decreases
B) increasesB) increases
C) stays constantC) stays constant
D) increases, then decreasesD) increases, then decreases
E) decreases, then increasesE) decreases, then increases
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.
Conceptual QuizConceptual Quiz
v
tA
v
tB
v
tCv
tD
Conceptual QuizConceptual Quiz
You drop a very bouncy You drop a very bouncy
rubber ball. It falls, and then it rubber ball. It falls, and then it
hits the floor and bounces hits the floor and bounces
right back up to you. Which right back up to you. Which
represents the velocity represents the velocity v v vs.vs. t t
graph for this motion?graph for this motion?
v
tA
v
tB
v
tCv
tD
You drop a very bouncy You drop a very bouncy
rubber ball. It falls, and then rubber ball. It falls, and then
it hits the floor and bounces it hits the floor and bounces
right back up to you. Which right back up to you. Which
of the following represents of the following represents
the velocity the velocity v v vs.vs. t t graph for graph for
this motion?this motion?
Initially, the ball is falling down, so its
velocity must be negativenegative (if UP is
positive). Its velocity is also
increasingincreasing in magnitude as it falls.
Once it bounces, it changes direction
and then has a positivepositive velocity,
which is also decreasingdecreasing as the ball
moves upward.
Conceptual QuizConceptual Quiz
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.
Conceptual Quiz
A) Between 00 and 900.B) Between 900 and 1800.C) Between 1800 and 2700.D) Between 2700 and 3600.E) Cannot be determined with information given.
The components of a vector A satisfy 0 and 0. What is the range of possible
direction angles for A?
x
y
AA
A
Which of the following are equal vectors?
A) B)
C) D)
E)
H and -B
A and -G
F and - I
A and C
A, J and H
Conceptual Quiz
Answer:If we look closely, we see they have the same magnitude and with the minus sign, they have the same direction.
F and - I
C)
Unit Vectors
More common to use ˆ ˆi and j or just i, j or ,
ˆ ˆthan x, y. All of thesemean the same thing.They are unit vectors.
i j
Vector Component Use
Unit vectors make vector addition and subtraction reasonably easy.
ˆ ˆ ˆ ˆA 3i + 4j B 2i -2j
ˆ ˆ ˆ ˆ ˆ ˆA B (3i+2i) (4j 2j) = 5i 2 j
ˆ ˆ ˆ ˆ ˆ ˆA - B (3i+4j) (2i 2j) i 6 j
ˆ ˆ ˆ ˆ ˆ ˆA - 2B (3i+4j) 2(2i 2j) i 8j
Vector Components
Draw this new vector. Find magnitude and angle of this vector. Special Extra Problem 3 on Vectors.
Consider
ˆ ˆ ˆ ˆ ˆ ˆA - 2B (3i+4j) 2(2i 2j) i 8j
Good review of vector use:
http://www.physics.uoguelph.ca/tutorials/vectors/vectors.html
If each component of a
vector is doubled, what
happens to the angle of
that vector?
A) it doubles
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 QuizConceptual Quiz
If each component of a
vector is doubled, what
happens to the angle of
that vector?
A) it doubles
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 QuizConceptual 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
Conceptual QuizConceptual 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 QuizConceptual Quiz
A certain vector has A certain vector has xx and and yy components components
that are equal in magnitude. Which of the that are equal in magnitude. Which of the
following is a possible angle for this vector following is a possible angle for this vector
in a standard in a standard x-yx-y coordinate system? coordinate system?
A) 30°
B) 180°
C) 90°
D) 60°
E) 45°
Conceptual QuizConceptual Quiz
A certain vector has A certain vector has xx and and yy components components
that are equal in magnitude. Which of the that are equal in magnitude. Which of the
following is a possible angle for this vector following is a possible angle for this vector
in a standard in a standard x-yx-y coordinate system? 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°.
Conceptual QuizConceptual Quiz