vectors 2d & 3d motion lecturer: professor stephen t. thornt on

Vectors2D & 3D Motion

Lecturer: Professor Stephen T. Thornton

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

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.

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.


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?


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.


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


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.


v

tA

v

tB

v

tCv

tD


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.


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

2 2 1

cos sin

tan

x y

yx y

x

A A A A

AA A A

A

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

Answer: C

y

C) Between 1800 and 2700

x

A

The Sum of Two Vectors

We can add vectors.

C = A + B

A+B = C = B+A??

Can we addvectors either way? (commutative)

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

We can add negative vectors.

HeadTail

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

Multiplying a Vector by a Scalar

We can multiply a vector by a scalar.

Vector Component Use

Unit vectors make vector addition and subtraction reasonably easy.

ˆ ˆ ˆ Â 3i + 4j B 2i -2j

ˆ ˆ ˆ ˆ ˆ Â B (3i+2i) (4j 2j) = 5i 2 j

ˆ ˆ ˆ ˆ ˆ Â - B (3i+4j) (2i 2j) i 6 j

ˆ ˆ ˆ ˆ ˆ Â - 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



Vector Addition & Subtraction. For the vectors given in the figure, determine . A B C

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


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).


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


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.


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°


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°.


vectors 2d & 3d motion lecturer: professor stephen t. thornt on

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