Review

DisplacementAverage VelocityAverage Acceleration

Instantaneous velocity (acceleration) is the slope of the line tangent to the curve of the position (velocity) -time graph

For constant acceleration… For constant gravitational acceleration …

v v0 gt

y v0t 1

2gt 2

v 2 v02 2gy

v x

t

v v0 at

x v0t 1

2at 2

v 2 v02 2ax

a v

t

x x f x i

v v0 gt

y v0t 1

2gt 2

v 2 v02 2gy

Chapter 2

Motion in two dimensions

2.1 :An introduction to vectors

Many quantities in physics, like displacement, have a magnitude and a direction. Such quantities are called VECTORS.

Other quantities which are vectors: velocity, acceleration, force, momentum, ...Many quantities in physics, like distance, have a magnitude only. Such quantities are called SCALARS.Other quantities which are scalars: speed, temperature, mass, volume, ...

P

Q

Initial Point

Terminal Point

magnitu

de is th

e length

direction is

this angle

How can we find the magnitude if we have the initial point and the terminal point? 22 , yx

11, yx

The distance formula

How can we find the direction? (Is this all looking familiar for each application? You can make a right triangle and use trig to get the angle!)

22 , yx

11, yx

Q

Terminal Point

direction is

this angle

Although it is possible to do this for any initial and terminal points, since vectors are equal as long as the direction and magnitude are the same, it is easiest to find a vector with initial point at the origin and terminal point (x, y).

yx,

0,0If we subtract the initial point from the terminal point, we will have an equivalent vector with initial point at the origin.

P

Initial Point

A vector whose initial point is the origin is called a position vector

Equality of Two Vectors

Two vectors are equal if they have

the same magnitude & direction

Are the vectors here equal?

A vector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the vector represents the magnitude and the arrow indicates the direction of the vector.

Two vectors are equal if they have the same direction and magnitude (length).

Blue and orange vectors have same magnitude but different direction.

Blue and green vectors have same direction but different magnitude.

Blue and purple vectors have same magnitude and direction so they are equal.

Addition of vectors

Given two vectors , what is?

A &

B

A

B

A

B

Graphical Techniques of Vector Addition

Two vectors can be added using these method:

1 -tip to tail method.

2 -the parallelogram method.

1“-Tip-to-Tail Method”

•Two vectors can be added by

placing the tail of the 2nd on

the tip of the 1st

A

B

R

A

B

R

Vector B50 m θ= 0O

Vector C30 m

Θ = 90O

Vector A30 m θ = 45O

A

B

C

Resultant = 9 x 10 = 90 meters

Angle is measured at 40o

To add the vectorsPlace them head to tail

A

B

C

D

A

BC

DR

A + + + =B C D R

ALL VECTORS MUST BE DRAWN TO

SCALE & POINTED INTHE PROPER DIRECTION

2 -the parallelogram method.

C = A + B2 2

Multiplying a Vector by a Scalar

• Given , what is ?

s

3s

s

s

s

s

s

s

Scalar multiplicationScalar multiplication: multiply vector by scalardirection stays samemagnitude stretched by given scalar(negative scalar reverses direction)

Vector Subtraction

A

B

C

DA + - - =B C D R

A+ + ( - ) + ( - ) =B C D R

-C

=

-D=

A

-D

R

B

-C

Example

Example

Example

Components of a Vector

Vector component:

A

A x

A y

where and are the components of the vector

Ax

Ay

A

A unit vector is a vector that has a magnitude of 1, with no units.

Its only purpose is to point

We will use x , y for our Unit Vectors

x means x – direction, y is y – direction, We also put little “hats” (^) on x , y to show that they are unit vectors

Unit Vectors

Notes about Components• The previous equations are valid only if Ѳ is

measured with respect to the X-axis.

• The components can be positive or negative and will have the same units as the original vector .

Vector component

at

• WHAT ARE THE X AND Y COMPONENTS OF A VECTOR 40 m θ=60O ?

• AX = 40 m x COS 600 = 20 m

• AY = 40 m x SIN 600 = 34.6 m

• WHAT ARE THE X AND Y COMPONENTS OF A VECTOR 60 m/s θ = 2450 ?

• BX = 60 m/S x COS 245 0 = - 25.4 m/S

• BY = 60 m/S x SIN 245 0 = - 54.4 m/S

VECTOR COMPONENTS

jiw 43

find the magnitude of the vector W

What is ?w

2 23 4 w 525

Example:

Example: The angle between where

and the positive x axis is :

1. 61°2. 29°3. 151°4. 209°5. 241°

A

A x

A y

Ax 25 & Ay 45

Vector component:

jiji 4352

If we want to add vectors that are in the form a i + b j, we can just add the i components and then the j components.

jiv 52

wv ji

Let's look at this geometrically:

i2

j5 v

i3

j4w

ij

When we want to know the magnitude of the vector (remember this is the length) we denote it

v 22 52

Can you see from this picture how to find the length of v?

29

jiw 43 Example:

ADDING & SUBTRACTING VECTORS USING COMPONENTS

Vector A30 m θ = 45O

Vector B 50 m θ = 0O

Vector C30 m Θ = 9 0O

ADD THE FOLLOWINGTHREE VECTORS USING

COMPONENTS

(1) RESOLVE EACH INTO X AND Y COMPONENTS

ADDING & SUBTRACTING VECTORS USING COMPONENTS

• AX = 30mx cos 450 = 21.2 m

• AY = 30 m x sin 450 = 21.2 m

• BX = 50 m x cos 00 = 50 m• BY = 50 m x sin 00 = 0 m

• CX = 30 m x cos 900 = 0 m• CY = 30 m x sin 900 = 30 m

(2) ADD THE X COMPONENTS OF EACH VECTOR ADD THE Y COMPONENTS OF EACH VECTOR

X = SUM OF THE Xs = 21.2 + 50 + 0 = +71.2 Y =SUM OF THE Ys = 21.2 + 0 + 30 = +51.2

(3) CONSTUCT A NEW RIGHT TRIANGLE USING THE X AS THE BASE AND Y AS THE OPPOSITE SIDE

X = +71.2

Y = +51.2

THE HYPOTENUSE IS THE RESULTANT VECTOR

(4) USE THE PYTHAGOREAN THEOREM TO THE LENGTH(MAGNITUDE) OF THE RESULTANT VECTOR

X = +71.2

Y = +51.2

(+71.2)2 + (+51.2)2 = 87.7

(5) FIND THE ANGLE (DIRECTION) USING INVERSETANGENT OF THE OPPOSITE SIDE OVER THE

ADJACENT SIDE

angle tan-1 (51.2/71.2)

Θ = 35.7 O

QUADRANT I

RESULTANT = 87.7 m θ = 35.7 O

SUBTRACTING VECTORS USING COMPONENTS

Vector A30 m θ = 45O

Vector C30 m θ = 90O

Vector B50 m θ = 0O

A - + =B C R

A + (- ) + =B C R

Vector A30 m θ = 45O

- Vector B

50 m θ = 180O

Vector C30 m θ = 90O

• RESOLVE EACH INTO X AND Y COMPONENTS

X-comp y-comp

AX = 30 m x cos 450 = 21.2 m AY = 30 m x sin450 = 21.2 m

• BX = 50 m x cos1800 = - 50 m BY = 50 m x sin 1800 = 0

• CX = 30 m x cos 900 = 0 m CY = 30 m x sin 900 = 30 m

X = SUM OF THE Xs = 21.2 + (-50) + 0 = -28.8 Y =SUM OF THE Ys = 21.2 + 0 + 30 = +51.2

X = -28.8

Y = +51.2

(2) ADD THE X COMPONENTS OF EACH VECTOR ADD THE Y COMPONENTS OF EACH VECTOR

X = SUM OF THE Xs = 21.2 + (-50) + 0 = -28.8 Y =SUM OF THE Ys = 21.2 + 0 + 30 = +51.2

(3) CONSTUCT A NEW RIGHT TRIANGLE USING THE X AS THE BASE AND Y AS THE OPPOSITE SIDE

X = -28.8

Y = +51.2

THE HYPOTENUSE IS THE RESULTANT VECTOR

X = -28.8

Y = +51.2angle

Θ=tan-1 (51.2/-28.8)θ = -60.6 0

(1800 –60.60 ) = 119.40

QUADRANT II

RESULTANT ( R) = 58.7 m θ = 119.4O

R = (-28.8)2 + (+51.2)2 = 58.7

jiji 4352

If we want to add vectors that are in the form a i + b j, we can just add the i components and then the j components.

jiv 52

wv ji

Let's look at this geometrically:

i2

j5 v

i3

j4w

ij

When we want to know the magnitude of the vector (remember this is the length) we denote it

v 22 52

Can you see from this picture how to find the length of v?

29

jiw 43 Example:

example

Example :

If we know the magnitude and direction of the vector, let's see if we can express the vector in a + b form.

5, 150 v

1505

As usual we can use the trig we know to find the length in the horizontal direction and in the vertical direction.

yxyx ˆ2

5ˆ

2

35ˆ150sinˆ150cos5 v

Example:

Example :F1 = 37N 54° N of E F2 = 50N 18° N of F3 = 67 N 4° W of S

F=F1+F2+F3

W

Ex : 2 – 10 A woman walks 10 Km north, turns toward the north west , and walks 5 Km further . What is her final position?

Example:

Example:

Example: