Review Displacement Average Velocity Average 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…

Motion in two dimensions 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, ...

How can we find the magnitude if we have the initial point and the terminal point? Q The distance formula Terminal Point magnitude is the length direction is this angle Initial Point P 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!)

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). Q Terminal Point A vector whose initial point is the origin is called a position vector direction is this angle Initial Point P If we subtract the initial point from the terminal point, we will have an equivalent vector with initial point at the origin.

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 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. Blue and orange vectors have same magnitude but different direction. Blue and purple vectors have same magnitude and direction so they are equal. Blue and green vectors have same direction but different magnitude. Two vectors are equal if they have the same direction and magnitude (length).

Addition of vectors Given two vectors , what is ?

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

Drawing Vectors to Scale Vector A 30 m θ = 45O Drawing Vectors to Scale To add the vectors Place them head to tail Vector B 50 m θ= 0O C Vector C 30 m Θ = 90O A B Angle is measured at 40o Resultant = 9 x 10 = 90 meters

Adding Vectors ALL VECTORS MUST BE DRAWN TO SCALE & POINTED IN THE PROPER DIRECTION A D R B C B C A D + + + = A B C D R

2- the parallelogram method.

2 2 C = A + B

Multiplying a Vector by a Scalar Given , what is ? Scalar multiplication Scalar multiplication: multiply vector by scalar direction stays same magnitude stretched by given scalar (negative scalar reverses direction)

Vector Subtraction

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

Example

Example

Example

Components of a Vector

Vector component: where and are the components of the vector

Unit Vectors 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

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

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VECTOR COMPONENTS 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

Example : find the magnitude of the vector W

Example : The angle between where and the positive x axis is: 61° 29° 151° 209° 241°

Vector component:

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. Example : When we want to know the magnitude of the vector (remember this is the length) we denote it Let's look at this geometrically: Can you see from this picture how to find the length of v?

ADDING & SUBTRACTING VECTORS USING COMPONENTS Vector A 30 m θ = 45O ADD THE FOLLOWING THREE VECTORS USING COMPONENTS RESOLVE EACH INTO X AND Y COMPONENTS Vector B 50 m θ = 0O Vector C 30 m Θ = 9 0O

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  Y = +51.2  X = +71.2 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 angle tan-1 (51.2/71.2) Θ = 35.7 O QUADRANT I (+71.2)2 + (+51.2)2 = 87.7 (5) FIND THE ANGLE (DIRECTION) USING INVERSE TANGENT OF THE OPPOSITE SIDE OVER THE ADJACENT SIDE RESULTANT = 87.7 m θ = 35.7 O

SUBTRACTING VECTORS USING COMPONENTS Vector A 30 m θ = 45O A - + = B C R A + (- ) + = B C R Vector A 30 m θ = 45O Vector B 50 m θ = 0O - Vector B 50 m θ = 180O Vector C 30 m θ = 90O Vector C 30 m θ = 90O

AX = 30 m x cos 450 = 21.2 m AY = 30 m x sin450 = 21.2 m 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  Y = +51.2  X = -28.8

(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  Y = +51.2  X = -28.8 THE HYPOTENUSE IS THE RESULTANT VECTOR

(-28.8)2 + (+51.2)2 = 58.7 angle Θ=tan-1 (51.2/-28.8) θ = -60.6 0  X = -28.8  Y = +51.2 angle Θ=tan-1 (51.2/-28.8) θ = -60.6 0 (1800 –60.60 ) = 119.40 QUADRANT II R = (-28.8)2 + (+51.2)2 = 58.7 RESULTANT ( R) = 58.7 m θ = 119.4O

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. Example : When we want to know the magnitude of the vector (remember this is the length) we denote it Let's look at this geometrically: Can you see from this picture how to find the length of v?

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

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