Presentation on theme: "Review Displacement Average Velocity Average Acceleration"— Presentation transcript:
1 Review Displacement Average Velocity Average Acceleration Instantaneous velocity (acceleration) is the slope of the line tangent to the curve of the position (velocity) -time graphFor constant acceleration… For constant gravitational acceleration…
2 Motion in two dimensions Chapter 2Motion in two dimensions
3 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, ...
4 How can we find the magnitude if we have the initial point and the terminal point? QThe distance formulaTerminal Pointmagnitude is the lengthdirection is this angleInitial PointPHow 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!)
5 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).QTerminal PointA vector whose initial point is the origin is called a position vectordirection is this angleInitial PointPIf we subtract the initial point from the terminal point, we will have an equivalent vector with initial point at the origin.
6 Equality of Two Vectors Two vectors are equal if they have the same magnitude & directionAre the vectors here equal?
7 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).
9 Addition of vectorsGiven two vectors , what is ?
10 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 byplacing the tail of the 2nd onthe tip of the 1st
12 Drawing Vectors to Scale Vector A30 m θ = 45ODrawing Vectors to ScaleTo add the vectorsPlace them head to tailVector B50 m θ= 0OCVector C30 mΘ = 90OABAngle is measured at 40oResultant = 9 x 10 = 90 meters
13 Adding Vectors ALL VECTORS MUST BE DRAWN TO SCALE & POINTED IN THE PROPERDIRECTIONADRBCBCAD=ABCDR
20 Multiplying a Vector by a Scalar Given , what is ?Scalar multiplicationScalar multiplication: multiply vector by scalardirection stays samemagnitude stretched by given scalar(negative scalar reverses direction)
31 Vector component:where and are the components of the vector
32 Unit VectorsA unit vector is a vector that has a magnitude of 1, with no units.Its only purpose is to pointWe will use x , y for our Unit Vectorsx means x – direction, y is y – direction,We also put little “hats” (^) on x , y to show that they are unit vectors
33 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 .
36 VECTOR COMPONENTSWHAT ARE THE X AND Y COMPONENTS OF A VECTOR 40 m θ=60O ?AX = 40 m x COS 600 = 20 mAY = 40 m x SIN 600 = 34.6 mWHAT ARE THE X AND Y COMPONENTS OF A VECTOR 60 m/s θ = 2450 ?BX = 60 m/S x COS = m/SBY = 60 m/S x SIN = m/S
40 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 itLet's look at this geometrically:Can you see from this picture how to find the length of v?
41 ADDING & SUBTRACTING VECTORS USING COMPONENTS Vector A30 m θ = 45OADD THE FOLLOWINGTHREE VECTORS USINGCOMPONENTSRESOLVE EACH INTOX AND Y COMPONENTSVector B50 m θ = 0OVector C30 m Θ = 9 0O
42 ADDING & SUBTRACTING VECTORS USING COMPONENTS AX = 30mx cos 450 = 21.2 mAY = 30 m x sin 450 = mBX = 50 m x cos 00 = 50 mBY = 50 m x sin 00 = 0 mCX = 30 m x cos 900 = 0 mCY = 30 m x sin 900 = 30 m
43 (2) ADD THE X COMPONENTS OF EACH VECTOR ADD THE Y COMPONENTS OF EACH VECTOR X = SUM OF THE Xs = = +71.2 Y =SUM OF THE Ys = = +51.2(3) CONSTUCT A NEW RIGHT TRIANGLE USING THE X AS THE BASE AND Y AS THE OPPOSITE SIDE Y = +51.2 X = +71.2THE HYPOTENUSE IS THE RESULTANT VECTOR
44 (4) USE THE PYTHAGOREAN THEOREM TO THE LENGTH (MAGNITUDE) OF THE RESULTANT VECTOR X = +71.2 Y = +51.2angletan-1 (51.2/71.2)Θ = 35.7 OQUADRANT I(+71.2)2 + (+51.2)2 = 87.7(5) FIND THE ANGLE (DIRECTION) USING INVERSETANGENT OF THE OPPOSITE SIDE OVER THEADJACENT SIDERESULTANT = 87.7 m θ = 35.7 O
45 SUBTRACTING VECTORS USING COMPONENTS Vector A30 m θ = 45OA=BCRA+ ( ) =BCRVector A30 m θ = 45OVector B50 m θ = 0O- Vector B50 m θ = 180OVector C30 m θ = 90OVector C30 m θ = 90O
46 AX = 30 m x cos 450 = 21.2 m AY = 30 m x sin450 = 21.2 m RESOLVE EACH INTO X AND Y COMPONENTSX-comp y-compAX = 30 m x cos 450 = 21.2 m AY = 30 m x sin450 = 21.2 mBX = 50 m x cos1800 = - 50 m BY = 50 m x sin = 0CX = 30 m x cos 900 = 0 m CY = 30 m x sin 900 = 30 m X = SUM OF THE Xs = (-50) + 0 = Y =SUM OF THE Ys = = +51.2 Y = +51.2 X = -28.8
47 (2) ADD THE X COMPONENTS OF EACH VECTOR ADD THE Y COMPONENTS OF EACH VECTOR X = SUM OF THE Xs = (-50) + 0 = -28.8 Y =SUM OF THE Ys = = +51.2(3) CONSTUCT A NEW RIGHT TRIANGLE USING THE X AS THE BASE AND Y AS THE OPPOSITE SIDE Y = +51.2 X = -28.8THE HYPOTENUSE IS THE RESULTANT VECTOR
49 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 itLet's look at this geometrically:Can you see from this picture how to find the length of v?
53 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.
54 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 SF=F1+F2+F3W
55 Ex : 2 – 10A woman walks 10 Km north, turns toward the north west , and walks 5 Km further . What is her final position?