VECTORS Speed is not velocity; It needs a small correction. When moving in a line or curve, Don’t forget direction!

THINK!! Suppose I told you that last night I buried a pot of gold exactly three kilometers from the flagpole in front of the building. How easy would it be to find the treasure? What other information do you need?

THINK AGAIN! Suppose you walk, in a straight line, a distance of ten meters then walk in another straight line a distance of eight meters. What is the greatest distance you could be from your starting point? What is the least distance? What other distances are possible? What determines where you end up?

VECTOR VOCABULARY VECTOR SCALAR COMPONENT RESULTANT A quantity that requires direction to be fully defined. Examples: Magnitude (size) and units only. Examples: The part of a vector that lies along an axis. The vector sum of two, or more, vectors. Max @ zeroo Min @ 180o

VECTOR NOTATION A vector quantity is typically described by an arrow drawn in a particular direction The length of the arrow represents the magnitude (size) of the vector The direction is where the arrow tip (the head) is pointing (The other end is called the tail) MAGNITUDE TAIL HEAD

DETERMINING A RESULTANT There are three methods that may be employed to add vectors 1. Graphically – vectors are drawn to scale and arranged head to tail. The resultant is measured from the beginning of the first vector to the head of the last vector. EXAMPLE: A bug, that walks 1.5 m/s, walks across a treadmill at an angle of 45o relative to the narrow side of the treadmill. The treadmill is running at 2.0 m/s. What is the resultant velocity of the bug with respect to the floor?

DETERMINING A RESULTANT Let the scale be 3cm = 1m/s Vtm=6cm, Vbug=4.5cm Angle = 45o Vbug = 1.5m/s Vbug Measure the length and angle of the resultant and compare it to the scale Vtm= 2.0m/s Vrel Vtm Vrel = 9.2cm or 3.1m/s Angle = 17o

DETERMINING A RESULTANT 2. Mathematically If vectors are parallel or anti-parallel, the resultant is equal to the algebraic sum. BE SURE TO WATCH THE SIGNS!! If vectors are perpendicular, the resultant may be calculated using the Pythagorean Theorem (C2=A2+B2) and the angle (q) is tan-1(B/A) C B q A

DETERMINING A RESULTANT EXAMPLE ONE: Buzz runs from the high school to Stony Point (1.25 Km west) turns left and runs to the Whitehaven Road intersection (2.0 Km south). Calculate Buzz’s resultant displacement. N dr2 = d12 + d22 dr = [d12 + d22]1/2 dr = [(1.25km)2+(2km)2]1/2 dr = [(5.56km2)]1/2 dr = 2.36km = tan-1(2/1.25) =58o S of W GIVEN d1=1.25 km W d2=2.0 km S d1 q d2 ASKED dr = ?

DETERMINING A RESULTANT EXAMPLE TWO While skydiving, Click experiences two accelerations; gravitational acceleration (g = 9.81m/s2) and an upward acceleration (a2) by an air current of 6.3m/s2. What is Click’s resultant acceleration? ar = a2 + g ar = (6.3m/s2) + (-9.81m/s2) ar = -3.51m/s2 g a2

DETERMINING A RESULTANT 3. By components Any vector can be divided into any number of vectors at various angles. (how many paths are there from your desk to your front door?) It is sometimes convenient to be able to separate a vector into components at right angles, place them on a coordinate system, add the components, and determine the resultant

DETERMINING A RESULTANT TRIG FUNCTIONS!

DETERMINING A RESULTANT In right triangle geometry: Sineq = opp/hyp so: opp = hyp sinq Cosq = adj/hyp adj = hyp cosq hyp opp q adj

DETERMINING A RESULTANT FINAL EXAMPLE: Buzz walks at a speed of 2.5 m/s on a sidewalk that makes an angle of 30o west of north. After 20 seconds, how far has he moved (a) toward the north and (b) toward the west?

And your life will have direction! Know vectors And your life will have direction!

DETERMINING A RESULTANT EXAMPLE THREE: A quarterback throws a backward pass 15 meters at an angle of 190o then the halfback throws the ball 35 meters downfield at an angle of 700. When the ball is caught, the receiver cuts at an angle of 130o and after running 10 meters is tackled. Calculate the displacement of the ball.

DETERMINING A RESULTANT A quarterback throws a backward pass 15 meters at an angle of 190o then the halfback throws the ball 35 meters downfield at an angle of 70o. When the ball is caught, the receiver cuts at an angle of 130o and after running 10 meters is tackled. Calculate the displacement of the ball. DETERMINING A RESULTANT Simplify it by putting all of the Displacement vectors at the origin of a single coordinate system SKETCH THE PROBLEM How can the components be determined?

DETERMINING A RESULTANT Component vectors are determined in the same manner y dx = 35m cos 70o = 12m dy = 35m sin 70o = 33m 35 m 70o x

DETERMINING A RESULTANT Complete the data table: Calculate the resultant. Disp (m) Angle (deg) dx (m) dy (m) 15 190 -14.8 -2.6 35 70 12 32.9 10 130 -6.4 7.7 sum -9.2 38 dr = 39 m q = 103.6o