Sclrs versus Vectors IMPOSSIBLE NAVIGATION The need for mgnitude AND direction Sclr: A quntity tht hs mgnitude (numer with units) ut no direction. Vector: A quntity tht hs oth mgnitude (displcement) nd direction. Vectors cn e represented grphiclly or lgericlly. Quick Conventions II The Glencoe Physics ook will define vectors y color: Displcement (m) Velocity (m/s) Accelertion (m/s 2 ) Force (N) Momentum/Impulse Representing Vectors Exmple: Displcement Grphiclly= ctully drwing n rrow Algericlly = numericl vlue Grphiclly Representing Vectors Mgnitude Mgnitude is drwn s n rrows length. Longer rrows, lrger mgnitude. Direction Grphiclly direction is represented y n rrow (tip to til). N, S, E, W: Crtesin Coordinte System. Direction cn e roken down into two prts: N/S Prt E/W Prt 1
Adding vectors=tip to til Sy we hve two vectors nd. To dd them: Reposition one of the vectors so tht the tip of one touches the til of the other. Mke new vector going from the til of the open end to the point of the other end. Grphiclly dding vectors r Mesure the vector you re going to move. Slide your ruler cross the pge. Drw the resultnt vector. Mke cross t the til of r to find direction. Adding multiple vectors c d Vector Exmples Rememer ll vectors need mgnitude nd direction. If necessry, mesure direction with protrctor. Every Arrow should flow in the sme direction except the resultnt vector. r It does not mtter which order or wht vectors you move when clculting the resultnt. 6 cm 25 S of E 3 cm 10 N of W Quick Conventions Algericlly Representing Vectors Tke look t this vector: 30 N of E 30 60 E of N While oth re correct lwys put the protrctor on the horizontl xis (EW) nd mesure to the north or south. Mgnitude mgnitude is ssigned numer vlue. The mgnitude should still hve unit such s 15-m. 2
Direction Algericlly direction is represented with degrees. use the form x N/S of E/W Protrctor should e lined up on the horizon No rdins (check your clcultor) 36 36 N of E N of W S of W N of E S of E The Resultnt The resultnt is the vector sum of two or more vectors. It is the result of dding two or more vectors together. If displcement vectors A, B, nd C re dded together, the result will e vector R. A + B + C = R Consider footll plyer who gets hit simultneously y three plyers on the opposing tem (plyers A, B, nd C). The footll plyer experiences three different pplied forces nd ech pplied force contriutes to totl or resulting force. If the three forces re dded together using methods of vector ddition, then the resultnt vector R cn e determined. Bsic mth nd vectors vector + or - vector = vector vector x vector = sclr vector / vector = not done vector + or - sclr = cn t do vector x or / sclr = vector To e hit y plyers A, B, nd C would result in the sme force s eing hit y one plyer pplying force R. Vectors nd sclrs Multiplying Vectors When multiplying or dividing vector y sclr you only need to know two things: Either multiply or divide the mgnitude of the vector. If the sclr is negtive numer, reverse the direction of the vector. Why do we need to know this? Force is defined s mss x ccelertion Mss is sclr, ccel is vector. Force therefore is vector. (needs direction) Energy is defined s Force x distnce. Force is vector nd distnce is vector. Energy therefore is sclr. (no direction) 3
Breking Down Vectors Using Trig for Direction If you hve vector you cn rek it down into n x nd y component. This is useful for dding. N-S comp: y TRIG PNEMONIC sin = o/h cos = /h tn = o/ Rememer: SOH CAH TOA o h E-W comp: x Using Trig for Direction WHEN WORKING WITH VECTORS Useful formuls for 90 vectors: 2 + 2 = c 2 sin = /c cos = /c tn = / c Rememer: DEGREE s on clcs! FYI: For non 90 use lw of cosines r 2 = 2 + 2-2cos. Exmple Displcement Vector Prolem A cr is driven 100 km due west, then 20 km due south. Wht is the mgnitude of its displcement? Non 90 exmple prolem A cyclist rides 5 km nd then mkes left hnd turn t 30 degrees from the horizontl. They then ride nother 10 km. Wht is the mgnitude of the resultnt vector, which is lso the mgnitude of their displcement? Lw of cosines: R 2 = A 2 + B 2 2ABcos 4
Prctice Prolems Red 63-66 Complete 1-4 on pge 67. 5