Vectors and 2D Motion. Vectors and Scalars

Transcription

1 Vectos and 2D Motion Vectos and Scalas Vecto aithmetic Vecto desciption of 2D motion Pojectile Motion Relative Motion -- Refeence Fames Vectos and Scalas Scala quantities: equie magnitude & unit fo complete desciption Eamples: mass, time, tempeatue, speed, (what othes?) 2.7 kg 57 C 60 m/s Vecto quantities: equie magnitude, unit & diection fo complete desciption Eamples: displacement, velocity, acceleation (what othes?) 500 m noth 50 m/s heading m/s 2 down Chapte 3 mateial 1

2 Vecto Notation Vecto quantities ae gaphically epesented as aows. y θ The length of the aow epesents the magnitude of the vecto and the diection is self-evident. Vectos quantities ae efeed to by symbol, such as o (aow used on blackboad, boldface in the tet, and on oveheads) simple, unbold, unaowed efes to the magnitude of vecto. = Vecto Math Vectos can be multiplied by scalas: B 3B -B 0.75B The esult is that the magnitude of the vecto changes, but not the diection, ecept in the case of muliplication by a negative numbe whee the vecto eveses diection. Chapte 3 mateial 2

3 dding Vectos (paallelogam method) R B + B = R dding Vectos ( tip-to-tail method) R B + B = R Chapte 3 mateial 3

4 dding Vectos ( tip-to-tail method) The esult is independent of the ode of addition! B + = + B B R B + = R B Subtacting Vectos -B R R + B = R = R - B R B Chapte 3 mateial 4

5 Vecto Components y y θ Vecto (with magnitude, ) is diected at an angle θ above the + ais = + y : y : vecto component of in the -diection vecto component of in the y-diection Scala Components y y θ Given, θ, the scala components of ae = cos (θ) y = sin (θ) Chapte 3 mateial 5

6 Scala Components Given, y, how do you find, θ? (-,+) y (+,+) Magnitude: (-,-) (+,-) ngle: = + θ = 2 2 y tan ( 1 ) y Be caeful! Calculato in degee mode? Look at the signs of and y. Does the angle make sense? Invese tangent only gives back a esult fom -90 to +90. How do you get the ight quadant? +y dding Vectos by Components +y +y C B B y +B=C B y + C B B y y + θ C C C y + 1. Choose coodinate system and daw a pictue. 2. Find scala components:, y, B, B y 3. Calculate scala components: C = + B and C y = y + B y Find: C = C + C 5. Find: y θ = tan ( C 1 ) Cy Chapte 3 mateial 6

7 dding Displacement Vectos: Eample N hikes walks D 1 = 6.0 km east on day 1 D 2 = 4.0 km noth on day 2 D 3 = 10.0 km at 30.0 noth of west on day 3. W D 3 D T 30 θ D 2 E Find he total displacement fo the Tip. D 1 By gaphical Methods D T = D 1 + D 1 + D 1 S nalysis Eample N -component y-component D 3 D km 0 km D2 0 km +4.0 km W D T 30 θ D 2 E D3-10 cos(30 ) +10 sin(30 ) = -8.7 km = +5.0 km D T -2.7 km +9.0 km D 1 By gaphical Methods 2 2 DT = ( 27. ) + ( 90. ) km = 94. km S o o o o θ = tan ( ) = = 107 (Needed to add 180 to get esult in 27. the coect quadant) Chapte 3 mateial 7

8 Vecto Desciption of Motion veage Vecto Displacement: = f i paticle moves fom P (at t i )to Q(at t f ) along the tajectoy shown above. ( tajectoy is a path of motion though space) veage Vecto Velocity: = t Instantaneous Vecto Velocity: = lim t 0 t Vecto Desciption of Motion Velocity vectos shown in ed How does velocity change? v f v i The velocity vecto s diection is always tangent to the tajectoy and in the diection of the motion. The length of the velocity vecto shows the instantaneous speed of the paticle. Longe means faste. v f v i = v v f i veage Vecto cceleation: a = t Instantaneous Vecto cceleation: a lim t 0 t = Chapte 3 mateial 8

9 Vecto Desciption of Motion Velocity vectos shown in ed, acceleation vectos in puple a i a f v i The velocity vecto s diection is always tangent to the tajectoy and in the diection of the motion. The length of the velocity vecto shows the instantaneous speed of the paticle. Longe means faste. v f How does velocity change? v f v i = v v f i veage Vecto cceleation: a = Instantaneous Vecto cceleation: a = t lim t 0 t Vecto Desciption of Motion s a paticle moves along its tajectoy, cceleation vecto shows how the velocity vecto changes. Velocity vecto shows how the position changes Both velocity and acceleation can change in both magnitude and diection (This is a quicktime animation in the sceen vesion ) Chapte 3 mateial 9

10 Pojectile Motion Pojectile Motion descibes the motion of any object thown into the ai at any abitay angle. To descibe pojectile motion mathematically, we assume: 1. Pojectile motion is unifomly acceleated motion. The acceleation vecto is diected vetically downwad and has a magnitude of g=9.8 m/s 2. Quicktime Movie of a juggle 2. The effects o ai esistance ae negligible. nalyzing Pojectile Motion +y v 0 θ 0 (,y) + ( 0,y 0 ) Subsequent to launch, the -component of velocity, emains constant; only the y-component changes. v The pojectile is launched at initial velocity, v 0 at angle θ 0 above the hoizontal. Initial components ae: v0 = v0cosθ0 v = v sinθ y motion y - motion v = v0 = v0cosθ0 = constant vy = vy0 gt = v0t = ( v0cos θ0) t y y = vy t gt 2 2 vy = vy0 2 g( y y0 ) The speed at time t is given by: v = v + v 2 2 y Chapte 3 mateial 10

11 Eample: Find components fist! o v = 30cos 30 m/ s = 26 m/ s 0 o v = 30sin 30 m/ s = 15 m/ s y0 ball is thown off a 25 m high oof at a speed of 30 m/s at an angle of 30 above hoizontal ) What is the maimum height? vy0 = 2g( y y0) y = y0 + vy0/ 2g 2) t what time does it land? 2 2 y = 25 m+ ( 15m/ s) /( 19. 6m/ s ) = 36 m 2 y y0 = vy0t 05. gt t 15t 25 = 0 t = 1. 2 s, t = s 3) How fa does it tavel? (What is the hoizontal ange?) = v t 0 0 5/24/01 Physics = 11, 0 Summe + ( 26)( ) = Motion m in a Plane Relative Motion How is a motion descibed by diffeeent obseves in diffeent efeence fames which ae in motion with espect to each othe? Relative Position Relative Velocity Chapte 3 mateial 11

12 Relative Velocities Conside the motion of a boat in a ive. v b : velocity of boat seen by ive v e : velocity of ive seen by eath v be : velocity of boat seen by eath Vecto equation: v be = v b + v e Sign Convention: v e = - v e Eample t what angle do you point the boat to go staight acoss the ive? v b : speed is 20 m/s, diection??? v e : cuent, 10 m/s, due east v be : speed is???, due noth N Vecto equation: v be = v b + v e E In components: East: (v b ) + (v e ) = (v be ) -20 m/s sin(θ)+10 m/s = 0 ---> θ =30 Noth: (v b ) y + (v e ) y = (v be ) y 20 m/s cos(30 )+0 m/s = (v be ) y = v be = 17 m/s Chapte 3 mateial 12