0 Vectors & 2D Motion Mr. Finn Honors Physics

Slide 1 Overview 1.VectorsVectors –What are they –Operations Addition Subtraction 2.Relative VelocityRelative Velocity –boats and planes 3.Projectile MotionProjectile Motion –Free fall + relative motion

Slide 2 1. Vectors What are they? –How to represent them? Math Operations –Graphical Vector Addition –Analytical Vector Addition cartesian coordinates trig: laws of sine/cosine

Slide 3 What are vectors Two types of physical quantities –Scalars –Scalars: quantities with no “sense” of direction  example: volume, temperature –Vectors –Vectors: quantities with a direction  example: displacement, force How to represent vectors –1D: direction = +/- sign as forward/backward –2D: direction = angle from reference line OR two components

Slide 4 Representing Vectors Graphically Analytically –Cartesian: (x, y) –Polar: (r,  ) angle  tail head (“tip”) magnitude r = A (Textbook = bold) x = r cos  y = r sin  r = (x 2 + y 2 ) 1/2  = tan -1 (y/x) x y

Slide 5 Practice 1.Break the following into components a)12.0 m at 35º b)42 m/s at 160º 2.Find magnitude/direction a)(-2.4 m, +4.8 m) b)(+500 m, -350 m)

Slide 6 Compass Headings North South EastWest 60° North of East 22.5  West of North

Slide 7 Mathematical Operations Scalar arithmetic operations: +/  /  /  Vector arithmetic operations –must be different  two numbers vs. one –vector addition = adding two arrows cannot add magnitudes so “1 + 1  2” unit vectors at right angles = sum of  2 Redefine arithmetic operation = addition –by drawing pictures –by doing calculations

Slide 8 Graphical Addition “Tip-to-Tail” Method Reposition vectors so they are aligned “tip-to-tail” Sum of A and B is arrow whose tail starts at tail of A and whose head ends at tip of B.

Slide 9 Graphical Addition “Parallelogram” Method Reposition vectors so they are aligned “tail-to-tail” Sum of A and B is diagonal of parallelogram from tail to head. Difference is other diagonal Complete parallelogram with two vectors as two adjacent sides

Slide 10 Practice 70º 2.0 m 4.0 m A B Find: A + B (graphically)

Slide 11 Solution length = 6.2 m at 22º

Slide 12 Analytical Addition Break each vector into its components –A = (A x, A y ) –B = (B x, B y ) Add corresponding components to get components of sum –A+B = (A x +B x, A y +B y ) Convert to polar coordinates, if needed

Slide 13 Vector Subtraction Inverse of vector A = -A –switch direction Subtract by adding inverse –A - B = A + (-B) –Subtraction is treated same as Addition

Slide 14 Practice Add the following and find magnitude, direction: –A: (4.5 m, -8.2 m) –B: (5.3 m, -3.1 m) Subtract the preceding (A - B) and find magnitude, direction

Slide Relative Velocity Application of Vector Addition Examples of adding vectors –used displacement vectors  “Race across Desert” –now use velocity vectors  file flight plan All motion is relative to FoR - Galileo –boat sailing across water –plane flying through air media defines one FoR to describe motion implied second FoR is ground relate two motions because 2 FoR are moving relative to each other

Slide 16 Relative Velocity Equation Let: –A = air (water); G = ground; P = plane (boat) –/ = “relative to” Relation among relative velocities: plane relative to gnd (AKA ground speed) plane relative to air (AKA air speed) air relative to gnd (AKA wind speed)

Slide 17 Relative Velocity “Triangle” Destination Represent equation graphically - as a triangle Air = FoR? Gnd = FoR

Slide Projectile Motion Acceleration = gravity, down –a x = 0 –a y = -9.8 m/s 2 Initial velocity at angle to vertical – v ox = v o cos  = constant –v oy = v o sin   vovo Free Fall in 2D or as seen in moving FoR

Slide 19 Kinematics Variables Position = Cartesian coordinates –x = x o + v ox t –y = y o + v oy t - 1/2 g t 2 Velocity constant –v x = v ox = constant –v y = v oy - gt Acceleration constant –a y = -g = constant Horizontal-vertical motions are independent

Slide 20 Direction of Acceleration

Slide 21 Relative Motion & Projectiles Projectile motion –free fall seen from a moving frame of reference vertical motion = constant acceleration down horizontal = constant velocity

Slide 22 Motion – Ground FoR Wagon moves relative to ground

Slide 23 Motion – Wagon FoR Wagon moves relative to ground – so tree “moves” toward wagon.

Slide 24 A ball rolls off a level /horizontal table with an initial velocity of 3.0 m/s. The table is 0.50 m high. Where does the ball strike the ground - i.e., how far from the end of the table? Example:

Slide 25 Numerical Solution time to fall from table to floor –horizontal-vertical motion independent –h = 1/2 gt 2 where h = 0.50 m, g = 9.8 m/s 2 –t = s (avoid rounding intermediate results) horizontal distance moved during fall –x = v ox t where v ox = 3.0 m/s (table horizontal) –x = m  0.96 m (proper SigFig in answer)

Slide 26 Baseball Problem Baseball is hit so that it leaves the bat at –initial speed of 45 m/s –angle of 30° above horizontal Find: –time in air –distance traveled (hit ground) Assume –no air resistance (?!) –ignore initial height (start on ground)

Slide 27 Solution - Range Equation Time in air - vertical velocity –v oy = v o sin  –v top = 0  0 = v o sin  - gt t = v o sin  / g –T = total time in air = 2t = 2 v o sin  / g T = 4.59 s

Slide 28 Solution - continued Distance traveled = “range” –R = x = x i + v ix T = 0 + (v o cos  )T But T = 2 v o sin  / g –R = (v o 2 /g)(2 cos  sin  ) trig identity for sin 2  = 2 cos  sin  –R = v o 2 /g sin 2  R = 179 m

Slide 29 Summary For projectiles –start and end on ground Time in air –T = 2 v o sin  / g Range or distance traveled –R = v o 2 /g sin 2  –ONLY –ONLY if x o = x = 0 R Rocket Challenge

Slide 30 Harder Problem Shoot cannon from castle wall –height of 20 m –velocity of 50 m/s –angle of 20° above horizontal Find –time in air –where hit ground below assume ground to be level Do not use “Range” equation!! 4.41 s 207 m

Slide 31 Summary Kinematics Projectile Linear Circular a = constant (free fall a = -g) a x = 0 a y = -g direction = +/  direction = vector, angle Kinematics of Uniform Acceleration dot or scalar product Projectile Motion = free fall plus relative motion Next Key Question: What causes acceleration or the motion of an object to change?

Slide 32 Air = Frame of Reference? x z y Air/balloon is a MOVING Frame of Reference