1. 0 Vectors & 2D Motion Mr. Finn Honors Physics

2. 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

3. 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

4. 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

5. 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

6. 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)

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

8. 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

9. 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.

10. 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

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

12. Slide 11 Solution length = 6.2 m at 22º

13. 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

14. 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

15. 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

16. Slide 15 2. 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

17. 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)

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

19. Slide 18 3. 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

20. 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

21. Slide 20 Direction of Acceleration

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

23. Slide 22 Motion – Ground FoR Wagon moves relative to ground

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

25. 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:

26. 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 = 0.3194 s (avoid rounding intermediate results) horizontal distance moved during fall –x = v ox t where v ox = 3.0 m/s (table horizontal) –x = 0.9583 m  0.96 m (proper SigFig in answer)

27. 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)

28. 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

29. 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

30. 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

31. 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

32. 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?

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