Two-Dimensional Motion and Vectors

1. Two-Dimensional Motion and Vectors

2. 3.1 Introduction to Vectors

3. Scalars & Vectors • Scalar quantities have magnitude only • Speed, volume, # students • Vectors have magnitude & direction • Velocity, force, weight, displacement

4. Representing Vectors • Boldface type: vis a vector; v is not • Symbol with arrow: • Arrow drawn to scale: • -----> 5m/s, 0° (east) • ----------> 10 m/s, 0° (east) • <--------------- 15 m/s, 180° (west)

5. Resultant Vectors • Are the sum of two or more vectors • Are “net” vectors • Can be determined by various methods • Graphical addition • Mathematically • Pythagorean theorem • Trigonometry

6. Graphical Addition of Vectors • Parallelogram method • Head-to-tail method • Draw vectors to scale, with direction, in head-to-tail fashion • Resultantdrawn from tail of first vector to head of last vector • Measure length and angle to determine magnitude and direction

7. Head-to-Tail Addition of Vectors

8. Head-to-Tail Addition of Vectors

9. Other Properties of Vectors • Vectors may bemoved parallel to themselves when constructing vector diagrams • Vectors may be added in any order • Vectors are subtracted by adding its opposite • Vectors multipliedby scalars are still vectors

10. Vectors may be added in any order

11. Direction of Vectors • Degrees are measured counterclockwise from x-axis

12. Direction of Vectors • Direction may be described with reference to N, S, E, or West axis • 30° West of North • 60° North of West • Or 120°

13. 3.2 Vector Operations • Adding parallel vectors • Simple arithmetic

14. Perpendicular Vectors & the Pythagorean Theorem • If two vectors are perpendicular, then you can use the Pythagorean theorem to determine magnitude only • Example: if Δx = 40 km/h East Δy = 100 km/h North Then d = (402 + 1002)½ d≈ 108 km/h But what is the direction?

15. Basic Trig Functions hyperphysics.phy-astr.gsu.edu

16. Tangent Function When two vectors are perpendicular: Use Pythagorean theorem to find the magnitude of the resultant vector Use the tangent function to find the direction of the resultant vector

17. Sample Problem • Indiana Jones climbs a square pyramid that is 136 m tall. The base is 2.30 x 102 m wide. What was the displacement of the archeologist? • What is the angle of the pyramid

18. Resolving Vectors • Vectors can be “resolved” into x- and y- components • Resolve = Decompose = Break down • Trig functions are used to resolve vectors

19. Resolving Vectors into X & Y Components • For the vectorA • Horizontal component =Ax = A·cos θ • Vertical component = Ay = A·sin θ hyperphysics.phy-astr.gsu.edu

20. Resolving a Vector • A helicopter travels 95 km/h @ 35º above horizontal. Find the x- and y- components of its velocity.

21. Adding Non-perpendicular VectorsA + B = R

22. Resolving Vectors into x and y Components hyperphysics.phy-astr.gsu.edu

23. Adding vectors that are not perpendicular • Two or more vectors can be added by decomposing each vector • Add all x components to determine Rx • Add all y components to determine Ry • Determine magnitude of the resultant R using Pythagorean theorem • Determine direction angle θof resultant using tan-1

24. Vector Analysis

25. 3.3 Projectile Motionphet.colorado.edu • Objectives • Recognize examples of projectile motion • Describe the path of a projectile as a parabola • Resolve vectors into components and apply kinematic equations to solve projectile motion problems

26. Projectile Motion • Motion of objects moving in two dimensions under the influence of gravity • Baseball, arrow, rocket, jumping frog, etc. • Projectile trajectory is a parabola

27. Projectile motion • Assumptions of our problems • Horizontal velocity is constant, i.e. • Air resistance is ignored • Projectile motion is free fall with a horizontal velocity

28. Motion of a projectile Equations relating to vertical motion: ∆y = vyi(∆t) + ½ag(∆t)2 vyf = vyi + ag∆t vyf2 = vyi2 + 2ag∆y Equations relating to horizontal motion: ∆x = vx∆t vx = vxi = constant (an assumption relating to Newton’s 1st law of motion)

30. Horizontal LaunchComparison of vx & vy vectors

31. Driving off a Cliff • A stunt driver on a motorcycle speeds horizontally of a 50.0m high cliff. How fast must the motorcycle leave the cliff in order to land on level ground below, 90.0m from the base of the cliff? Ignore air resistance. • Sketch the problem • List knowns & unknowns • Apply relevant equations

32. Driving off a Cliff • Known: ∆x = 90.0m; ∆y = -50.0m; ax = 0; ay = -g = -9.81 m/s2; vyi = 0 • Unknown: vx; ∆t • Strategy: vx = ∆x/∆t • Since ∆tx must = ∆ty determine ∆t from the vertical drop Now solve for vx using ∆t

33. Effect of Gravity on Ballistic Launch [physicsclassroom.com]

34. Projectile Motion: Horizontal vs Angled Horizontal Launch Ballistic Launch www.ngsir.netfirms.com/englishhtm/ThrowABall.htm

35. Sample Projectile MotionProblem • A ball is thrown with an initial velocity of 50.0 m/s at an angle of 60º. • How long will it be in the air? • How high will it go? • How far will it go?

36. Sample Problem • A ball is thrown with an initial velocity of 50.0 m/s at an angle of 60º. • How long will it be in the air? • Known: vi = 50.0 m/s, θ = 60º, a = -g = -9.81 m/s2; • Find: Δt, total time in the air

37. Sample Problem • A ball is thrown with an initial velocity of 50 m/s at an angle of 60. • How high will it go?

38. Sample Problem • A ball is thrown with an initial velocity of 50 m/s at an angle of 60º. • How far will it go?

39. 3.4 Relative MotionObjectives • Describe motion in terms of frames of reference • Solve problems involving relative velocity

40. Frames of ReferenceMotion is relative to frame of reference To an observer in the plane, the ball drops straight down (vx= 0) To an observer on the ground, the ball follows a parabolic projectile path (vx≠ 0) Frame of reference: a coordinate (defined by the observer) system for specifying the precise location of objects in space A frame of reference is a “point of view” from which motion is described

41. Relative Velocity • Relative velocity of one object to another is determined from the velocities of each object relative to another frame of reference

42. Example • See problem 1, page 109 • Use subscripts to indicate relative velocities • vbe = vbt + vte • vbe = -15 m/s + 15 m/s • vbe = 0 m/s

43. Example • Car A travels 40 mi/h north; Car B travels 30 mi/h south. What is the velocity of Car A relative to Car B? • vae = 40; vbe = -30; veb = +30 • Find vab • vab =vae +veb • vab = 40 + 30 • vab = 70 mi/h North

44. Relative Velocity • One car travels 90 km/h north, another travels 80 km/h north. What is the speed of the fast car relative to the slow car? • vfe = 90 km/h north; vse = 80 km/h north • vfs = vfe + ves • vfs = 90 + (-)80 • vfs = 10 km/h north