Unit 2: Motion in 2D

1. Unit 2: Motion in 2D Textbook: Chapter 3 & Chapter 4

2. Unit Objectives: Motion Models • Recognize that an object in free fall will accelerate at a constant rate of 9.8 m/s2 downward near the surface of the earth. • Use kinematic equations to determine velocity or position at any time • Determine which model (constant velocity or constant acceleration, or varying acceleration) is appropriate to describe the horizontal and vertical component of motion of an object

3. Unit Objectives: Projectiles • Use appropriate kinematic equations to determine the position or velocity of a projectile at a specific point. Sketch the graph of motion for projectiles a) y-x, y-t, x-t, vx – t, vy- t, ax-t, ay-t • Given information about the initial velocity and height of a projectile, determine a) time of flight, b) the point where a projectile lands, and c) velocity at impact

4. Unit Objectives: Vectors • Graphical representation of Vectors • Given a vector, draw its components • Recognize the magnitude and direction of a vector from a vector diagram • Determine the sum of 2 or more vectors graphically • Numerical Analysis of Vectors • Given the magnitude and direction of a vector, determine the components using trig • Given the components of a vector, determine the magnitude and direction using Pythagorean Theorem and trig • Determine the sum of 2 or more vectors using Pythagorean Theorem and trig • Represent by using unit vectors i, j, & k.

5. Textbook: Chapter 2: Section 26 Chapter 3 & Chapter 4 Unit Objectives: Relative Motion • Use vectors to perform relative velocity calculations

6. Free Fall Acceleration • Acceleration due to the force of Earth’s gravity • Acceleration due to gravity at the surface of the Earth is -9.8 m/s2. Negative because it points down.

7. Graphs: x-t & v-t

8. In 1971, the commander of Apollo 15 confirmed this concept by dropping a hammer and a feather. Both hit the ground at the same time. Free Fall on the Moon • Acceleration of a falling object is constant regardless of mass or density

9. Free Fall – Key Points • At max height, velocity is zero. • At a given height, velocity up is equal to velocity down. • Time up equals time down

10. Vectors answer both questions: 1 – How much (magnitude)? 2 – Which way (direction)? Vectors: How much & which way? • When describing motion, often the questions asked are “How far?” or “How fast?” • However, for a person that is lost, “which way?” becomes more valuable.

11.  R head tail Scalars vs Vectors • Scalars have magnitude only • Quantity of something • Distance, speed, time, mass, temperature • Vectors have both magnitude and direction • displacement, velocity, acceleration

12.  x A Direction of Vectors • The direction of a vector is represented by the direction in which the ray points. • This is typically given by an angle. • Can also be given by using unit vectors

13. If vector A represents a displacement of three kilometers to the north… A B Then vector B, which is twice as long, would represent a displacement of six kilometers to the north! Magnitude of Vectors • The magnitude of a vector is the size of whatever the vector represents. • The magnitude is represented by the length of the vector. • Symbolically, the magnitude is often represented as │A │

14. v = 5 m/s at 135˚ Polar Notation • Magnitude and direction of the vector are stated separately. • Magnitude is a positive number and the angle is made with the positive x-axis

15. Vx Vy Rectangular Notation • Defining a vector by its components • y-component: vector projection parallel to y-axis • x-component: vector projection parallel to x-axis

16. Converting Polar & Rectangular

17. B A R Graphical Addition of Vectors • Vectors are added graphically together head-to-tail. • The sum is called the resultant. • The inverse, is called the equilibrant . A + B = R

18. Component Addition of Vectors • Resolve each vector into its x- and y-components. Ax = Acos Ay = Asin Bx = Bcos By = Bsin • Add the x-components together to get Rx and the y-components to get Ry. • Use the Pythagorean Theorem to get the magnitude of the resultant. • Use the inverse tangent function to get the angle.

19. Sample: What is the value of “a” and “b”? a = -3 & b = 10

20. Sample Problem • Add together the following graphically and by component, giving the magnitude and direction of the resultant and of the equilibrant. • Vector A: 300 m @ 60o • Vector B: 450 m @ 100o • Vector C: 120 m @ -120o Resultant: 599 m @ 1o Equilibrant: 599 m @ 181o

21. Yet another sample!!! Sprint (-6, -2) blocks

22. Unit Vectors • Unit vectors are quantities that specify direction only. They have a magnitude of exactly one, and typically point in the x, y, or z directions.

23. Unit Vectors z k j y i x

24. Unit Vectors • Instead of using magnitudes and directions, vectors can be represented by their components combined with their unit vectors. • Example: displacement of 30 meters in the +x direction added to a displacement of 60 meters in the –y direction added to a displacement of 40 meters in the +z direction yields a displacement of:

25. Adding Vectors Using Unit Vectors • Simply add all the i components together, all the j components together, and all the k components together.

26. C = -2.20 i + 9.90 j Sample Problem • Consider two vectors, A = 3.00 i + 7.50 j and B = -5.20 i + 2.40 j. Calculate C where C = A + B.

27. 5 i + 10 j + 3 k Sample Problem • You move 10 meters north and 6 meters east. You then climb a 3 meter platform, and move 1 meter west on the platform. What is your displacement vector? (Assume East is in the +x direction).

28. Suppose I need to convert unit vectors to a magnitude and direction? • Given the vector

29. 11.56 m Back to Sample Problem • You move 10 meters north and 6 meters east. You then climb a 3 meter platform, and move 1 meter west on the platform. How far are you from your starting point?

30. x: position x: displacement v: velocity a: acceleration r: position r: displacement v: velocity a: acceleration In Unit Vector Notation • r = x i + y j + z k • r = x i + y j + z k • v = vxi + vyj + vzk • a = axi + ayj + azk 1 Dimension 2 or 3 Dimensions

31. v = 2 i - 10t k a = -10 k freefall Sample Problem • The position of a particle is given byr = (80 + 2t)i – 40j - 5t2k. Derive the velocity and acceleration vectors for this particle. What does motion “look like”?

32. v = 3t2 i + 5 j v(2) = 12 i + 5 j a = 6t i a(2) = 12 i Another Sample • A position function has the form r = x i + y j with x = t3 – 6 and y = 5t - 3. • What are the velocity and acceleration functions? • What are the velocity and acceleration at t=2s?

33. Practice Problems 1- A baseball outfielder throws a long ball. The components of the position are x = (30 t) m and y = (10 t – 4.9t2) m • Write vector expressions for the ball’s position, velocity, and acceleration as functions of time. Use unit vector notation! • Write vector expressions for the ball’s position, velocity, and acceleration at 2.0 seconds. 2- A particle undergoing constant acceleration changes from a velocity of 4i – 3j to a velocity of 5i + j in 4.0 seconds. What is the acceleration of the particle during this time period? What is its displacement during this time period?

34. Projectiles • An object that moves in two dimensions under the influence of only gravity • Accomplished by usually launching at an angle or going off a flat surface with initial horizontal velocity. • Neglect air resistance • Follow parabolic trajectory

35. Launch Angle The components vix & viy are not necessarily positive. If an object is thrown downward, then viy is negative.

36. Projectiles & Acceleration • If you take an object and drop, it will fall straight down and not sideways • ax = 0 & • ay=g = -9.8 m/s2 • The vertical component of acceleration is just the familiar g of free fall while the horizontal is zero

37. g g g g g Trajectory of Projectile • This shows the parabolic trajectory of a projectile fired over level ground. • Acceleration points down at 9.8 m/s2 for the entire trajectory.

38. Trajectory of Projectile • The velocity can be resolved into components all along its path. Horizontal velocity remains constant; vertical velocity is accelerated. vx vx vy vy vx vy vx vy vx

39. Trajectory Path of a Projectile

40. y y x x t t Position graphs for 2-D projectiles. Assume projectile fired over level ground.

41. Acceleration graphs for 2-D projectiles. Assume projectile fired over level ground. ay ax t t

42. Lets think about this!!! • A heavy ball is thrown exactly horizontally at height h above a horizontal field. At the exact instant that ball is thrown, a second ball is simply dropped from height h. Which ball hits firsts? (demo-x-y shooter)

43. Two Independent Motions 1) Uniform motion at constant velocity in the horizontal direction 2) Free-Fall motion in the vertical direction

44. Vo,y = Vo sin  Vo  Vo,x = Vo cos  Remember…To work projectile problems… • …resolve the initial velocity into components.

45. Practice Problems • A soccer player kicks a ball at 15 m/s at an angle of 35o above the horizontal over level ground. How far horizontally will the ball travel until it strikes the ground? • A cannon is fired at 100m/s at an 15o angle above the horizontal from the top of a 120 m high cliff. How long will it take the cannonball to strike the plane below the cliff? How far from the base of the cliff will it strike? • Students at an engineering contest use a compressed air cannon to shoot a softball at a box being hoisted straight up at 10 m/s by a crane. The cannon, tilted upward at 30 degree angle, is 100 m from the box and fires by remote control the instant the box leaves the ground. Students can control the launch speed of the softball by setting air pressure. What launch speed should the students use to hit the box?

46. Range Equation • Derive the range equation for a projectile fired over level ground.

47. Acceleration in 2-D • A runner is going around a track. She is initially moving with a velocity vector of (0.00, -8.00) m/s and her constant acceleration is (1.10, 1.10) m/s2. What is her velocity 7.23 seconds later. Round the final velocity components to the nearest 0.01 m/s.

48. Multidimensional Motion - Calculus What is the velocity function of the plane? What is the velocity at t = 2 seconds? Just like in 1-D, take the derivative of the position function, to get the velocity function. Take the double derivative to find acelleration…

49. Unit Vectors & Calculus • Treat the same way as you do with one dimensional motion • Take the derivative or integral for each unit vector

50. Reference Frames • Coordinate system used to make observations. The woman is using the surface of the Earth as her reference frame. She considers herself and the train platform to be stationary, while the train is moving to the right with positive velocity.