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  1. Do now – on a new piece of paper • The diagrams below represent two types motions. One is constant motion, the other, accelerated motion. Which one is constant motion and which one is accelerated motion? explain your answer.

  2. Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors using the graphical method. 3. Multiply and divide vectors by scalars. Homework – vector 1

  3. Pointing the Way Vectors

  4. Representing Vectors Vectors on paper are simply arrows Direction represented by the way the ARROW POINTS Magnitude represented by the ARROW LENGTH Examples of Vectors Displacement Velocity Acceleration

  5. Vector Diagrams • a scale is clearly listed • a vector arrow (with arrowhead) is drawn in a specified direction. The vector arrow has a head and a tail. • the magnitude of the vector is clearly labeled. head tail Vectors can be moved parallel to themselves in a diagram

  6. Directions of Vector Reference Vector Uses due EAST as the 0 degree reference, all other angles are measured from that point 20 meters at 190° 34 meters at 48° Compass Point • The direction of a vector is often expressed as an angle of rotation of the vector about its "tail" from east, west, north, or south 20 meters at 10° south of west 34 meters at 42° east of north 90° N 180° W E 0° S 270°

  7. Changing Systems What is the reference vector angle for a vector that points 50 degrees east of south? What is the reference vector angle for a vector that points 20 degrees north of east? 50° 20° 270° + 50° = 320° 20°

  8. Practice • Guided notes – page 2 – 1a, 2a

  9. What we can DO with vectors ADD/SUBTRACTwith a vector To produce a NEW VECTOR MULTIPLY/DIVIDEby a vector or a scalar To produce a NEW VECTOR or SCALAR

  10. Vector Addition • Two vectors can be added together to determine the sum (or resultant). The resultant is the vector sum of two or more vectors. It is the result of adding two or more vectors together

  11. + = ? A B A B Two methods for adding vectors • Graphical method: using a scaled vector diagram • The head-to-tail method • Parallelogram method • Mathematical method - Pythagorean theorem and trigonometric methods

  12. A B C A B C Vector addition: head-to-tail method • A cart is pushed in two directions, as the result, the cart will move in the resultant direction + = + = (Resultant)

  13. A B A B 30o 60o • Frank is walking from his house to Adam’s house, which 200 m away at 30o north of east from his house. Frank and Adam then walk together to their school. The school is 200 m away at 60o west of north from Adam’s house. Determine the resultant (vector sum) of the two vectors ( : Frank’s house to Adam's house; : Adam's house to school) school The resultant is not from the head to tail, it is from beginning to end. Adam’s house resultant Frank’s house Magnitude: measure with ruler, determine using scale Direction: measure with protractor with East

  14. A B C Parallelogram method • A cart is pushed in two directions, as the result, the cart will move in the resultant direction

  15. examples R R Parallelogram method Head and tail method

  16. 10/5 do now Determine resultant graphically • Vector A: 3 m East; Vector B: 6 m North • You need to • Indicate your scale • show magnitude and direction of the resultant

  17. objectives • Lab 6 – motion graphs • Vector addition – graphically and mathematically • Homework • Castle learning • Vector HW 2

  18. Lab 6 – graph matching • OBJECTIVES • Analyze the motion of a student walking across the room. • Predict, sketch, and test position vs. time kinematics graphs. • Predict, sketch, and test velocity vs. time kinematics graphs. • MATERIALS • Computer; Motion Detector; Vernier computer interface; Logger Pro

  19. PRELIMINARY QUESTIONS • An object at rest • An object moving in the positive direction with a constant speed • An object moving in the negative direction with a constant speed • An object that is accelerating in the positive direction, starting from rest match the graphs with the following motion position position position position velocity velocity velocity velocity Time Time Time Time Time

  20. Position vs. Time and Velocity vs. Time Graph Matching • Sketch a Position vs. Time graph and Velocity vs. Time graph that that represents the motion of the object: • Speeds up from rest to the positive direction; Moves at constant speed to positive direction; Slows down to a stop; Speeds up from rest to the negative direction; Moves at constant speed to the negative direction; Slows down to a stop.

  21. Compare your sketch with the experimental data. If you had made mistakes, sketch the correct graph and explain your mistakes.

  22. ANALYSISPART I – POSITION VS. TIME GRAPH • Explain the significance of the slope of a Position vs. Time graph. Include a discussion of positive and negative slope. • What type of motion is occurring when the slope of a Position vs. Time graph is zero? • What type of motion is occurring when the slope of Position vs. Time graph is constant? • What type of motion is occurring when the slope of a Position vs. Time graph is decreasing? What type of motion is occurring when the slope of a Position vs. Time graph is increasing?

  23. AnalysisPART II – VELOCITY VS. TIME GRAPH • What type of motion is occurring when the slope of a Velocity vs. Time graph is zero? • What type of motion is occurring when the slope of a Velocity vs. Time graph is positive? What type of motion is occurring when the slope of a Velocity vs. Time graph is negative? • What does the area under a Velocity vs. Time graph represent?

  24. 3rd pd. - 10/4 - Do now – on a new piece of paper • The diagrams below represent two types motions. One is constant motion, the other, accelerated motion. Which one is constant motion and which one is accelerated motion? explain your answer.

  25. 10/5 do now Determine resultant graphically • Vector A: 3 m East; Vector B: 6 m North • You need to • Indicate your scale • show magnitude and direction of the resultant

  26. objectives • Vector addition – graphically and mathematically • Homework • Castle learning • Vector HW 2

  27. Steps for adding vectors using head and tail method • Choose a scale and indicate it on a sheet of paper. The best choice of scale is one that will result in a diagram that is as large as possible, yet fits on the sheet of paper. • Pick a starting location and draw the first vector to scale in the indicated direction. Label the magnitude and direction of the scale on the diagram (e.g., SCALE: 1 cm = 20 m). • Starting from where the head of the first vector ends, draw the second vector to scale in the indicated direction. Label the magnitude and direction of this vector on the diagram. • Repeat steps 2 and 3 for all vectors that are to be added • Draw the resultant from the tail of the first vector to the head of the last vector. Label this vector as Resultant or simply R. • Using a ruler, measure the length of the resultant and determine its magnitude by converting to real units using the scale (4.4 cm x 20 m/1 cm = 88 m). • Measure the direction of the resultant using the counterclockwise convention.

  28. practice • A man walks east for 3 meters, then south for 5 meters, then west for 6 meters. • Draw his path in the area below using a scale of 1 centimeter = 1 meter. • Draw the man’s final displacement vector. • Measure the length of the vector on your paper. • Calculate the man’s final displacement in meters • Devise a way to solve this problem using your knowledge of geometry. Explain your method and show your work. • How do the results of the two methods compare to one another?

  29. The commutative property of vectors A + B = B + A

  30. - + (- ) = B A B A - = + Vector subtraction - = ?

  31. vector addition vs. subtraction Resultant A + B A Difference A - B B

  32. example • A 5.0-newton force and a 7.0-newton force act concurrently on a point. As the angle between the forces is increased from 0° to 180°, the magnitude of the resultant of the two forces changes from • 0.0 N to 12.0 N • 2.0 N to 12.0 N • 12.0 N to 2.0 N • 12.0 N to 0.0 N

  33. example • A 3-newton force and a 4-newton force are acting concurrently on a point. Which force could not produce equilibrium with these two forces? • 1 N • 7 N • 9 N • 4 N

  34. example • As the angle between two concurrent forces decreases, the magnitude of the force required to produce equilibrium • decreases • increases • remains the same

  35. 10/9 do now • Write all you know all about vector • Definition: • Examples (3): • Representation: • Ways to add vectors • Head to tail: (sketch) • Parallelogram method: (sketch)

  36. objectives • Homework questions? • How to add vectors mathematically? • Homework: castle learning • No post session today • Homework quiz is on Friday

  37. Add vectors mathematically Apply the Pythagorean theorem and tangent function to calculate the magnitude and direction of a resultant vector The procedure is restricted to the addition of two vectors that make right angles to each other.

  38. opp. opp. tanθ = adj. adj. ( ) θ = tan-1 Using tangent function to determine a Vector's Direction Hyp. opp. θ adj.

  39. example • Example: Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement.

  40. Note: The measure of an angle as determined through use of SOH CAH TOA is not always the direction of the vector. R2 = (5.0km)2 + (10km)2 R = 11 km Or at 26 degrees south of west

  41. example • An archaeologist climbs the Great Pyramid in Giza, Egypt. If the pyramid’s height is 136 m and its width is 2.30 x 102 m, what is the magnitude and the direction of the archaeologist’s displacement while climbing from the bottom of the pyramid to the top?

  42. B A B A R R A Equilibrant Equilibrant B Equilibrant • The equilibrant vectors of A and B is the opposite of the resultant of vectors A and B. • Example: Head to tail Parallelogram

  43. Vector Components • In situations in which vectors are directed at angles to the customary coordinate axes, a useful mathematical trick will be employed to transform the vector into two parts with each part being directed along the coordinate axes.

  44. Any vector directed in two dimensions can be thought of as having an influence in two different directions. • Each part of a two-dimensional vector is known as a component. • The components of a vector depict the influence of that vector in a given direction. • The combined influence of the two components is equivalent to the influence of the single two-dimensional vector. • The single two-dimensional vector could be replaced by the two components.

  45. Vectors can be broken into COMPONENTS X-Y system of components AX = A cos θ AY = A sin θ Example vi = 5.0 m/s at 30° vix = 5.0 m/s (cos 30°) = 4.33 m/s viy = 5.0 m/s (sin 30°) = 2.5 m/s Any vector can be broken into unlimited sets of components

  46. EXAMPLE Calculate the x and y components of the following vectors. a. A = 7 meters at 14° b. B = 15 meters per second at 115° c. C = 17.5 meters per second2 at 276°

  47. Adding with Components Vectors can be added together by adding their COMPONENTS Results are used to find RESULTANT MAGNITUDE RESULTANT DIRECTION Adding Vectors Algebraically

  48. Example Add vectors D and F by following the steps below. a. Calculate the components of vectors D and F. D = 35 meters at 25° F = 55 meters at 190° b. Calculate the sum of the x-components of vectors D and F. c. Calculate the sum of the y-components of vectors D and F. d. Sketch the resultant x and y vectors on the axes below. e. Calculate the length of the resultant generated by the resultant components f. Calculate the direction of the resultant generated by the resultant components

  49. Exit slip – hand in by the end of class A bus heads 6.00 km east, then 3.5 km north, then 1.50 km at 45o south of west. What is the total displacement? A: 6.0 km, 0° CCWB: 3.5 km, 90° CCWC: 1.5 km, 225° CCW + + A B C Cx = Ccos225o = -1.06 km Cy = Csin225o = - 1.06 km