Physics Vector Used to Represent Quantities

PHET – Vectors Name: ______________________ Introduction 1. The displacement between two points is the straight-line distance (shortest distance) between them. 2. The distance traveled by an object moving from the first point to the second is simply the total length of the path taken to get from the first point to the second point. The distance is always greater than or equal to the displacement. 3. Vectors are used to represent quantities that have a number and a direction associated with them. For example, velocity is a vector, because it has both a number part and a direction part (e.g. 30 mph to the north). 4. The number part of a vector is called its magnitude. 5. A vector is represented graphically by an arrow which points in the direction of the vector quantity, and the length of the arrow corresponds to the magnitude of the vector. For example, 30 mph to the north could be represented on a map as an arrow upwards. 15 mph to the west could be represented by an arrow to the left, that was half the length of the 30 mph arrow. 6. The mathematical notation for vectors is as follows: a. Vectors are written with an arrow over them: 𝑣⃗ b. 𝑥̂ (called x-hat) is a unit vector that points along the x-axis (length of 1). Some textbooks notate 𝑥̂ as 𝑖̂. c. 𝑦̂ (called y-hat) is a unit vector that points along the y-axis (length of 1). Some textbooks notate 𝑦̂ as 𝑗̂. d. Vectors can be expressed as the sum of their x-components and their y-components as follows: 𝑣⃗ = 𝑣𝑥 𝑥̂ + 𝑣𝑦 𝑦̂ = (𝑣𝑥 , 𝑣𝑦 ) e. 𝑟̂ is a unit vector that points away from the origin (length of 1) f. 𝜃̂ is a unit vector that points counterclockwise around the origin (length of 1) g. Vectors can be expressed as their magnitude (r) and direction (𝜃 – angle they make with the positive x-axis): 𝑣⃗ = 𝑣𝑟 𝑟̂ + 𝑣𝜃 𝜃̂ = (𝑣𝑟 , 𝑣𝜃 ) 7. Scalars are simply numbers and have no direction associated with them. For example, an object’s mass has no direction associated with it, so mass would be represented by a scalar. 8. Go to https://phet.colorado.edu/sims/html/vector-addition/latest/vector-addition_en.html (or go to http://phet.colorado.edu, click ‘Physics’, and click the simulation ‘Vector Addition’, and click on the arrow to make it run) . 1 9. Select “Lab”. Vector Representation Basics 10. Make sure the values, angles, and grid are showing. Also, make the vector components show. 11. Drag a blue vector out of the box in the lower right-hand corner. 2 12. PHET Controls: Move a vector by dragging it. Resize or redirect a vector by dragging the tip of the vector. To get rid of a single vector, drag it back to the box. To delete all vectors, click on the eraser. 13. Move your vector around. Resize it. Change the angle. Note how | 𝑣⃗|, 𝜃, 𝑣𝑥 , and 𝑣𝑦 change as you resize and redirect the vector. 14. Click on the other component view options to see other views of the x and y components of your vector. These all are ways to represent the same thing: the part of a vector that projects along a given axis. As previously stated, a vector can be represented as a vector sum of its x and y components. 15. To what does | 𝑣⃗| correspond? (select all that apply) a. The length of the vector b. The angle the vector makes with the x-axis c. The angle the vector makes with the y-axis d. The x-component of the vector e. The y-component of the vector f. The projection of the vector onto the x-axis g. The projection of the vector onto the y-axis h. The magnitude of the vector 16. To what does 𝜃 correspond? (select all that apply) a. The length of the vector b. The angle the vector makes with the x-axis c. The angle the vector makes with the y-axis d. The x-component of the vector e. The y-component of the vector f. The projection of the vector onto the x-axis g. The projection of the vector onto the y-axis h. The magnitude of the vector 17. To what does 𝑣𝑥 correspond? (select all that apply) a. The length of the vector b. The angle the vector makes with the x-axis c. The angle the vector makes with the y-axis d. The x-component of the vector e. The y-component of the vector f. The projection of the vector onto the x-axis g. The projection of the vector onto the y-axis h. The magnitude of the vector 18. To what does 𝑣𝑦 correspond? (select all that apply) a. The length of the vector b. The angle the vector makes with the x-axis c. The angle the vector makes with the y-axis d. The x-component of the vector 3 e. The y-component of the vector f. The projection of the vector onto the x-axis g. The projection of the vector onto the y-axis h. The magnitude of the vector 19. Next move the vector without resizing or redirecting it by left clicking the middle of the arrow and dragging the arrow to another position. This moving a vector from one location to another without resizing or redirecting the vector is called translating the vector (to move a vector is to translate the vector). 20. Do the values of | 𝑣⃗|, 𝜃, 𝑣𝑥 , and 𝑣𝑦 change when you translate a vector? a. Yes b. No Vector Components 21. Make three different sizes and directions of vectors (your choice) in the simulation, and complete the table below. Remember to change your calculator to degree-mode. | 𝑣⃗| 𝜃 𝑣𝑥 𝑣𝑦 Compute: | 𝑣⃗| cos 𝜃 Compute: | 𝑣⃗| sin 𝜃 Vector 1 Vector 2 Vector 3 22. What is the mathematical relationship relating | 𝑣⃗|, 𝜃, and 𝑣𝑥 ? (select all that apply) a. 𝑣𝑥 = | 𝑣⃗| sin 𝜃 b. 𝑣𝑥 = | 𝑣⃗| cos 𝜃 c. 𝑣𝑥 = | 𝑣⃗| tan 𝜃 d. | 𝑣⃗| = 𝑣𝑥 sin 𝜃 e. | 𝑣⃗| = 𝑣𝑥 cos 𝜃 f. | 𝑣⃗| = 𝑣𝑥 tan 𝜃 23. What is the mathematical relationship relating | 𝑣⃗|, 𝜃, and 𝑣𝑦 ? (select all that apply) a. 𝑣𝑦 = | 𝑣⃗| sin 𝜃 b. 𝑣𝑦 = | 𝑣⃗| cos 𝜃 c. 𝑣𝑦 = | 𝑣⃗| tan 𝜃 d. | 𝑣⃗| = 𝑣𝑦 sin 𝜃 e. | 𝑣⃗| = 𝑣𝑦 cos 𝜃 f. | 𝑣⃗| = 𝑣𝑦 tan 𝜃 24. Given that 𝑣𝑥 and 𝑣𝑦 are the lengths of the sides of the right triangle with the hypotenuse 𝑣, answer the two following questions. 25. What is the mathematical relationship relating| 𝑣⃗|, 𝑣𝑥 , and 𝑣𝑦 ? a. | 𝑣⃗| = √𝑣𝑥2 + 𝑣𝑦2 4 b. | 𝑣⃗| = 𝑣𝑥 + 𝑣𝑦 c. | 𝑣⃗| = 𝑣𝑥 − 𝑣𝑦 d. | 𝑣⃗| = 𝑣𝑥 ∗ 𝑣𝑦 e. | 𝑣⃗| = 𝑣𝑥 /𝑣𝑦 f. | 𝑣⃗| = 𝑣𝑦 /𝑣𝑥 26. What is the mathematical relationship relating 𝜃, 𝑣𝑥 , and 𝑣𝑦 ? 𝑣 a. 𝜃 = 𝑣𝑥 𝑦 b. 𝜃 = 𝑣𝑦 𝑣𝑥 𝑣 c. tan 𝜃 = 𝑣𝑥 𝑦 𝑣𝑦 d. tan 𝜃 = 𝑣 𝑥 e. None of the above Vector Addition Basics 27. Turn off the component display and the angles. Make two vectors line up tip to tail, similar to what is shown below. 28. Check the blue Sum option. 29. Drag the dark blue summation vector to where it completes the triangle. 30. Now move the tip of the second vector around, resizing and redirecting it. Notice that the sum of two vectors always connects the vectors when lined up tail-to-tip. This is how you can add vectors graphically. 31. For your two vectors, fill in the values for the following table: 5 𝑣𝑥 𝑣𝑦 𝑣𝑥 𝑣𝑦 Vector 1 Vector 2 Sum of Column Summation Vector 32. From deductions based upon the table above, what is a method for adding two vectors if you know their components? ⃗⃗ = (𝑏𝑥 , 𝑏𝑦 ) = (7,1) and 𝐶⃗ = 𝐴⃗ + 𝐵 ⃗⃗. 33. Given the following vectors, 𝐴⃗ = (𝑎𝑥 , 𝑎𝑦 ) = (4,3) and 𝐵 34. What is the value of the x-component of 𝐶⃗? 35. What is the value of the y-component of 𝐶⃗? 36. Using the equations determined in Vector Components above, answer the following questions. 37. What is the length (or magnitude) of 𝐶⃗, i.e. |𝐶⃗|? 38. What angle in degrees does 𝐶⃗ make with the x-axis? Application Problem 1 39. To get to school from your house, you drive 6 miles south and then 10 miles west. Model this trip in the vector simulation and paste a screenshot of your vector model below. Use the model of your trip from the simulation to answer the next two questions. 40. What is the displacement from your house to the school? 41. From your house, at what positive angle with respect to east should you point in order to point to the school? 6 42. Use the mathematical relationships inferred earlier to check your work on the previous two questions. Application Problem 2 43. Starting from your home, suppose you cycle 10.0 km east and then 15.0 km north to get to a shopping mall. After shopping, you cycle 5.0 km west and then 14.1 km south-west to meet your friend for dinner at a restaurant. You may solve this set of problems by modeling this trip in the simulation, or by solving them by hand on paper. 44. When you arrived at the restaurant for dinner, what total distance had you cycled throughout the day on your trip? 45. What is your displacement from your house when you are at the restaurant? 46. In what direction should you travel to get home? Give this direction as a positive angle with respect to east. 47. Describe the difference between distance and displacement. 48. Are distance and displacement ever equal in value? If so, under what conditions are they equal? Which one is usually larger in value? Application Problem 3 49. A force of 10 N in the positive x-direction is applied to an object, and a force of 7 N in the negative x-direction is applied to the same object. You may solve this set of problems by modeling these forces in the simulation, or by solving them by hand on paper. 50. What is the magnitude of a third force that if applied in the correct direction would create a net force of zero on the object? 51. What is the direction of a third force that if applied with the correct magnitude would create a net force of zero on the object? Application Problem 4 7 52. A force of 13 N is applied on an object in the negative x-direction, and a force of 15 N is applied to the same object in the negative y-direction. You may solve this set of problems by modeling these forces in the simulation, or by solving them by hand on paper. 53. What is the magnitude of a third force that if applied in the correct direction would create a net force of zero on the object? 54. What is the direction of a third force that if applied with the correct magnitude would create a net force of zero on the object? Application Problem 5 55. A man exerts a force of 47.2 N on a box in the direction 32o North of East. 56. What is the magnitude of the force in the x-direction? 57. What is the magnitude of the force in the y-direction? Application Problem 6 58. The diagram shows the top view of three motorboats crossing a river (left bank to right bank). All three motors can move the boats at a speed relative to the surface of the water of 30 m/s but in different directions, and all experience the same river current at 10 m/s flowing south. 59. You may solve this set of problems by modeling this in the simulation, or by solving them by hand on paper. 8 60. Find the magnitude of the resultant velocity relative to its starting point (i.e. the resultant speed) for each of the three motorboats. a. What is the resultant speed for motorboat a? b. What is the resultant speed for motorboat b? c. What is the resultant speed for motorboat c? 61. Which motorboat travels at the fastest speed relative to the surface of the water? a. Motorboat a b. Motorboat b c. Motorboat c d. They’re all equal e. Impossible to determine 62. Which motorboat travels at the fastest speed relative to its starting point? a. Motorboat a b. Motorboat b c. Motorboat c d. They’re all equal e. Impossible to determine 63. Which motorboat will take the least time to reach the other side of the river? a. Motorboat a b. Motorboat b c. Motorboat c d. They’re all equal e. Impossible to determine 64. Which motorboat will travel the shortest distance to reach the other side of the river? a. Motorboat a b. Motorboat b c. Motorboat c d. They’re all equal e. Impossible to determine 65. Which motorboat will travel the longest distance to reach the other side of the river? a. Motorboat a b. Motorboat b c. Motorboat c d. They’re all equal e. Impossible to determine 9 66. Which motorboat takes longer to reach the other side, a or c? a. Motorboat a b. Motorboat c c. They’re equal d. Impossible to determine 10

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