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1. General Physics 1 Science, Technology, Engineering, and Mathematics Lesson 2.4 Vector Addition through Analytical Method

2. ‹#› Consider a race car shown below.

3. ‹#› What are the forces present in this car?

4. ‹#› The car has weight. Weight acts downward. weight

5. ‹#› While running, the tires of the car experience friction with the pavement. friction running in this direction

6. ‹#› The sum of these forces, among others, determine whether the car will move or not.

7. ‹#› You learned in the past lessons that resultant vectors can be solved graphically. But can we do vector addition analytically?

8. ‹#› How are two or more vectors added analytically?

9. ‹#› Perform addition of vectors (STEM_GP12V-Ia-9).

10. ‹#› Understand the rules in adding vectors using the analytical method. Add two or more vectors using the analytical method.

11. ‹#› V ectors can be added by placing them head to tail. Graphical method, however, is prone to measurement errors. Recall: Graphical Method of Adding Vectors

12. ‹#› Consider the word problem below. Analytical Method of Adding Vectors Denise walks every day from her house to the school. First, she covers 10 m, 20° north of east. Then, she walked 15 m in a direction 50° north of east. What is her resultant displacement?

13. ‹#› We can represent the displacements as vectors. Analytical Method of Adding Vectors Denise walks every day from her house to the school. First, she covers 10 m, 20° north of east. Then, she walked 15 m in a direction 50° north of east. What is her resultant displacement?

14. ‹#› Analytical Method of Adding Vectors

15. ‹#› Analytical Method of Adding Vectors What is Denise’s resultant displacement? Step 1 : Determine whether the angles given were measured from the + x -axis. Vector A is 10 m, 20º NE.

16. ‹#› Analytical Method of Adding Vectors What is Denise’s resultant displacement? Step 1 : Determine whether the angles given were measured from the + x -axis. Vector B is 15 m, 50º NE.

17. ‹#› Analytical Method of Adding Vectors What is Denise’s resultant displacement? Step 2 : Resolve each vector into its x - and y -components.

18. ‹#› Analytical Method of Adding Vectors What is Denise’s resultant displacement? Step 3 : Add all components together.

19. ‹#› Analytical Method of Adding Vectors Distance Angle x -component y -component A = 10 m 𝜃 = 20° 9.40 m 3.42 m B = 15 m 𝜃 = 50° 9.64 m 11.49 m R x = 19.04 m R y = 14.91 m

20. ‹#› Analytical Method of Adding Vectors What is Denise’s resultant displacement? Step 4 : Calculate the magnitude of the vector using the Pythagorean theorem.

21. ‹#› Analytical Method of Adding Vectors What is Denise’s resultant displacement? Step 5 : Calculate the angle 𝜃 using the inverse tangent function.

22. ‹#› Analytical Method of Adding Vectors Denise walks every day from her house to the school. First, she covers 10 m, 20° north of east. Then, she walked 15 m in a direction 50° north of east. What is her resultant displacement? The resultant vector of Denise is 24.18 m, 38.06° or 38.06° north of east.

23. ‹#› Always check whether your calculator is in the degree mode before proceeding to the calculations. All your calculations would be different if this is not addressed beforehand.

24. ‹#› A car covered 25 km, 60° north of east on its initial route. Afterwards, it covered 50 km in the direction 30° north of west. What is its resultant displacement?

25. ‹#› A car covered 25 km, 60° north of east on its initial route. Afterwards, it covered 50 km in the direction 30° north of west. What is its resultant displacement? The resultant displacement is 55.90 km, 123.43° or 56.57° north of west.

26. ‹#› ‹#› During his early morning training, Louie jogged 10 km, 20° south of west. He then covered another 15 km in the direction of 60° south of east before resting. What is his resultant displacement?

27. ‹#› A person covered three displacement vectors as shown below. What is his resultant displacement?

28. ‹#› A person covered three displacement vectors as shown below. What is his resultant displacement? The resultant displacement is 532.99 m, 6.36° or 6.36° north of east.

29. ‹#› ‹#› Find the resultant vector R if A is 95 N, 30° north of east, B is 50 N, south while C has a magnitude of 75 N and a direction of 45° south of west.

30. ‹#› Four sled dogs are pulling a 1000 kg load. Sled dog A is pulling the load at 20 N, 10° north of east. Sled dog B is pulling the load at 55 N, 70° north of east. Sled dog C is pulling the load at 45 N, 33° north of west, while sled dog D exerts 30 N at a direction of 80° south of west. What is the resultant force acting on the load?

31. ‹#› Four sled dogs are pulling a 1000 kg load. Sled dog A is pulling the load at 20 N, 10° north of east. Sled dog B is pulling the load at 55 N, 70° north of east. Sled dog C is pulling the load at 45 N, 33° north of west, while sled dog D exerts 30 N at a direction of 80° south of west. What is the resultant force acting on the load? The resultant force is 50.32 N, 95.06° or 84.94° north of west.

32. ‹#› ‹#› A plane covered four routes with the following velocities: A = 50 m/s, 30° south of east, B = 20 m/s, 60° south of west, C = 70 m/s, 25° south of west, and D = 65 m/s, 15° north of west. What is the resultant velocity of the plane?

33. ‹#› How can you check whether the angle 𝜃 from the inverse tangent function was measured from the + x -axis?

34. ‹#› Write the correct word(s) in the space provided to complete the sentence. The ___________ function is used to calculate the x -component of a vector. The ___________ function is used to calculate the y -component of a vector. The ___________ function is used to calculate the direction of the resultant vector.

35. ‹#› Calculate the resultant vector using the analytical method. = 650 N, 270°; = 550 N, 32° north of west = 130 m, 19° north of east; = 200 m, 70° north of east = 22 m/s, 70° north of west; = 53 m/s, 60° south of east

36. ‹#› The analytical method of adding vectors utilizes the trigonometric functions and the Pythagorean theorem. It is more accurate and less time consuming than the graphical method. Before resolving the components, ensure first that the angle is measured from the +x -axis.

37. ‹#› Steps in adding vectors using the analytical method: Calculate the x - and y -components of all the vectors. Use the cosine function to calculate the x -component, and the sine function to determine the y -component.

38. ‹#› Steps in adding vectors using the analytical method: 2. After this, add all the x -components together. Do the same for the y -components. The sums are the x - and y -components of the resultant vector. 3. Use the Pythagorean theorem to determine the magnitude of the resultant vector.

39. ‹#› Steps in adding vectors using the analytical method: 4. Use the inverse tangent function to determine the direction of the resultant vector. Check first whether the angle 𝜃 from your calculation is measured from the + x -axis or not. Express it in the correct notation.

40. ‹#› Concept Formula Description Vector Addition through Analytical Method where A x is the x -component of the vector A is the magnitude of the vector 𝜃 is the angle measured from the + x -axis Use this formula to calculate the x -component of a vector.

41. ‹#› Concept Formula Description Vector Addition through Analytical Method where A y is the y -component of the vector A is the magnitude of the vector 𝜃 is the angle measured from the + x -axis Use this formula to calculate the y -component of a vector.

42. ‹#› Concept Formula Description Vector Addition through Analytical Method where R is the magnitude of the resultant vector R x is the x -component of the resultant vector R y is the y -component of the vector Use this formula to calculate the magnitude of the resultant vector.

43. ‹#› Concept Formula Description Vector Addition through Analytical Method where 𝜃 is the angle R x is the x -component of the resultant vector R y is the y -component of the resultant vector Use this formula to determine the direction of the resultant vector.

44. ‹#› ‹#› What are examples of nonzero vectors with either of its x - or y -component as zero? Explain your answer.

45. ‹#› Faughn, Jerry S. and Raymond A. Serway. Serway’s College Physics (7th ed) . Singapore: Brooks/Cole, 2006. Giancoli, Douglas C. Physics Principles with Applications (7th ed). USA: Pearson Education, 2014. Knight, Randall D. Physics for Scientists and Engineers: A Strategic Approach (4th ed) . USA: Pearson Education, 2017. Serway, Raymond A. and John W. Jewett, Jr. Physics for Scientists and Engineers with Modern Physics (9th ed) . USA: Brooks/Cole, 2014. Young, Hugh D., Roger A. Freedman, and A. Lewis Ford. Sears and Zemansky’s University Physics with Modern Physics (13th ed) . USA: Pearson Education, 2012.