10/11/2013

1. 10/11/2013 • Today I will use trigonometry to solve right triangles. • Warm up- • Define and give an example of: • Scalar Quantity • Vector Quantity

2. Chapter 3 Projectile Motion Part 1 - Vectors

3. Trigonometry • When working with right triangles, if we know two sides, we can find the other using the Pythagorean theory A2 + B2 = C2 hypotenuse

4. Trigonometry a2 + b2 = c2 (8 m)2 + (x)2 = (12 m)2 64m2 + x2 = 144m2 x2 = 80m2 x = 8.9 m 12 m ? m 8 m

5. Trigonometry! • Right Triangles θ adjacent hypotenuse opposite

6. Trigonometry • Trigonometric Functions SINE - COSINE – TANGENT Formulas-

7. Trigonometry • How can I remember these equations? O h H eck A nother H our O f A lgebra SohCahToa

8. Trigonometry Have – hypotenuse Want – adjacent Which function uses both?

9. Trigonometry x 5 30° Let’s find x Have – opposite Want – hypotenuse Which function uses both? Be careful when the variable is on the bottom!

10. Trigonometry B 10 5 X 30° Find B We are now looking at a different angle, so opposite, and adjacent are different! Have – hypotenuse & adjacent Want – angle B Which function can we use?? To solve for an angle, you must use the inverse functions! Does this number make sense?

11. Trigonometry B 10 5 X 30° y Find y Since we have both other sides, we can use the Pythagorean theory.

12. Trigonometry • You can use trig with measurements: 23° 47° 345 m/s ? 34 m ? 18 km/hr 12 km/hr Ɵ

13. 10/15/13 • Today I will demonstrate vector addition • Warm Up – Find the a and b in the triangle below: 20° 150 m a b

14. Trig HW

15. Review… • What is a scalar quantity? • a quantity with only magnitude

16. Review… • What is a vector quantity? • a quantity with both magnitude and direction

17. Are the following quantities vectors or scalars? • Time • Acceleration • Distance • Velocity • Displacement • Speed • Scalar • Vector • Scalar • Vector • Vector • Scalar

18. Intro to Vectors Mini-Lab • Part 1 & 2a

19. 10/17/13 • Today I will draw vectors, add them and find the resultant. • Warm Up – If I walk 10 m North, 5 m East, 10 m South and then 5 m West, what is my displacement?

20. Drawing vectors… • To draw a vector you must represent both the direction and the magnitude • Direction is represented by the angle and arrow • Magnitude is represented by the length

21. Parts of a vector Tip – ending point (arrow) Tail – starting point

22. Drawing Vectors… • Length of the vector is the magnitude • Angle and arrow indicate direction • Angle represented by Greek letter theta θ

23. Drawing Vectors - scale • When drawing vectors, you must use some scale. Ex. 100 m You clearly cannot draw 100 m on a piece of paper. You must set up a workable scale. If 1 cm = 10 m, 10 cm = 100 m… You can draw 10 cm!

24. Drawing Vectors - scale • Scaling rules of thumb… Centimeters is a good scale to work in If your numbers are too large… Divide by something and make 1 cm = that many units If your numbers are too small… Multiply by something and make that many cm = 1 unit *Remember to scale your unit back up at the end by doing the opposite!

25. Drawing Vectors - scale • 120 m, 90 m, 70 m • 3500 km/h, 7200 km/h, 6000 km/h • 0.6 mi, 0.3 mi, 0.2 mi

26. Adding Vectors • We can add two or more vectors together; when we do this we are finding the resultant

27. Two methods of adding vectors Solving Graphically – Tip to Tail method • Draw first vector • Start next vector where the last one ended (so its tail is connected to the previous vectors tip) • Draw your resultant vector • Find the direction (including angle)

28. Solving Graphically Example: 2nd Vector 1st Vector

29. Solving Graphically • Resultant - Start where the first vector starts and end where the last vector ends

30. Solving Graphically Example: 2nd Vector Resultant 1st Vector

31. Vector Lab – Day 2 • Drawing your motion to scale If you make 1 cm = 1 m, that would be OK, but quite small. If you make 2 cm = 1 m, that would be better. You can even make 5 cm = 1 m Give the resultant. We will deal with direction tomorrow!

32. 10/18/13 • Today I will find direction in vector addition problems • Warm Up – Draw to scale and find the resultant: 1. You drive 27 miles North to the new Wal-mart to pick up some gift cards. You turn and drive 48 miles West to see your cousin’s baseball game. You then drive South for 62 miles for a graduation party. Find your displacement graphically. 2. What would be the best way to get home?

33. Vector Direction Look at this example: What is the direction of the resultant vector? Is it enough to say Northeast? No, because we could be at any angle between North and East. hypotenuse 123 km/h Opposite (y) 50° Adjacent (x)

34. Vector Direction Look at our previous example: Mrs. Nairn’s house OHIO RIVER OHIO RIVER My house is North of the Ohio River. 123 km/h What if the river were the x-axis? 50°

35. Vector Direction Look at our previous example: N Mrs. Nairn’s house W E 123 km/h My house would be North of the East line OR North of East (N of E) 50° S

36. Vector Direction Look at our previous example: If we put in the directional axis…. Our resultant is 50° North of the East line. (N of E) N 123 km/h 50° E

37. Vector Direction Look at our previous example: It would also be appropriate to say… N Our resultant is 40° East of the North line. (E of N) 123 km/h 50° E

38. Vector Direction • What is the direction on the following vectors? W W 60° 30° S S 30° W of S 60° S of W

39. Vector Direction • What is the direction on the following vectors? N E 80° 38° W S 38° N of W 80° S of E

40. Vector Direction • Another way of recording direction is to say that an angle so many degrees clock-wise or counter-clockwise from a directional line. E 280° counterclockwise from East 80° S 80° S of E

41. Measuring Angles Imagine a mini-coordinate system at the tail of your vector Place the center point of the protractor along one direction of the axis Measure the angle making sure to go from zero Determine the direction 50° N of E

42. Measuring Angles Imagine a mini-coordinate system at the tail of your vector Place the center point of the protractor along one direction of the axis Measure the angle making sure to go from zero Determine the direction 40° E of N

43. Vector Direction • Use your ruler to measure the angles on the Direction WS

44. Vector Lab – Part 3 • Go back and add direction into you lab!

45. 10/21/13 • Today I will solve vector addition problems graphically • Warm Up – Find x 25° x 127 km

46. Solving Graphically • A plane is flying East at 100 km/h • A Westward wind is blowing at 20 km/h When adding vectors, place them tip to tail to scale to find the resultant vector. A good scale here might be 1cm = 10 km/h 100 km/h 20 km/h Remember to scale your answer back up! What is the resultant vector? 80 km/h East

47. Solved Examples • Example 1: Every March, the swallows return to San Juan Capistrano, CA after the winter in the south. In the swallows fly due North and cover 200 km on the first day, 300 km on the second day, and 250 km on the third day, draw a vector diagram of their trip and find the total displacement for the three day journey. 1 cm = 100 km 250 km 750 km North 300 km 200 km

48. Solved Examples • Example 2: Suppose a car pulling with a force of 20000N was pulled back by a rope that Joe held in his teeth. Joe pulled the car with a force of 25000 N. Draw a vector diagram and find the resultant force. +20000 N -25000 N 5000 N in the direction Joe is pulling

49. Solved Examples • Example 3: If St. Louis Cardinals homerun king, Mark McGuire, hit a baseball due West with a speed of 50.0 m/s, and the ball encountered a wind that blew it north at 5.00 m/s, what was the resultant velocity of the baseball? 5m/s θ 50m/s The resultant is about 50.3 m/s at some angle Ɵ. Measure the angle with the protractor

50. Solved Examples • Example 4: The Maton family begins a trip driving 700 km west. Then the family drives 600 km south, 300 km east, and 400 km north. Measure the length and the angle with your ruler to find the answer!