1. PHYSICS: Vectors

2. Today’s Goals Students will: 1.Be able to describe the difference between a vector and a scalar. 2.Be able to draw and add vector’s graphically and use Pythagorean theorem when applicable. Day 1

3. Today’s Objective By the end of today… 1.You must be able to find the magnitude and direction of a resultant mathematically.

4. A couple of quick definitions! This should be review, but just in case….. Scalars Vectors Remember: Force, Velocity, Displacement, Acceleration are ALL VECTORS!!!

5. Representing Vectors Because of direction, vectors do not add the same way scalars do. 2 + 2 does not always equal 4!!!! We will discuss 2 Ways to Add Vectors 1.Graphically 2.Component Addition

6. Adding Vectors When two vectors are added together, the results are a vector called a RESULTANT. In order to find the resultant, you must draw the two vectors “head to tail.” Example: http://www.stmary.ws/highschool/physics/home/note s/forces/animations/AngleResultants.html http://www.stmary.ws/highschool/physics/home/note s/forces/animations/AngleResultants.html Another Reason why vectors are important: http://www.stmary.ws/highschool/physics/home/notes/kine matics/MotionTerms/animations/boatsRiver.swf

10. Find the Following Resultants 1. 5 N + 5 N = ? 2. + 6 N3 N = ?

11. Components Every vector can be separated into two equivalent vectors. These two vectors are known as components. In physics, we often break vectors down into horizontal (x) and vertical (y) components. X - component Y - component

12. Calculating Components Use the Pythagorean Theorem to calculate the missing component for each velocity below. a 2 + b 2 = c 2 3 m/s 4 m/s - 8 m/s ? ? 5 m/s ? 8 m/s 5 m/s 8.9 m/s 6.24 m/s

13. a 2 + b 2 = c 2 (-8) 2 + 4 2 = c 2 64 + 16 = c 2 80 = c 2 c = 8.9 m/s 4 m/s ? - 8 m/s

14. Right Triangle Trig. Q: WHAT ARE TRIG FUNCTIONS? A: Sin CosTan Three functions that are useful when dealing with right triangles. These functions are used to determine missing sides of right triangles when other information is known.

15. Sine (sin) The Sin of any angle is equal to the ratio of the length of the “opposite side” and the length of the “hypotenuse.” Sin (ø) = H O O ø H

16. Cosine (cos) The Cosine of any angle is equal to the ratio of the length of the “adjacent side” and the length of the “hypotenuse.” A ø H Cos (ø) = H A

17. Tangent (tan) The Tangent of any angle is equal to the ratio of the length of the “opposite side” and the length of the “adjacent side.” O A ø Tan (ø) = A O

18. An easy way to remember… What should you do when you have dirty feet? »SOHCAHTOA (pronounced: Soak A Toe.. Uhhh….) SOH CAHTOA Sine Opposite Hypotenuse Cosine Adjacent Hypotenuse Tangent Opposite Adjacent

19. What is a Function? 2x = Y What do I get for Y as I plug in numbers for x? x y 0 1 2 2 4 Sin (x)Cos (x)and Tan (x) do the same thing!!!! Plug in a number for the angle: (x degrees) and you get back a different number (usually a decimal). Next, following SOHCAHTOA, the number you get back from the function, is equal to O/H, A/H, or O/A!

20. Trig. Practice Problems #1 a. Find x. b. Find y. c. Find q. d. Find r. q r 25 o 9 sin (30) = x/20.5 = x/20 (.5)20 = x x = 10 cos (30) = y/20.866 = y/20 (.866)20 = y y = 17.32 sin (25) = 9/q.423 = 9/q.423q = 9 q = 21.3 tan (25) = 9/r.466 = 9/r.466r = 9 r = 19.3

21. Trig. Problems #1 HINT: Draw a diagram showing the information. m From a point on the ground 25 meters from the foot of a tree, the angle of elevation of the top of the tree is 32º. (Meaning you have to look up 32 o to be looking at the top of the tree.) Find to the nearest meter, the height of the tree.

22. A ladder leans against a building. The foot of the ladder is 6 feet from the building. The ladder reaches height of 14 feet on the building. a. Find the length of the ladder to the nearest foot. Trig. Practice #2

23. b. What is the angle between the top of the ladder and the wall it leans on? Trig. Practice #2

24. Trig. Practice Problem #3 3. A plane travels 25 km at an angle 35 degrees to the ground and then changes direction and travels 12.5 km at an angle of 22 degrees to the ground. What is the magnitude and direction of the total displacement?

25. Quiz! The Vectors listed below represent the velocity of an airplane and the velocity of wind. Each Plane is trying to Go North. Draw a Vector Diagram and Calculate the magnitude and direction of the resulting velocity. 1.Wind: 50 m/s East Plane Engines: 200 m/s North 2. Wind 30 m/s East Plane Engines: 100 m/s North 3. Wind 120 m/s West Plane Engines: 175 m/s North