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Vectors and Scalars in Physics

Explore the concepts of vectors and scalars in physics, learning about magnitude, direction, vector addition, subtraction, graphical methods, and more with practical applications and examples.

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Vectors and Scalars in Physics

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  1. Vectors and Scalars AP Physics B

  2. Scalar A SCALAR is ANY quantity in physics that has MAGNITUDE, but NOT a direction associated with it. Magnitude – A numerical value with units.

  3. Vector A VECTOR is ANY quantity in physics that has BOTH MAGNITUDE (size) and DIRECTION. Vectors are typically illustrated by drawing an ARROW above the symbol. The arrow is used to convey direction and magnitude.

  4. Vectors don’t always have to be in a straight line but may be oriented at angle to each other such as. Vector resultants

  5. Applications of Vectors VECTOR ADDITION – If 2 similar vectors point in the SAME direction, add them. • Example: A man walks 54.5 meters east, then another 30 meters east. Calculate his displacement relative to where he started? + 54.5 m, E 30 m, E Notice that the SIZE of the arrow conveys MAGNITUDE and the way it was drawn conveys DIRECTION. 84.5 m, E

  6. Applications of Vectors VECTOR SUBTRACTION - If 2 vectors are going in opposite directions, you SUBTRACT. • Example: A man walks 54.5 meters east, then 30 meters west. Calculate his displacement relative to where he started? 54.5 m, E - 30 m, W 24.5 m, E

  7. Graphical Addition of Vectors • Using a ruler, draw all vectors to scale and connect them head to tail. The resultant is the vector that connects the tail of the 1st vector with the head of the last. (Hint: using graph paper makes this method even easier)

  8. Non-Collinear Vectors When 2 vectors are perpendicular, you must use the Pythagorean theorem. A man walks 95 km, East then 55 km, north. Calculate his RESULTANT DISPLACEMENT. Finish The hypotenuse in Physics is called the RESULTANT. 55 km, N Vertical Component Horizontal Component Start 95 km,E The LEGS of the triangle are called the COMPONENTS

  9. BUT……what about the direction? In the previous example, DISPLACEMENT was asked for and since it is a VECTOR we should include a DIRECTION on our final answer. N W of N E of N N of E N of W E W N of E S of W S of E NOTE: When drawing a right triangle that conveys some type of motion, you MUST draw your components HEAD TO TOE. W of S E of S S

  10. BUT…..what about the VALUE of the angle??? Just putting North of East on the answer is NOT specific enough for the direction. We MUST find the VALUE of the angle. To find the value of the angle we use a Trig function called TANGENT. 109.8 km 55 km, N q N of E 95 km,E So the COMPLETE final answer is : 109.8 km, 30 degrees North of East

  11. What if you are missing a component? Suppose a person walked 65 m, 25 degrees East of North. What were his horizontal and vertical components? The goal: ALWAYS MAKE A RIGHT TRIANGLE! To solve for components, we often use the trig functions since and cosine. H.C. = ? V.C = ? 25 65 m

  12. Example #1 • Every March, the swallows return to San Juan Capristrano, California after their winter in the south. If the swallows fly due north and cover 200km on the first day, 300km on the second day, and 250 km on the third day, draw a vector diagram of their trip and find their total displacement for the three-day journey.

  13. Every March, the swallows return to San Juan Capristrano, Californa after their winter in the south. If the swallows fly due north and cover 200km on the first day, 300km on the second day, and 250 km on the third day, draw a vector diagram of their trip and find their total displacement for the three-day journey. 250km 300km 750km 200km Example #1

  14. Example #2 • In record books, there are men who claim that they have such strong teeth that they can even use them to move cars, trains, and helicopters. Joe Ponder of Love Valley, North Carolina is one such man. Suppose a car pulling forward with a force of 20,000N was pulled back by a rope that Joe held in his teeth. Joe pulled the car with a force of 25,000N. Draw a vector diagram of the situation and find the resultant.

  15. In record books, there are men who claim that they have such strong teeth that they can even use them to move cars, trains, and helicopters. Joe Ponder of Love Valley, North Carolina is one such man. Suppose a car pulling forward with a force of 20,000N was pulled back by a rope that Joe held in his teeth. Joe pulled the car with a force of 25,000N. Draw a vector diagram of the situation and find the resultant. 25,000N 20,000N 5,000N 5,000N in the direction Joe is pulling Example #2

  16. Example # 3 • If Boston Red Sox baseball legend, Carl Yaztremski, hit a baseball due west with a speed of 50 m/s, and the ball encountered a wind that blew it north at 5 m/s, what was the resultant velocity of the baseball?

  17. If Boston Red Sox baseball legend, Carl Yaztremski, hit a baseball due west with a speed of 50 m/s, and the ball encountered a wind that blew it north at 5 m/s, what was the resultant velocity of the baseball? 5m/s 50m/s Solve using the Pythagorean theorem: Tan θ = opp = 5m/s = .100 adj 50m/s Example #3

  18. Example #4 • The Maton Family begins a vacation trip by driving 700km west. Then the family drives 600km south, 300km east, and 400km north. Where will the Matons end up in relation to their starting point? Solve graphically.

  19. 700 km west 600 km south 300 east 400 km north Fy Fx

  20. Example # 5 A bear, searching for food wanders 35 meters east then 20 meters north. Frustrated, he wanders another 12 meters west then 6 meters south. Calculate the bear's displacement. 23 m, E - = 12 m, W - = 14 m, N 6 m, S 20 m, N 14 m, N 35 m, E R q 23 m, E The Final Answer: 26.93 m, 31.3 degrees NORTH or EAST

  21. Example #6 A boat moves with a velocity of 15 m/s, N in a river which flows with a velocity of 8.0 m/s, west. Calculate the boat's resultant velocity with respect to due north. 8.0 m/s, W 15 m/s, N Rv q The Final Answer : 17 m/s, @ 28.1 degrees West of North

  22. Example #7 A plane moves with a velocity of 63.5 m/s at 32 degrees South of East. Calculate the plane's horizontal and vertical velocity components. H.C. =? 32 V.C. = ? 63.5 m/s

  23. Example #8 A storm system moves 5000 km due east, then shifts course at 40 degrees North of East for 1500 km. Calculate the storm's resultant displacement. 1500 km V.C. 40 5000 km, E H.C. 5000 km + 1149.1 km = 6149.1 km R 964.2 km q The Final Answer: 6224.14 km @ 8.91 degrees, North of East 6149.1 km

  24. Example #9 • Ralph is mowing the backyard with a push mower that he pushes down with a downward force of 20.0 N at an angle of 300 to the horizontal. What are the horizontal and vertical components of the force exerted by Ralph?

  25. Ralph is mowing the backyard with a push mower that he pushes down with a downward force of 20.0 N at an angle of 300 to the horizontal. What are the horizontal and vertical components of the force exerted by Ralph? Example #9 20N Fy 30° Fx

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