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Vectors and Scalars

Learn about vectors and scalars in physics, including their definitions, applications, and mathematical operations. Understand how to calculate displacement, velocity components, and resultant displacement using vectors.

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Vectors and Scalars

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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 and DIRECTION. Vectors are typically illustrated by drawing an ARROW above the symbol. The arrow is used to convey direction and magnitude.

  4. Applications of Vectors VECTOR ADDITION (Graphically) – If 2 similar vectors point in the SAME direction, draw them tip to tail, then measure RESULTANT. • Example: A man walks 54.5 meters east, then another 30 meters east. Calculate his displacement relative to where he started? Notice that the SIZE of the arrow conveys MAGNITUDE and the way it was drawn conveys DIRECTION.

  5. Applications of Vectors VECTOR ADDITION (mathematically) – 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 (Graphically) - If 2 vectors are going in opposite directions, you draw them tip to tail them measure the RESULTANT. • Example: A man walks 54.5 meters east, then 30 meters west. Calculate his displacement relative to where he started?

  7. 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

  8. Non-Collinear Vectors - Graphically When 2 vectors are perpendicular, you must use the Tip to tail method still. 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 Tip to tail. 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. 55 km, N q N of E 95 km,E

  11. Non-Collinear Vectors - Mathematically 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

  12. 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. 109.77 km 55 km, N q N of E 95 km,E So the COMPLETE final answer is : 109.77 km, 30.07 degrees North of East

  13. 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 sine and cosine. H.C. = ? V.C = ? 25 65 m

  14. Example 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 of EAST

  15. Example 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, @ 28.07 degrees West of North

  16. Example 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

  17. Example 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

  18. First, let’s talk about Mr Morris’s boat… . If the boat has a speed of 10 m/s in still water and the current is 5 m/s, what is the resultant velocity of the boat?

  19. If the boat has a speed of 10 m/s in still water and the current is 5 m/s, what is the resultant velocity of the boat with respect to the riverbank? 10 m/s across 5 m/s downstream Θ VR VR = √10 2 + 52 = 11.18 m/s Θ = Tan-1(10/5) = 63.43º w/riverbank

  20. How long to travel across the 120 m wide river? The time to cross depends on the speed across the river. t = d v = 120 m 10m/s = 12 sec How far downstream will the boat land on the far bank? The distance downstream depends on the downstream current speed and the time in the water. d = vt = (5 m/s)(12sec) = 60 m downstream

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