1 / 30

Vectors and Scalars

Vectors and Scalars. AP Physics. Scalar. A SCALAR is ANY quantity in physics that has MAGNITUDE , but NOT a direction associated with it. Magnitude – A numerical value with units. Vector. A VECTOR is ANY quantity in physics that has BOTH MAGNITUDE and DIRECTION.

gabela
Download Presentation

Vectors and Scalars

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Vectors and Scalars AP Physics

  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. y Ax+ Ay= A A = A sin θ Ay θ x Ax = A cos θ

  5. Scalar Multiplication Multiplying a vector by a scalar will ONLY CHANGE its magnitude. Multiplying a vector by “-1” does not change the magnitude, but it does reverse it's direction or in a sense, it's angle. Thus if A = 12 < 105, Then 2A = 24 < 105 Thus if A = 12 < 105, then -A = 12 < 285 If A = 12 < 105, then (-1/2)A = 6 < 285 -1/2A

  6. Unit Vector Notation An effective and popular system used in engineering is called unit vector notation. It is used to denote vectors with an x-y Cartesian coordinate system.

  7. Unit Vector Notation =3j J = vector of magnitude “1” in the “y” direction = 4i i = vector of magnitude “1” in the “x” direction The hypotenuse in Physics is called the RESULTANT or VECTOR SUM. The LEGS of the triangle are called the COMPONENTS 3j Vertical Component NOTE: When drawing a right triangle that conveys some type of motion, you MUST draw your components HEAD TO TOE. 4i Horizontal Component

  8. Unit Vector Notation The proper terminology is to use the “hat” instead of the arrow. So we have i-hat, j-hat, and k-hat which are used to describe any type of motion in 3D space. How would you write vectors J and K in unit vector notation?

  9. 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

  10. 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

  11. 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. 55 km, N 95 km,E

  12. 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 :

  13. Example 1 A boat moves with a velocity of 15 m/s, north in a river which flows with a velocity of 8.0 m/s, west. Calculate the boat's resultant velocity. 8.0 m/s, W 15 m/s, N Rv q The Final Answer :

  14. How could you add the following vectors? A = 15 m/s, 45o N of E B = 15 m/s, 20o N of E C = 20 m/s, 65o N of E

  15. GRAPHICALLY: A 450

  16. GRAPHICALLY: A+B B 200 A

  17. GRAPHICALLY: A+B+C C = 650 B A

  18. = A+B+C A GRAPHICALLY: Ay 450 Ax Draw and Find the Components of A Note: A = Ax + Ay R By Components: How can R be determined mathematically?

  19. By 200 Bx Draw and Find the Components of B Note: B = Bx + By B

  20. Cy 650 Cx Draw and Find the Components of C Note: C = Cx + Cy C

  21. Add the x-components together Cx Bx Ax

  22. Add the y-components together Cy By Ay

  23. Combine all x components together to get Rx and all y components together to get Ry. = Ay + By + Cy Ry Rx = Ax + Bx + Cx

  24. R Ry Rx

  25. Mathematically determine R R = Rx2 + Ry2  = tan -1 (Ry/Rx) R Ry  Rx

  26. So let’s do it! Solve for the sum of these velocity vectors mathematically: A = 15 m/s, 45o N of E B = 15 m/s, 20o N of E C = 20 m/s, 65o N of E

  27. What’s the correct answer? V = 47.40 m/s, 45.6 N of E

  28. Example 2 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:

  29. 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

More Related