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Vectors

Vectors. Scalars require only magnitude (a number with a unit) to fully describe them. Examples of scalars include mass, time, distance, speed, and energy. Vectors require both magnitude and direction. Examples of vectors include displacement, velocity, acceleration, force, and momentum.

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Vectors

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  1. Vectors

  2. Scalars require only magnitude (a number with a unit) to fully describe them. • Examples of scalars include mass, time, distance, speed, and energy.

  3. Vectors require both magnitude and direction. • Examples of vectors include displacement, velocity, acceleration, force, and momentum.

  4. Vectors are drawn as arrows whose lengths represent magnitude, and whose points indicate direction. Representing Vectors

  5. q = 30° q = 30° 30° N of E 60o E of N

  6. q = 60o f = 240° f = - 120° 60° S of W 30° W of S q = 60°

  7. The resultant of two or more vectors can be found graphically using a procedure called the HEAD-TO-TAIL method. Graphical Vector Addition

  8. A Resultant B

  9. When vectors are added, the order of addition and the number of vectors does not matter.

  10. The magnitudes of colinear (“in-line”) vectors are added when the angle between them is 0° and subtracted when that angle is 180°. Basic Vector Addition

  11. B R = A + B A The maximum value for the resultant is obtained when both vectors point in the same direction.

  12. B R = A – B = A + (-B) A The minimum value for the resultant is obtained when the vectors point in the opposite direction.

  13. Concurrent vectors do not act in line. • Perpendicular vectors form a right angle when added head to tail, and their resultant forms the hypotenuse of a right triangle.

  14. A R = A² + B² B Remember Pythagoras? His theorem says : A2 + B2 = C2

  15. hyp opp q adj sin q = opp/hyp cos q = adj/hyp tan q = opp/adj

  16. hyp opp q adj SohCahToa sin q = opp/hyp cos q = adj/hyp tan q = opp/adj

  17. Any angled vector can be resolved into two vectors that are 90° to each other. These components are completely independent of each other. Vector Components

  18. F Fv Fh

  19. Fv F q Fh = F cosq Fh Fv = F sinq

  20. Motion in Two Dimensions Projectile Motion

  21. A projectile is given an initial velocity and is then allowed to move under the influence of gravity.

  22. Horizontal Projection

  23. The horizontal and vertical motions of a projectile are independent of each other.

  24. The horizontal motion of a projectile has constant velocity, while its vertical motion is accelerated.

  25. Projectile Equations The projectile's initial horizontal and vertical velocities (v1x and v1y) depend on its initial velocity (v1) and its launch angle (q).

  26. v1 v1y q v1x

  27. v1y v1 q v1x v1x = v1 cosq v1y = v1 sinq

  28. General Kinematic Equations vf = vi + a(Dt) Dr = ½a(Dt)² + vi(Dt) Dr = ½(vi + vf)Dt vf² = vi² + 2aDr

  29. Horizontal Equations vfx = vix Dx = vixt

  30. Vertical Equations vfy = viy + (-g)(Dt) Dy = viy(Dt) + ½(-g)(Dt)² Dy = ½(viy + vfy)Dt vfy² = viy² + 2(-g)Dy

  31. Dx Dy

  32. Dy Dx Dr

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