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Understanding Vectors and Scalars: Measurements and Operations

Learn about vectors and scalars, measurements with direction, graphical representation, addition methods, and mathematical operations. Discover how to combine vectors, solve for resultant velocity, and apply tail-to-tip and parallelogram methods.

jamesnjohnson
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Understanding Vectors and Scalars: Measurements and Operations

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  1. I live exactly 15 miles from here. Meet me there at 4 PM. I’ll cook dinner for you. • What is wrong with the following statement?

  2. Vectors & Scalars Measurements can be vectors or scalars. vector = magnitude (size) & direction. Scalar = magnitude (size) only. Both have units.

  3. List some measurements that might have a direction. • Velocity • Time • Acceleration • Force • mass

  4. Some Vector & Scalars:Vectors (direction)Scalars (number unit) no directionDisplacement distanceVelocity speedAcceleration temperatureForce timeMomentum mass Displacement, d (s) and distance, d.

  5. Displacement, d = change in position, Dx Delta means subtract Dx = xf – xi. • Start at 2 cm end at 10 cm. • Dx = xf – xi. • d = 10 cm – 2 cm = 8 cm • Start at 10 cm end at 2 cm. • Dx = xf – xi. • d = 2 cm – 10 cm = - 8 cm. • The + or – sign says which direction. • Same distance dif displacement.

  6. Displacement (d) • Straight line distance with direction from starting point. • Distance is the path length and no direction.

  7. Take a walk.

  8. Representation of vectors:Vectors represented diagrammatically (graphically) by sketching scaled arrows.

  9. 1. Sketch & label a vector arrow to represent 9 m/s left. Use a scale of 1 cm = 1m/s sketch 9 cm arrow pointing left Label 9 m/s left -9 m/s

  10. scale labeled arrowhead pointsin vector direction Magnitude shown by length of arrow Scaled vector arrows can be sketched onto graph axes to show direction: d = 8 m

  11. Direction can be stated as compass directions or angles in degrees (azimuth). Zero degrees is to the right or east. Sketch into notes. Sketch the axes.

  12. State this direction 3 different ways. N. • 120o (azimuth) • 30o W of N • 60o N of W. 30o. E W S.

  13. Positive & Negative Displacementsdirection from start point Positive Negative Left West South Down • Right • East • North • Up

  14. +30 m could be E, N, up, Right • - 25 could be • W, • S, • down, • Left • A negative sign negates the direction: • -40 km W • +40 km, E.

  15. Ex: State the vector – 25 m, W as a positive. • +25 m, E • Or simply 25 m, E

  16. Hwk Worksheet Intro Vectors.

  17. Addition or Combining Vectors Sometimes, more than one vector quantity are combined. In that case, we must combine (add or subtract) the individual vector components to find a resultant vector.

  18. Examples of combining vector quantiti: • Walk west and turn and walk north. • Drive 20 km/h south then 50 km/h SE.

  19. The resultant velocity depends on 2 pushes: the engine push & the wind push.

  20. Combining Parallel Vectors at 0o or 180o. • Simply add then. • 20 m right and 5m right. • +20 m + 5m = + 25 m or 25 m right. • Object is 25 m right of start point. • 20 m left and 5m right. • - 20 m + 5m = - 15 m or 15 m left. • Object is 2 m left of start point.

  21. We can also use scaled arrows sketched head to tail to represent components and find resultant vectors.

  22. 3. When vectors are at 0o or 180o (straight line). I walked 10 km, E, then 5 km, E. Find displacement. Sketch the component and the resultant displacement vector arrows with a scale 1 cm = 1 km . 0o Same direction 5 km, E = 15 km, E = Resultant 10 km, E + R = length from tail to tip of new arrow.

  23. 180o Opposite directions. I walked 10 km, E, then 5 km, W. 10 km, E - 5 km, W = 5 km, E = R = 5 km, E = R 10 km, E 5 km, W

  24. Mathematical operations can be done on non-linear vectors – but not in the usual way. Their direction has to be taken into account. We cannot simply add 40 km North + 20 km North East. The resulting displacement is not 60 km. Combining (adding) Vectors not parallel:

  25. Graphical analysis: a scaled diagram is used for any number of, & combination of vectors. Methods: 1) Parallelogram (Two vectors only) 2) Tail to tip (head to tail). For any number of vectors to be added.

  26. Parallelogram Method • Football Vectors 3 min • http://science360.gov/obj/tkn-video/0ca015f8-0d4c-4d0b-a31e-257ba1445c32/science-nfl-football-vectors

  27. Tail to Tip/Head to Tail Graphical Method to find Resultant • Sketch scaled arrows one after the other.

  28. 1. Two people kick a ball at the same time or concurrently. One gives it a velocity of 6.5 m/s east, the other gives it a velocity of 4.5 m/s 30o N of E. What is the final resultant velocity? Both methods of solving. 4.5 m/s 30o 6.5 m/s

  29. Tail to Tip:Sketch a diagram with a scale of 1cm = 1 m/s.Sketch each vector separately 1 at a time.Place the tail of one of the vectors to the tip of the other. 4.5 cm 30o 30o 6.5 cm

  30. Now connect a straight line, the resultant, from the tail of the unmoved (1st)vector to the tip of the 2nd vector (moved). 4.5 cm R 30o 6.5 cm

  31. Measure the resultant with your ruler to get the magnitude.Measure the angle to get the direction of the resultant. R=10.6 m/s b=13o

  32. Negativevectors are in the opposite direction of positive ones. 10 m/s East = +10 m/s West 36 km 20o N of E = +36 km 20o S of W. What does –10 m South mean? +10 m North

  33. Subtraction: Just reverse the direction of the negative vector & add graphically (make your scaled diagram). • 12 km East – 6 km south • 12 km East + 6 km north. 13 m/s north – 5 m/s 20o N of E = 13 m/s north +5 m/s 20o S of W

  34. Equilibrant is a vector that “neutralizes” the resultant.It is equal and opposite the resultant.Ex: R = 25 m/s South, Equilibrant = 25 m/s North or (-25m/s S)

  35. Wksht Prb Vector Sketching

  36. Parallelogram Method The parallelogram method is similar, but instead of moving a vector & sketching a triangle, you turn your two vector components into a parallelogram.

  37. You may sketch both vectors from the origin. But you must turn the shape into a parallelogram

  38. 4.5 km 40o 8.0 km When the parallelogram is complete, sketch the resultant between the two original component vectors corner to corner. Measure the resultant and the new angle.

  39. Try this: A hiker walks 4.5 km at 40o. He then turns and walks 8 km due east. Find his resultant displacement and direction. 12 m/s, 14 o.

  40. Note: the tail to tip (head to tail) vector diagram may be to resolve any components even more than two.The parallelogram method may be used to resolve only two vector components.The Pythagorean theorem may only be used for vectors at right angles.

  41. Graphical Method • Make a scale – plan the line lengths • Sketch graph axes • Sketch 1st vector to scale in appropriate direction from origin – place arrowhead at the end. • Sketch the 2nd vector in the appropriate direction to the tip (arrowhead) of the 1st vector. • Continue until all vectors sketched. • Connect the beginning (origin) to the tip of the last vector sketched w/ a straight line. • Measure the line (resultant) and convert the scale back to appropriate units. Use protractor to find direction.

  42. Hwk read 3-1 do page 87 show scaled vector sketches. • https://www.youtube.com/watch?v=7p-uxbu24AM Youtube Vector Diagram Lesson 13 minutes

  43. Situations where No Diagrams Needed • Vectors along a line-add or subtract. • Vectors at right angles. Use Pythagorean Theorem.

  44. If vectors at 0o or 180o to each other – simply add or subtract.

  45. What is the resultant displacement for the following: • 15 m North + 5 m North. • 15 m North + 5 m South. 20 m, N 10 m, N

  46. Vectors at right angles a and b are the components, and c is the resultant.

  47. Ex 1: Find Resultant displacement & direction of R using Pythagorean: 10 km q 5 km 52 + 10 2 = R2 R = 11 km tan-1 0.5= q tanq = 5 km 10 km q = 26.6o S of W

  48. Ex 2: A dog walks 4 km to the east and then 9 km to the north. What is his resultant displacement? • 9.8 km • 66o N of E

  49. Ex 3: A pirate in search of treasure follows a map and walks 45-m north, turns and walks 7.5-m east. Use trigonometry to find the displacement. • 45.6-m, 9.5o E of North

  50. Ex 4: Spiderman sprints forward for 115 m and then scales a vertical building straight up for 136 m. A. Make a rough sketch.B. Find his resultant displacement.

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