Scalars and Vectors

1. Scalars and Vectors MR. KEMUEL DE CASTRO “You can’t control the direction of the WIND but you can always adjust your SAIL”.

2. Definition of Terms • It is the space between two points. • It is change of place or position. • It is the continuous change of place or position, a.k.a. non-stop displacement. • DISTANCE • DISPLACEMENT • MOTION

3. Scalar Quantity • Scalar Quantity A scalar is a simple physical quantity that is not changed by coordinate system rotations or translations. MAGNITUDE Ex. Dimension (length, width, height) Age Mass Time Temperature 17.86 km 14 min 37.5 °C

4. Scalar Quantity • One characteristic of scalar quantities is that they add up or subtract like ordinary numbers. Examples: 4 km + 3 km = 7 km 90 °C – 15 °C = 75 °C

5. Scalar a quantity described by magnitudeonly examples include: time, length, speed, temperature, mass, energy Vector a quantity described bymagnitudeanddirection examples include: velocity, displacement, force, momentum, electric and magnetic fields

7. Vector Quantity • Vector A vector is a geometric object that has both a magnitude, direction and sense, i.e., orientation along the given direction. MAGNITUDE + DIRECTION Ex. Velocity Force Displacement Acceleration

8. Vector Quantity • A vector is represented by an arrow. • Vector quantities are important in the study of physics. If scalar quantities follow ordinary arithmetic rules, vector quantities do not. This is one important characteristic of vectors. A displacement of 20 m to the East 1 cm: 5 m

10. A 16-point nautical compass

11. N 35 E of N E W 55 N of E 35 W of S S

12. N 58 E of N E W S

13. Resultant Vector • Resultant Vector It is the sum of two or more given vectors. Given: Unknown: Equation: Solution: a. Analytical b. Graphical AnSwer:

14. Vector Resolution • The same direction • Opposite direction • Different Direction at an angle • Three directions

15. Importance of Vectors • Sports • Travel Time • Marine Vessels • Forces in an object

16. Travel time…

17. Displacement acting in a straight line

18. Displacement acting in a straight line b. Graphical 5 blocks: 1 cm Given: V1 = 10 blocks ↑ V2 = 20 blocks ↑ Unknown: VR = ? Equation: VR = V1 + V2 Solution: a. Analytical VR = 10 blocks ↑ + 20 blocks ↑ VR = 30 blocks ↑ Answer: 30 blocks ↑ V2 = 20 blocks ↑ VR = 30 blocks ↑ V1 = 10 blocks ↑

19. Displacement acting in a straight line but opposite direction

20. Displacement acting in a straight line but opposite direction Solution: a. Analytical VR = 25 m →+ (-40 m ←) VR = -15 m ← Given: V1 = 25 m → V2 = 40 m ← Unknown: VR = ? Equation: VR = V1 + V2 b. Graphical 5 m: 1 cm Answer: -15 m ← VR = -15 m ← V1 = 25 m → V2 = 40 m ←

21. Problem Set A Solve for the resultant vectors. • Chacha walks 300 m East, stops to rest and then continues 400 m East. • Mimi walks home from school 300 m East and remembers that she has to bring home her Science book which a classmate borrowed. She walks back 500 m West to her classmate’s house.

22. Seatwork • To stay fit, Avelino jogs around Villamor Airbase every morning. Calculate his total displacement from his house if he jogged 45 m to the East and 60 m to the West. • Compute for the total distance a Komodo dragon has travelled if it walked 6 km Northward for 4 hours, took some rest for an hour or so, and walked 4 km in the same direction before sleeping.

23. Displacement acting on different directions

24. Displacement acting on different directions Given: V1 = 50 m East V2 = 30 m North Unknown: VR = ? Equation: VR = V1 + V2 Answer: 58 m 31° North of East VR = 58 m 31 ° North of East V2 = 30 m N Θ ≈ 31° V1 = 50 m E Graphical Solution: 10 m: 1 cm

25. c2 = a2 + b2 Hypotenuse Opposite Adjacent

26. Getting the Angle Sin = Soh Hypotenuse Opposite Cos = Cah Tan =Toa Adjacent

27. Problem Set After reading a book, Steh stands up from the bench she was sitting on. She walks 600 m East, then turns 400 m North. What is her total displacement from the bench?

28. Problem Set Ann and Steh saw a bird outside Ann’s bedroom window. They saw that the bird moved 30 m North and 60 m West before it disappeared. What was the displacement of the bird from Ann’s window?

29. Try… • A car is driven 125 km W then 65 km S. What is the magnitude of its displacement? • A shopper walks from the door of the mall to her car 250 m down the lane of cars, then turns 900 to the right and walks an additional 60 m. What is the magnitude of his displacement of her car from the mall door?

30. Graded Seatwork • An ant crawls on a tabletop. It moves 2 cm East then turns 3.5 cm West. What is the ant’s total displacement? • A boat leaves shore and travels 20 km North and then 15 km West. Find the boat’s resultant displacement vector.

31. Vector at an angle… Sin = Soh ? Cos = Cah ? Tan =Toa

32. Example…. 20 km/hr N 350 W E S

33. Try… • A golf ball hit from the tee travels 325 m in a direction 250 South of East axis. What are the East and North components of its displacement? • What are the components of a vector with a magnitude of 1.5 m at an angle of 300 to the positive x axis? • A hiker walks 14 km at an angle 360 south of east. Find the esat and north component of this walk.

34. Three Vectors… • A hiker walks 50 m E and turn right and walk 50 m S and finally covered 60 m E. What is the total displacement of the hiker?