2. VECTORS v Fp

3. Scalar quantities – that can be completely described by a number with the appropriate units. ( They have magnitude only. ) Such as length, mass, speed, energy, temperature… {notation : Δd, m, v, E, t o …}

4. Vector quantities – that require both a magnitude AND a direction for their complete description. Such as displacement, velocity, force… {Notation : Δd, v, F… } In a diagram, vectors are represented by arrows which point in the direction of the vector and whose length represents the magnitude.

5. TRIGONOMETRY REFRESHER ANATOMY OF A RIGHT TRIANGLE Hypotenuse (HYP) θ Reference angle (θ, THETA) Right angle Adjacent (ADJ) Opposite (OPP)

6. TRIGONOMETRY REFRESHER TRIGONOMETRIC FUNCTIONS θ HYP OPP ADJ SOH CAH TOA

7. TRIGONOMETRY REFRESHER I II III IV A ll are + S in is + C os is + T an is + 0o0o 90 o 180 o 270 o 360 o

8. For a vector, V, when θ is to the positive x-axis : θ V VxVx VyVy

9. Resolving vectors into their components {finding the x- and y components of a vector} To find the x-component To find the y-component V y = V sin θ V x = V cos θ

10. Sample problems  Resolve this velocity into its components. V = 15 m/s at 60.0 o V = 15 m/s θ = 60.0 o GIVEN V θ VyVy VxVx V y = V sin θ V y = 15 m/s sin 60.0 o V x = V cos θ V x = 15 m/s cos 60.0 o V x = + 7.5 m/s V y = + 13 m/s

11. Sample problems  Resolve this velocity into its components. V = 80.0 m/s at 140.0 o θ V VyVy VxVx V = 80.0 m/s θ = 140.0 o GIVEN V y = V sin θ V y = 80.0 m/s sin 140.0 o V y = + 51.4 m/s V x = V cos θ V x = 80.0 m/s cos 140.0 o V x = -61.3 m/s

12. GRAPHIC RESOLUTION Use a ruler and protractor to draw the two previous vectors to scale. Graph paper is helpful in establishing axes but it is not needed to complete this assignment. Draw the perpendicular to the x-axis and label the components. Measure the components in cm and use your scale to convert back to the appropriate units.

13. GRAPHIC RESOLUTION For V = 15 m/s at 60.0 o, if I let 1 cm = 0.5 m/s ……I draw V 30 cm long at 60.0 o VyVy VxVx Draw the perpendicular to the x-axis Label the components Measure the components and convert V θ = +7.750 m/s = +13.12 m/s

14. Your turn Do both vectors and submit

15. TRIGONOMETRIC VECTOR ADDITION (FINDING RESULTANTS) 1. Resolve vectors into x- and y-components. 2. Add all of the x-components to find the x-component of the resultant. 3. Add all of the y-components to find the y-component of the resultant. 4. Use tan -1 (arctan) to find the direction, θ R. The signs of the components determine the quadrant. 5. Use cos θ R or sin θ R to find the magnitude.

16. θ ref 0o0o 360 o 270 o 180 o 90 o θ R = θ ref θ R = 180 o - θ ref θ R = 180 o + θ ref θ R = 360 o - θ ref θRθR θRθR θRθR θRθR I II III IV

17. ADVANCE TO SAMPLE PROBLEM

18. SAMPLE PROBLEM (from sheet) V 1 = 320 N at 240.0 o V 2 = 35 N at 30.0 o R = V 1 + V 2 Given V 1 = 320 N V 2 = 35 N θ 1 = 240.0 o θ 2 = 30.0 o θ2θ2 θ1θ1 V1V1 V2V2

19. R x = V 1x + V 2x R = V 1 + V 2 R x = V 1 cos θ 1 + V 2 cos θ 2 R x = 320 N cos 240.0 o + 35 N cos 30.0 o R x = (-160 N) + (+30 N)R x = -130 N Find the x-component From

20. R = V 1 + V 2 R y = V 1y + V 2y R y = V 1 sin θ 1 + V 2 sin θ 2 R y = 320 N sin 240.0 o + 35 N sin 30.0 o R y = (-280 N) + (+18 N)R y = -260 N Find the y-component From

21. R x = -130 NR y = -260 N R = R x + R y RxRx RyRy RyRy R θRθR Draw the components and resultant Find the direction, θ R Using

22. RxRx RyRy RyRy R θRθR Your calculator only gives you the reference angle to the x-axis θ ref = 63 o θ ref Since the x- and y-components are both negative, R is in Quadrant III 63 o beyond 180 o on the x-axis. θ R = 180 o + 63 o θ R = 243 o From

23. RxRx RyRy RyRy R θRθR θ ref Find the magnitude, R --- using the sin θ R = 290 N

24. RxRx RyRy RyRy R θRθR θ ref Find the magnitude, R --- using the cos θ R = 290 N

25. RxRx RyRy RyRy R θRθR θ ref Find the magnitude, R --- using pythagorean theorem R = 290 N

26. RxRx RyRy RyRy R θRθR θ ref And so, the final answer is R = 290 N at 243 o Do all of the practice problems on the sheet to prepare yourself for your next quiz.

27. GRAPHIC VECTOR ADDITION When adding vectors graphically : 1. Draw the vectors to scale. 2. Move one of the vectors so that it is “head-to-toe” with the other. (be sure to maintain its magnitude and direction) 3. Draw the resultant from the origin to the head of the last vector you moved. 4. Measure the resultant and use your scale to convert back to the appropriate units. 5. Use a protractor to measure the direction.

28. GRAPHIC VECTOR ADDITION Consider a plane flying with a velocity of 220 m/s at 170 o in a wind blowing at 45 m/s at 270 o. What is the ground velocity of the plane? The problem is to find the resultant, V g, of the velocity of the plane, V p,in the air and the velocity of the wind, V w. V g =V p + V w V p = 220 m/s at 170 o V w = 45 m/s at 270 o

29. GRAPHIC VECTOR ADDITION VpVp VwVw VwVw VgVg Let 1 cm = 10 m/s V p = 220 m/s ( 1cm / 10 m/s ) = 22 cm V w = 45 m/s ( 1cm / 10 m/s ) = 4.5 cm V g = 21.70 cm ( 10m/s / 1cm ) = 217.0 m/s The protractor measures 3 o beyond 180 o so θ g =183 o θpθp θgθg

30. BONUS PROBLEM A pilot wishes to maintain a ground velocity of 150.0 m/s at 270 o and the wind is blowing at 50.0 m/s at 120 o. What air speed must he have and at what bearing (direction)? Hint : This is actually a vector subtraction. Just as in math, subtraction is the same as adding a negative. A negative vector points in the opposite direction. Add or subtract 180 o to the direction of the vector to be subtracted and then add as usual. (solve both graphically and trigonometrically to earn maximum credit)

31. THIS IS THE END OF VECTORS