Chapter 3: Two Dimensional Motion

1. Chapter 3: Two Dimensional Motion Section 1: Vectors

2. scalar – a quantity with a magnitude, but no direction. • Speed is an example of a scalar. • vector - a quantity with both a magnitude and direction. • Velocity & acceleration are vectors. • Vectors are drawn with arrows on diagrams.

4. vector component – one portion of a 2-dimensional vector. • For example… Sarah is walking southeast at 1.7 m/s. • Her motion has a SOUTHWARD component, and an EASTWARD component. • SKETCH THIS: • Vectors are resolved by separating them into their components. Actual Motion “Resultant Vector” South Component East Component

5. Vectors may be added together… • resultant – a vector representing the sum of two or more vectors. • When adding vectors, they must have the same unit. • You cannot add 47.0 km west to 31 mi west.

6. Consider the two vectors shown below. When you add them together, you get… 9 cm + (-5 cm) = +4 cm OR -5 cm + 9 cm = +4 cm +4 cm is the resultant vector. ∆x = +9 cm Positive Negative ∆x2= -5 cm

7. Bellringer 9/13 Which of the following quantities are scalars, and which are vectors? • The acceleration of a plane as it takes off • Duration of flight • Displacement of flight • Number of passengers on the plane

8. Vectors can be added graphically as well by drawing the vectors. • Suppose Shannon walked 400 meters north to the store, and then 700 meters east to school. • To resolve vectors graphically, we use the head to tail method. 2) Then slide the next vector’s tail to the first vector’s head. ∆x = 700m head ∆y = 400m d = 806m tail 3) Once all vectors are added, draw the resultant. 1) ID the head & tail of the first vector.

9. ∆x = 700m ∆y = 400m ∆xresultant= 806m

10. Consider this scenario: • A toy car is moving south at .80 m/s straight across a walkway that moves at 1.4 m/s towards the east. What will the car’s resultant speedbe? • Draw this scenario. Use the scale 1cm = .1 m/s. • Measure the resultant to find the answer. • Try to find the answer using math…

11. The car has a resultant speed of: • 1.61 m/s (toward the southeast)

12. It is important to note that vectors can be moved parallel to themselves in a diagram. • Moving vectors will be vital to drawing the correct resultant vector! • Vectors can act at the same point at the same time.

13. Please note that displacement may now be represented by “d”. y N • Coordinate System -x x W E -y S

14. Vector Practice • A car is driven 125 km west, and then 65 km south. Solve this problem both graphically & mathematically. Compare your answers. (141 km) • Two shoppers exit the mall & walk 250.0 m down a lane of cars, then turn 90 degrees and walk an additional 60.0 m. What is their displacement from the mall door? (257 m)

15. Bellringer 9/13 • A plane can travel with a speed of 80mi/hr with respect to the air. Determine the resultant velocity of the plane (magnitude only) if it encounters a • 10 mph headwind • 10 mph tailwind • 10 mph crosswind • 60 mph crosswind

16. Practice • Add the following vectors and determine the resultant. Use the graphical method Part A: 10 km, North + 5 km, West Part B: 30 km, West + 40 km, South

17. Chapter 3: Two Dimensional Motion Section 2: Vector Operations

18. Although finding vectors graphically is “fun”, it is not very practical. • Vectors can also be determined using geometry  Pythagorean’s Theorem for Rt. Triangles: c2 = a2 + b2 c represents the hypotenuse; a & b represent legs The Pythagorean theorem will only give you the magnitude of the resultant vector! c a b

19. Pythagorean’s theorem can only be used when a right triangle is present!

20. Direction is represented with angles. • Theta (θ) is used to represent the angle when working with vectors. • Trigonometry is applied in order to find the angles of right triangles. We’ll begin with tangent. Tangent Function for Rt. Triangles: tan θ = opp/adj (opp = opposite leg of angle; adj = leg adjacent to angle) hyp opp θ adj

21. Vectors will often need to be broken down into components. • Definition: vector component – the projection of a vector along the x & y axis. The vector C can be broken down into it’s X component and Y component. C

22. Sine Function for Rt. Triangles: sin θ = opp/hyp (opp = opposite leg of angle; adj = leg adjacent to angle) Cosine Function for Rt. Triangles: cos θ = adj/hyp (opp = opposite leg of angle; adj = leg adjacent to angle) hyp opp θ

23. Practice • An archaeologist climbs the Great Pyramid in Giza, Egypt. The pyramid’s height is 136m and its width is 230 m. What is the magnitude and the direction of the displacement of the archaeologist after she has climbed from the bottom of the pyramid to the top?

24. Bellringer 9/16 • While following the directions on a treasure map, a pirate walks 45.0m north and then turns and walks 7.5 m east. What single straight-line displacement could the pirate have taken to reach the treasure? (include displacement and angle)

25. Review • A roller coaster moves 85m horizontally, then travels 45 m at an angle of 30.0 degrees above the horizontal. What is its displacement from its starting point? (use graphical techniques)

26. Review • A novice pilot sets a plane’s controls, thinking the plane will fly at 2.50x102 km/hr to the north. If the wind blows at 75 km/hr toward the southeast, what is the plane’s resultant velocity? Use graphical techniques. 204 km/hr, 75 degrees north of east

27. Resolving vectors into components • X – component – parallel to x axis • Y – component – parallel to y axis • Example: A plane travels at 95 km/hr at an angle of 20 degrees relative to the ground. Attempting to film the plane from below, a camera team travels in a truck that is directly beneath the plane. What velocity must the truck maintain to stay beneath the plane? 89.3 km/hr

28. Adding three or more right angle vectors • A student drives his car 6.0k North before making a right hand turn and driving 6.0 km to the East. Finally, the student makes a left hand turn and travels another 2.0 km to the north. What is the magnitude of displacement of the student?

29. Adding three or more right angle vectors • Draw a diagram of the motion: • Add together using head to tail:

30. Adding three or more right angle vectors • Rearrange the order: • Now, apply the Pythagorean theorem 10 km

31. Example • Mac and Tosh are doing a Vector Walk Lab. Starting at the door of their physics classroom, they walk 2.0 meters, south. They make a right hand turn and walk 16.0 meters, west. They turn right again and walk 24.0 meters, north. They then turn left and walk 36.0 meters, west. What is the magnitude of their overall displacement? 56.5m

32. Adding vectors that are not perpendicular • Cannot apply Pythagorean theorem or trig functions • Determining magnitude and the direction of the resultant can be achieved by resolving each displacement vectors into its x and y components • Components along each axis can be added together

33. Example • A hiker walks 27.0 km from her base camp at 35 degrees south of east. The next day, she walks 41.0 km in a direction 65 degrees north of east and discovers a forest ranger’s tower. Find the magnitude and direction of her resultant displacement between the base camp and the tower.

34. Example • A football player runs directly down the field for 35 m before turning to the right at an angle of 25 degrees from his original direction and running an additional 15m before getting tackled. What is the magnitude and direction of the runner’s total displacement?