Chapter 3: Vectors & 2D Motion. Brent Royuk Phys-111 Concordia University

Transcription

1 Chapter 3: Vectors & 2D Motion Brent Royuk Phys-111 Concordia University

2 Vectors What is a vector? Examples? Notation:! a or! a or a 2

3 Vector Addition Graphical Methods Triangle, parallelogram, polygon 3

4 Graphical Vector Addition Resultant Construction vs. Analytical Right vs. Oblique Vector mobility Physical Diagrams vs. Vector Diagrams Vector subtraction 4

5 Trig Review Remember your trigonometry? 5

6 Trig Review For right triangles: c 2 = a 2 + b 2 sin A = a c cos A = b c tan A = a b 6

7 Vector Addition Magnitudes have units Directions have angles Directional systems Heading, 30 o N of W Bearing, Degrees clockwise from N Cartesian, 20 o below the x-axis 7

8 Vector Addition Directional systems Compass, heading, NW, SSE, etc. 8

9 Graphical Addition Examples If you travel north for 12 km and then west 8.0 km, what is the magnitude and heading of your displacement? A plane flying with a velocity of 120 m/s due south experiences a crosswind of velocity 38 m/s west. What is the plane s resultant velocity? Add these force vectors: 8.0 N at 20 N of E and 10.0 N at 20 W of N. 10

10 Vector Components Resolving a vector into components Expressing vectors in terms of carefully chosen orthogonal vector components 11

11 Vector Components!!! A = A + A x y A x =? A y =? 12

12 Component Vector Addition Vector addition made easy (or at least algorithmic) 1. Find x-components for vectors A and B 1. A x = A cos θ 2. B x = B cos θ 2. Find y-components for A and B 1. A y = A sin θ 2. B y = B sin θ 3. Find components for the resultant R 1. R x = A x + B x 2. R y = Ay + B y 4. Find the magnitude and direction of R 1. θ= tan -1 (R y /R x ) 2. R = R x 2 + R y 2 14

13 Component Vector Addition Pictorial Representation 15

14 Component Addition Examples A plane leaves an airport and is later sighted 215 km away, at 22 o E of N. How far east and north is the plane from the base? Add these forces: 58 N at 60 o W of S and 67 N at 15 o E of N. Give answer as components and also as magnitude and direction. 16

15 Projectile Motion What path does the ball follow when dropped? 17

16 Projectile Motion Horizontal Launch What happens if you kick a ball off a cliff? 18

17 Projectile Motion Falling Comparison 19

18 Projectile Motion Horizontal Launch 20

19 100-ft Cliffdiving 21

20 Projectile Motion Velocity Changes 22

21 Projectile Motion 24

22 Projectile Motion Launch at an angle What is the range? A useful identity: sin 2 = 2 sin cos v 25

23 Projectile Motion R = v2 sin 2θ g 26

24 Projectile Motion Air Resistance Data: 100 mph at 60 o ; vacuum = 581 ft., air = 323 ft. How about the moon? 27

25 ConcepTest A battleship simultaneously fires two shells at enemy ships. If the shells follow the parabolic trajectories shown below, which ship gets hit first? 1. A 2. both at the same time 3. B 4. need more information 28

26 Projectile Motion 29

27 Projectile Motion a 30

28 Projectile Motion Examples Find the range of a projectile launched with v o = 35 m/s at 52 with the ground. How high does it rise? With what velocity does it land? At what angle? An artillery shell with a muzzle velocity of 125 m/s is fired at an angle of 35.0 with the horizon. If the shell explodes 10.0 s after being fired, where does the blast occur? 31

29 Projectile Motion Examples A police officer is chasing a burglar across a rooftop. Both are running at 4.5 m/s. When the burglar reaches the end of the roof he jumps horizontally toward the next building, which is 6.2 m away but 4.8 m lower. Should the policeman jump to pursue or take the elevator to clean up the mess? Justify your answer. A plane flies at a velocity of 85.2 m/s at 43 W of N. The velocity of the wind is 24.3 m/s at 18 S of E. Find a) the velocity of the plane in component form and b) the distance the plane travels in 2.5 hours. 32

30 Relative Velocities Frames of reference Everyday examples of velocity addition baseball from pickup closing velocity of cars moving walkway in airport Example Consider a 500 m wide river with flow rate of 0.85 m/s. The boat can travel at rate of 2.3 m/s and is steered directly across the river. Find a) v of boat relative to observer on shore, b) distance traveled downstream while boat crosses, c) total actual distance traveled. 33

31 Relative Velocities 34

32 Relative Velocities 35

33 Relative Velocities 36

34 Addition of Velocities What if velocity is not in the direction of x or y axes? Compare: v, v x, v y, and v v x =?, v y =? x = v x t, y = v y t, etc. Example An object travels at an angle of 32 with the x-axis at a speed of 3.2 m/s. In 2.0 seconds, how far does it travel a) in the x- direction, b) in the y-direction and c) total. 37

35 Addition of Velocities Example A plane flies due west toward a destination 600 km away. The plane can fly at 200 km/hr and it points straight west and flies that fast. The total time is 3 hours, right? No: plane experiences a headwind at 24 S of E with speed 23 km/hr. How far away from the destination is the plane after 3 hours? 38