Transcript to Vectors
Vectors Cross Wind of 60 km/hr Memphis Plane traveling at 80 km/hr Houston Flying Airplanes Requires an Understanding of Vectors Steering a Boat Requires an Understanding of Vectors Campsite Across the Shore You can’t just steer your boat directly towards the campsite when the current is flowing east Boat Dock Vectors • A vector has both magnitude and direction. • Vectors are represented by arrows. An arrow will show: • magnitude of the vector by its length • direction of the vector by its angle measured counterclockwise from east. The Resultant A resultant is a single vector that acts as the sum of two or more vectors. The Resultant is the RESULT of adding two vectors If you add two displacement vectors, the result is the resultant displacement If you add two velocity vectors, the result is the resultant velocity If you add two force vectors, the result is….. • Direction of a vector is measured by its angle counterclockwise from east. • • • • East is 0 degrees North is 90 degrees West is 180 degrees South is 270 degrees • These are polar coordinates Vector Direction 2 Methods of Vector Addition 1. Graphical method – physically measuring a scale drawing for resultant and angle 2. Analytical method using trigonometry Keys to Successful Graphical Addition •Clearly define your scale factor – 100m = 1cm •Always add tip to tail Sample Problems • Using a treasure map, a pirate walks 200m east then 400m north. What single line could the pirate have taken instead? Scale: 100m = 1cm ANALYTICAL METHOD Writing Vectors in Polar Coordinates • A polar vector has both a magnitude [R] and a direction [degrees from east] • A Polar vector is written in polar coordinates [R,θ] • Example: [50 m/s, 40°] • Polar vectors must be changed to horizontal and vertical components (x,y) in order to be added. θ = 40° Trigonometric Functions • Sin θ = opposite/hypotenuse • Cos θ = adjacent/hypotenuse • Tan θ = opposite/adjacent a2 + b2 = c2 sin-1, cos-1, and tan-1 functions give θ With any 2 values, you can find all sides and all angles θ Changing from Polar to X and Y Components • Since we always measure our angles from east (0 degrees) when in polar format, we can use the following formulas: • x = R cos θ • y = R sin θ • This is Vector Resolution When you solve for the Magnitude of X and Y Example Problems • Convert the following polar coordinates to horizontal and vertical components: • 54 m/s, θ = 60 degrees • 4.5 N, θ = 235 degrees x = R cos θ y = R sin θ Example Vector Resolution • Barnard’s Star is closest to Earthafter the sun and the triple star Alpha Centauri. Barnard’s Star has a velocity of 165.2 km/s at an angle of 32.7⁰ away from its forward motion. What are the forward and sideward components of this velocity? Changing from X and Y Components to Polar • A Rectangular coordinate shows the x and y components of a vector. • Example: [38.3 m/s, 32.1 m/s] • For converting from rectangular to polar, we use the following formulas: • x2 + y2 = R2 • θ = tan-1 (y/x) Example Problems • Convert the following vertical and horizontal components, rectangular coordinates, to polar coordinates: • (36 m/s, 22 m/s) • (60 m, 35 m) x2 + y2 = R2 θ = tan-1 (y/x) Example Problems • A helicopter flies 165m east, then moves down to land 45m below. What is the magnitude and direction of the helicopter’s resultant displacement? x2 + y2 = R2 θ = tan-1 (y/x) Adding Polar Vectors • Convert each polar vector to x and y coordinates. • x1 = R1 cos θ • x2 = R2 cos θ • x3 = R3 cos θ y1 = R1 sin θ y2 = R2 sin θ y3 = R3 sin θ • Add all the x coordinates to get a single x coordinate. xtot = x1 + x2 + x3….. • Add all the y coordinates to get a single y coordinate. . ytot = y1 + y2 + y3….. • Convert to rectangular coordinates [15, 120°] • Convert to polar coordinates (4.2,-6.8) • Add [30, 115°] and [18, 255°] • An airplane flying due south at 230 km/hr experiences a crosswind from the east of 35 km/hr. What is the magnitude and direction of the resultant velocity? Quick Review • How do you know if a pair of coordinates is polar or rectangular? • Which formulas do you use to convert from polar to rectangular? • Which formulas do you use to convert from rectangular to polar? • What are the three steps for adding polar vectors? Examples • A ladder is used to climb a building that is 9 meters tall. The ladder makes an angle of 65 degrees with the ground. How long is the ladder? • A skateboard ramp is 3.3 meters long and makes an angle of 35 degrees with the ground. How high is the skateboarder when she leaves the ramp? x = R cos θ y = R sin θ • Add the following vectors graphically to find the magnitude and direction. Then, add them analytically to find magnitude and direction. • 3 km to the East. 6 km at 220° from East. 4 km at 150° from East. • A mover is loading a refrigerator into a truck with a ramp that is 8 feet long at an angle of 25 degrees with the ground. How far off the ground is the back of the truck? How far away from the truck is the ramp positioned?