Vectors Chapter 4

Flying Airplanes Requires an Understanding of 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 Boat Dock You can’t just steer your boat directly towards the campsite when the current is flowing east

Vectors A vector has both magnitude and direction. (20 m East, 40 m/s North) Vectors are represented by arrows. An arrow will show the vector’s: magnitude: Length of the vector direction: Angle of the vector measured from east. A resultant vector: One vector that acts as the sum of two or more vectors. Also called the Displacement Vector

Vector Direction Direction of a vector is measured by its angle from East. East is 0 degrees North is 90 degrees West is 180 degrees South is 270 degrees Sometimes direction is given as so many degrees _______ of ________. ( North of East, etc.) Substitute “from” for “of”.

Vectors://www.physicsclassroom.com/mmedia/vectors/vd.cfm

2 Methods of Vector Addition Graphical method – align vectors tip to tail and then draw the resultant from the tail of the first to the tip of the last. Analytical- Algebraic method Both give displacement or final position from the point of origin.

Head and Tail Winds: Tip to Tail Head Wind 20 km/hr 100 km/hr 120 km/hr 80 km/hr 100 km/hr 20 km/hr A tail wind increases velocity and a head wind decreases velocity

Pythagorean Theorem http://screencast.com/t/ZDg3NzM3Nm http://www.physicsclassroom.com/Class/vectors/u3l1b.cfm#vamethods

A man walks 11 km N, then turns and walks 11 km E A man walks 11 km N, then turns and walks 11 km E. What is his final displacement from his starting point? The final displacement is found by solving for the length of R using the pythagorean theorem. This only works for RIGHT TRIANGLES- must have a 90 degree angle!! http://www.physicsclassroom.com/Class/vectors/u3l1b.cfm#vamethods

Algebraic Method using Trig Functions Sine θ = opposite/hypotenuse Cosine θ = adjacent/hypotenuse Tangent θ = opposite/adjacent θ

Trigonometric Functions Sin θ = opposite/hypotenuse Soh Cos θ = adjacent/hypotenuse Cah Tan θ = opposite/adjacent Toa 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

Keys to Successful Graphical Addition Clearly define your scale factor Always add tip to tail

Sample Problems Using a treasure map, a pirate walks 35 m east then 15 m north. What single line could the pirate have taken instead? A pilot sets a plane’s controls at 250 km/hr to the north. If the wind blows at 75 km/hr toward the southeast, what is the plane’s resultant velocity?

Sample Problems Using a treasure map, a pirate walks 35 m east then 15 m north. What single line could the pirate have taken instead? A2 + b2 = c2 C= 38.1 m

A pilot sets a plane’s controls at 250 km/hr to the north A pilot sets a plane’s controls at 250 km/hr to the north. If the wind blows at 75 km/hr toward the southeast, what is the plane’s resultant velocity? a2 + b2 = c2

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 rectangular form (x,y) in order to be added. 50 m/s θ = 40°

Changing from Polar to Rectangular 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

Example Problems Convert the following polar coordinates to rectangular coordinates: 54 m/s, θ = 60 degrees 4.5 N, θ = 235 degrees x = R cos θ y = R sin θ

Changing from Rectangular 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 rectangular coordinates to polar coordinates: (36 m/s, 22 m/s) (-60 m, 35 m) x2 + y2 = R2 θ = tan-1 (y/x)

Warm Ups 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

Adding Polar Vectors Convert each polar vector to rectangular coordinates. x = R cos θ y = R sin θ Add all the x coordinates to get a single x coordinate. Add all the y coordinates to get a single y coordinate. Convert back to polar coordinates. x2 + y2 = R2 θ = tan-1 (y/x)

Warm Ups 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?

Warm Ups 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?

Trigonometric Functions Angle 1 Sin θ = 6/10 = .60 Cos θ = 8/10 = .80 Tan θ = 6/8 = 0.75 Angle 2 Sin θ = 8/10 = .80 Cos θ = 6/10 = .60 Tan θ = 8/6 = 1.33 Use sin-1 to get θ Sine θ = opposite/hypotenuse Cosine θ = adjacent/hypotenuse Tangent θ = opposite/adjacent

2 Methods of Vector Addition 2. Analytical Method – use Pythagorean Theorem and SOH CAH TOA Sin = Opp / Hyp Cos = Adj / Hyp Tan = Opp / Adj