VECTORS

Do you remember the difference between a scalar and a vector? Scalars are quantities which are fully described by a magnitude alone. Vectors are quantities which are fully described by both a magnitude and a direction.

Vector Representation 1. The length of the line represents the magnitude and the arrow indicates the direction. 2. The magnitude and direction of the vector is clearly labeled.

Scaling!!! The magnitude of a vector in a scaled vector diagram is depicted by the length of the arrow. The arrow is drawn precisely to length in accordance with a chosen scale.

Compass Coordinate System Direction!!! Sometimes vectors will be directed due East or due North. However we will encounter vectors in all sorts of directions and be forced to find the angle!

Compass Coordinate System Use the same scale for all vector magnitudes Δx = 30 m @ 20º E of N V = 20 m/s @ 30º W of N a = 10 m/s2@ 40º W of S F = 50 N @ 10º S of E N W E To Draw direction: Ex. 20º E of N: Start w/ North and go 20° East S Navigational System?

What factor is affecting their velocity? All these planes have the same reading on their speedometer. (plane speed not speed with respect to the ground (actual speed) What factor is affecting their velocity?

A B C

Vector Addition

Easy Adding…

Resultant Vectors The resultant is the vector sum of two or more vectors.

Vector Addition Graphical Method

Vector Addition: Head to Tail Method or (tip to tail) Select an appropriate scale (e.g., 1 cm = 5 km) Draw and label 1st vector to scale. *The tail of each consecutive vector begins at the head of the most recent vector* Draw and label 2nd vector to scale starting at the head of the 1st vector. Draw the resultant vector (the summative result of the addition of the given vectors) by connecting the tail of the 1st vector to the head of the 2nd vector. (initial to final pt.) Determine the magnitude and direction of the resultant vector by using a protractor, ruler, and the indicated scale; then label the resultant vector.

The speed of an airplane relative to the ground is affected by wind The speed of an airplane relative to the ground is affected by wind. (actual speed w/respect to the ground If we know the speed and direction of the plane and the speed and direction of the wind, we can calculate the resulting speed and direction of the airplane.

A B C Predict what will happen to the plane’s actual speed and direction for the above situations:

A B C

Velocity Vectors = = B. Headwind Tailwind (against the wind) 120 km/h 20 km/h 80 km/h 100 km/h = = 100 km/h 20km/h B. Headwind (against the wind) Tailwind (with the wind)

Velocity Vectors C. 90º crosswind Resultant Using a ruler and your scale, you can determine the magnitude of the resultant vector. Or you could use the Pythagorean Theorem. Then using a protractor, you can measure the direction of the resultant vector. Or you could use trigonometry to solve for the angle. Resultant 80 km/h 100 km/h 60 km/h

The Parallelogram Method Find the resultant force vector of the two forces below. 25 N due East, 45 N due South 25 N, East Decide on a scale!!! 51 N 59º S of E 51 N 31º E of S 45 N, South

An airplane is flying 200mph at 50o N of E. Wind velocity is 50 mph due S. What is the velocity of the plane? N 180o 0o Scale: 50 mph = 1 inch 270o

An airplane is flying 200mph at 50o N of E. Wind velocity is 50 mph due S. What is the velocity of the plane? N W E Scale: 50 mph = 1 inch S

An airplane is flying 200mph at 50o N of E. Wind velocity is 50 mph due S. What is the velocity of the plane? N 50 mph 200 mph W E VR = 165 mph @ 40° N of E Scale: 50 mph = 1 inch S

700 m/s @35 degrees E of N; 1000 m/s @ 30 degrees N of W 2. Find the resultant velocity vector of the two velocity vectors below. 700 m/s @35 degrees E of N; 1000 m/s @ 30 degrees N of W V2 Vr V1

Intro to Vectors Warm-up A bear walks one mile south, then one mile west, and finally walks one mile north. After his brisk walk, the bear ends back where he started. What color is the bear???

Vector Addition Component Method

In what direction is the leash pulling on the dog?

Breaking Down Vectors into Components

What would happen to the upward and rightward Forces if the Force on the chain were smaller?

Breaking Down Vectors into Components

Breaking Down Vectors into Components

Breaking Down Vectors into Components

Breaking Down Vectors into Components

Adding Component Vectors

Adding Component Vectors

Adding Component Vectors

Adding Component Vectors

Warm Up sohcahtoa 24.99 N 34.90º N of W VR=? V2 V1 1) Find the resultant Magnitude:__________ of the two vectors Direction:___________ Vector #1 = 20.5 N West Vector #2 = 14.3 N North Vector Diagram 34.90º N of W V2 V1 VR=?

Warm Up sohcahtoa + - + - 115.8 m South (-) 227.2 m East (+) V2 V1 Vr 2) Find the component of the resultant = 255m 27º South of East Vector # 1 _______ Direction__________ Vector # 2________ Direction__________ Vector Diagram: 115.8 m South (-) 227.2 m East (+) Conventions: V2 V1 + Vr - + -

Vector Addition: The Method of Components Practice: Find FR = Fnet =? 200 N due South, 100 N at 40º N of W Skip Draw vector diagram. (Draw axis) Resolve vectors into components using trig: Vadj = V cos θ Vopp = V sin θ 3. Sum x and y components: ΣVxi ΣVyi Redraw!! Determine resultant vector using Pythagorean’s Theorem and trig: Magnitude= √(Σ Vxi)² + (Σ Vyi)² Direction Action: θ = tan-1(opp/adj) Answer: Fnet = N @ ˚ W of S

Solve using the components method!! An airplane flies at an engine speed of 100 m/s at 50º W of S into a wind of 30 m/s at 200 E of N. What is the airplane’s resultant velocity? Solve using the components method!! How far has the plane traveled after 1 hr? a) Km b) Mph Answer: 75.52 m/s @ 28.54˚ S of W 271 km or 168.89 miles per 1 hour

You Try!!! A motor boat traveling 4.0 m/s, East encounters a current traveling 3.0 m/s, North. a. What is the resultant velocity of the motor boat? b. If the width of the river is 80 meters wide, then how much time does it take the boat to travel shore to shore? c. What distance downstream does the boat reach the opposite shore?

Another look, from a different perspective

Non-Collinear Vectors When 2 vectors are perpendicular, you must use the Pythagorean theorem. A man walks 95 km, East then 55 km, north. Calculate his RESULTANTDISPLACEMENT. Finish The hypotenuse in Physics is called the RESULTANT. 55 km, N Vertical Component Horizontal Component Start 95 km,E The LEGS of the triangle are called the COMPONENTS

BUT……what about the direction? In the previous example, DISPLACEMENT was asked for and since it is a VECTOR we should include a DIRECTION on our final answer. N W of N E of N N of E N of W W E N of E S of W S of E NOTE: When drawing a right triangle that conveys some type of motion, you MUST draw your components HEAD TO TOE. W of S E of S S

BUT…..what about the VALUE of the angle??? Just putting North of East on the answer is NOT specific enough for the direction. We MUST find the VALUE of the angle. To find the value of the angle we use a Trig function called TANGENT. 109.8 km 55 km, N q N of E 95 km,E So the COMPLETE final answer is : 109.8 km, 30 degrees North of East

What if you are missing a component? Suppose a person walked 65 m, 25 degrees East of North. What were his horizontal and vertical components? The goal: ALWAYS MAKE A RIGHT TRIANGLE! To solve for components, we often use the trig functions since and cosine. H.C. = ? V.C = ? 25 65 m

Example - = - = 23 m, E 12 m, W 14 m, N 6 m, S 20 m, N 35 m, E R A bear, searching for food wanders 35 meters east then 20 meters north. Frustrated, he wanders another 12 meters west then 6 meters south. Calculate the bear's displacement. 23 m, E - = 12 m, W - = 14 m, N 6 m, S 20 m, N 35 m, E R 14 m, N q 23 m, E The Final Answer: 26.93 m, 31.3 degrees NORTH or EAST

Example A boat moves with a velocity of 15 m/s, N in a river which flows with a velocity of 8.0 m/s, west. Calculate the boat's resultant velocity with respect to due north. 8.0 m/s, W 15 m/s, N Rv q The Final Answer : 17 m/s, @ 28.1 degrees West of North

Example A plane moves with a velocity of 63.5 m/s at 32 degrees South of East. Calculate the plane's horizontal and vertical velocity components. H.C. =? 32 V.C. = ? 63.5 m/s

Example A storm system moves 5000 km due east, then shifts course at 40 degrees North of East for 1500 km. Calculate the storm's resultant displacement. 1500 km V.C. 40 5000 km, E H.C. 5000 km + 1149.1 km = 6149.1 km R 964.2 km q The Final Answer: 6224.14 km @ 8.91 degrees, North of East 6149.1 km

Add vectors that are NOT perpendicular If the original displacement vectors do not form a right triangle 1. Resolve each vector into its x- and y-components 2. Find the sum of the x- and y-components 3. Use the Pythagorean Theorem to find the magnitude of the resultant 4. Use the tangent function to find the direction of the resultant

Adding non-perpendicular vectors

Adding non-perpendicular vectors

Adding non-perpendicular vectors

Practice #1 A hiker walks 27.0 km from her base camp at 35 south of east. The next day, she walks 41.0 km in a direction 65 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. Answer ~ 45 km at 29 degrees N of E