Vector Direction

• A vector quantity is a quantity that is fully described by both magnitude and direction.

• On the other hand, a scalar quantity is a quantity that is fully described by the magnitude.

• The emphasis is to understand vectors in order to understand motion in 2D.

• Examples of vector quantities are displacement, velocity, acceleration, and force.

• Vector diagrams depict vector by use of an arrow drawn to scale in a specific direction.

Vector Diagrams

• Scale clearly listed

• A vector arrow (head and tail) drawn in a specified direction.

• The magnitude and direction of the vector is clearly labeled.

• Ex: magnitude 20 m and direction is 30 degrees West of North

Describing Directions

• Vectors can be directed due East, due West, due South, and due North.

• But some vectors are directed northeast (at a 45 degree angle) and some vectors are even directed northeast, yet more north than east.

2 Conventions to Use

• The direction of a vector is often expressed as an angle of rotation of the vector about its tail from east, west, north, or south.

• Ex: 40 degrees North of West

65 degrees East of South

• The direction of the vector is often expressed as a counterclockwise angle of rotation of the vector about its tail from due East.

• Ex: 30 degrees

160 degrees

Representing Magnitude

• Magnitude of a vector is depicted by the length of the arrow.

• The arrow is drawn a precise length to a chosen scale.

• Ex: scale: 1 cm = 5 miles

• Vector of 20 miles would be 4 cm

You Try It!!!! (Paden)

Vector Addition

• Two vectors can be added together to determine the result (or resultant).

• Two methods for adding vectors:

• Pythagorean theorem and trig functions

• Head to tail method using a scaled vector diagram

Pythagorean Theorem

• Useful for adding only 2 vectors which make a right angle to each other.

• Not applicable when adding 2 vectors together that do not make a right angle with each other.

• Mathematical equation which relates the length of the sides of a right triangle to the length of the hypotenuse of a right triangle.

a2 + b2 = c2

ex: Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric’s resulting displacement.

• R = 15.6 km

More practice

• 10 Km, North + 5 Km, West

• 30 Km, West + 40 Km, South

• 1. R = 11.2 km

• 2. R = 50 km

Trig functions to determine Direction

• The direction of a resultant vector can often be determined by use of trigonometric functions.

• SOH CAH TOA is a mnemonic which helps remember the meaning of the 3 common trig functions - Sine, cosine, and tangent functions

• Sine = opposite/hypotenuse

• Cosine = adjacent/hypotenuse

• Tangent = opposite/adjacent

• 11 Km, North + 11 Km, East

• Find the direction of the hiker’s displacement.

• R= 45 degrees

Practice

• 10 Km, North + 5 Km, West

• Find the magnitude and direction of the resultant vector.

• R = 11.2 km, 116.6 degrees

Practice

• 30 Km, West + 40 Km, South

• Find the magnitude and direction of the resultant.

• R = 50, 233.1 degrees

Practice 3A

• An archeologist climbs the Great Pyramid in Giza, Egypt. If the pyramid’s height is 136 m and its width is 2.30 X 102 m, what is the magnitude and the direction of the archeologist’s displacement while climbing from the bottom of the pyramid to the top?

Answer

• R = 178 m

• θ = 49.8 degrees

Head to Tail Method

• The head to tail method is employed to determine the vector sum or resultant when two or more vectors are drawn to scale.

• Involves drawing a vector to scale at a beginning position. Where the head of the first vector ends, the tail of the second vector begins, etc.

• Once all of the vectors have been drawn head to tail, the resultant is then drawn from the tail of the first vector to the head of the last vector (from start to finish).

• Then, the length and direction of the resultant can be measured and determined using a ruler and protractor.

Practice

• Scale 1 cm = 5 m

• Add the vectors:

• 20 m, 45 degrees

• 25 m, 300 degrees

• 15 m, 210 degrees

• R= 22 m, 310 degrees

• The order of adding vectors doesn’t change the magnitude or direction of the resultant.

Vector Components

• A vector resultant can be transformed into two parts (x and y).

• For example, a vector pointed northwest can be directed as having a northward vector and a westward vector.

• Each part of a 2D vector is known as a component.

Resolving Vectors

• The influence of the 2 components is equivalent to the influence of a single 2D vector.

Practice 3B

• Find the component velocities of a helicopter traveling 95 km/h at an angle of 35° to the ground.

Answer

• Y = 54 km/h

• X = 78 km/h

Projectile Motion

• Objects that are thrown or launched into the air and are subject to gravity are called projectiles.

• Ex: throwing a ball, arrows projected through the air

Path of a Projectile

• The path of a projectile forms a curve called a parabola

• If a projectile has a horizontal velocity, it will have horizontal velocity throughout the flight.

• For our samples and problems, the horizontal velocity of a projectile will be considered constant.

• With air resistance, the horizontal velocity would not be constant.

• With air resistance, an object would travel along a shorter path, which would not be a parabola.

• Projectile motion is free fall with an initial horizontal velocity.

• A ball dropped straight down has no initial velocity.

• If air resistance is disregarded, one ball dropped and one launched horizontally will hit the ground at the same time.

• Projectiles can be analyzed as having both horizontal and vertical components of motion (2D)

• In any time interval, a launched ball undergoes the same vertical displacement as a ball that falls straight down, thus hitting the ground at the same time.

• The horizontal acceleration dimension (x) is zero but the acceleration in the vertical dimension (y) will be equal to acceleration due to gravity (g).

• We will analyze problems in each dimension separately and list givens for the x and y components separately similar to our one dimensional acceleration problems.

Vertical Motion of a Projectile that falls from rest

• Vy,f = -gΔt

• Vy,f2 = -2gΔy

• Δy = -1/2g(Δt)2

Horizontal Motion of a Projectile

• Vx = vx,i = constant

• Δx = vxΔt

• To find the velocity of a projectile at any point during its flight, find the vector sum of the components of the velocity at that point. Use the pythagorean theorem and tangent function to find the direction of the velocity.

Sample Problem 3D

• The Royal Gorge Bridge in Colorado rises 321 m above the Arkansas River. Suppose you kick a little rock horizontally off the bridge. The rock hits the water such that the magnitude of its horizontal displacement is 45.0 m. Find the speed at which the rock was kicked.

• Givens:

• Δy = -321 m

• Δx = 45 m

• ay = g = 9.81 m/s2

• Vi = ?

Formulas

• Δx = vxΔt

• Δy = -1/2g(Δt)2

• Answer = 5.56 m/s

Projectiles launched at an angle

• If a projectile is launched at an angle, then it has an initial vertical component as well as a horizontal component of velocity.

• We will use sine and cosine to find the horizontal and vertical components of the initial velocity.

• vx,i = vi(cos θ) and vy,i = vi(sin θ)

Projectiles Launched at an Angle

• vx = vi(cos θ) = constant

• Δx = vi(cos θ)Δt

• vy,f = vi(sin θ) – gΔt

• vy,f2 = vi

2(sin θ)2 – 2gΔy

• Δy = vi(sin θ)Δt – 1/2g(Δt)2

Sample 3E

• A zookeeper finds an escaped monkey hanging from a light pole. Aiming her tranquilizer gun at the monkey, the zookeeper kneels 10.0 m from the light pole, which is 5.00 m high. The tip of her gun is 1.00 m above the ground. The monkey tries to trick the zookeeper by dropping a banana, then continues to hold onto the light pole. At the moment the monkey releases the banana, the zookeeper shoots. If the tranquilizer dart travels at 50.0 m/s, will the dart hit the monkey, the banana, or neither one?

• Answer: the banana is 4.77 m above the ground

• The dart is 4.76 m above the ground so the dart hits the banana.