VECTORS

BIG IDEA: Horizontal and vertical motions of an object are independent of one another

Scalars – a quantity described by a magnitude (number) ; How much How many No direction! Ex: Temperature 23°C Time 37 s Speed 60 km/hr Distance 5 miles

Vector – a quantity described by magnitude and direction Ex:displacement velocity acceleration Force Magnitude - the length of the vector; the amount and the unit of a quantity

Vectors are represented 2 ways: 1.Graphically – drawing an arrow; The arrow is drawn on a coordinate system (x-y axis)  magnitude - arrow length is proportional  direction – the way the arrow points Defined by length and direction, not its starting point!!

Chester Dunsville Baker Acme E N 100 km SE 200 km East

Chester Dunsville Baker Acme E N 200 km East ?

2. Mathematically – vectors are represented by either: bold print symbol Ex: v, F arrow over the symbol Ex: v, F Magnitude only of the vector is in italics ex: v, F

Polar Notation: V = (V, Θ ) defines vectors by magnitude (always positive) angle (positive or negative) Ex: 3 km due north (3 km=magnitude, north=direction) A physicist would say: 3km directed at 90 degrees

Angles: positive = counterclockwise from x-axis negative = clockwise from x-axis Ex: 90° angle = -270° angle

Rectangular Notation: V = (V x, V y ) defines vectors by its components Cartesian coordinates x component= horizontal dimension y component = vertical dimension components are scalar direction is indicated by sign Can have z direction; it would be “to” or “from” you. Altitude is an example

Unit Vector Notation: V = (V x i ̂, V y j ̂) defined by each of its components in the direction of a “unit vector” Similar to rectangular notation

“unit vector” – a vector with a magnitude of one in a particular direction (no units) Allows us to mathematically +, - vectors Multiply the unit vector by a scalar (number with units)

Ex: displacement unit vector î would represent a 1 meter vector on the x-axis. Ex: velocity unit vector ĵ represents a 1 m/s vector on the y-axis.

2D: A = (A x ) î +(A y ) ĵ 3D: A = (A x ) î +(A y ) ĵ +(A y ) ẑ

Triangle sides  Given a right triangle, and reference angle  : opposite hypotenuse adjacent For a given angle  the ratio of any two sides is always the same number. 

Sine function  Given a right triangle, and reference angle A: sin A = A opposite hypotenuse The sin function specifies these two sides of the triangle, and they must be arranged as shown.

Cosine function The next trig function you need to know is the cosine function (cos): cos A = A adjacent hypotenuse Cosine Function

Tangent function The last trig function you need to know is the tangent function (tan): tan A = A adjacent opposite

Resolving Vectors  To resolve a vector means to find its components; a vector's components are the lengths of the vector along specified directions.  A vector may be defined by either its length and its direction angle (polar notation) or by its x component and its y component (rectangular notation).

Vectors can be resolved 2 ways: 1. Graphically – components determined by drawing a) triangle → head to tail indicating sequence b) rectangle → to show components on the axes indicating a continuous property.

2. Analytically (mathematically) – components determined by: r x = r cosθ andr y = r sinθ. Where θ = angle from the +x axis

© 2014 Pearson Education, Inc. Vectors in Physics  In a right triangle, the cosine of an angle is defined as the length of the adjacent side over the length of the hypotenuse.  The sine of an angle is defined as length of the opposite side over the length of the hypotenuse.  Therefore, in the figure below, r x = r cosθ and r y = r sinθ.

Vector resolution: Graphical d x = ______ cm = ______ km d y = ______ cm = ______ km

A) Express the vector in polar notation Example 1: _____________________ B) Calculate the components C) Express the vector in rectangular notation D) Express the vector in unit vector notation

Vector resolution: Graphical v x = ______ cm = ______ m/s v y = ______ cm = ______ m/s

A) Express the vector in polar notation Example 2: _____________________ B) Calculate the components C) Express the vector in rectangular notation D) Express the vector in unit vector notation

A vector can be converted from rectangular (V X,V Y ) to polar notation (V,  ) using the equations: V = √ V X 2 + V Y 2  = tan -1 ( V Y /V X )

Note: If the angle ends in the II or III quadrant (left side of graph), add 180 degrees to your calculator answer. If the angle is in the I or IV quadrant, the calculator answer is correct.

V = √ V X 2 + V Y 2  = tan -1 ( V Y /V X )

 Vector addition - process used to find the total effect of a set of vectors.  A resultant - single vector which has the same effect as two or more other vectors acting together.  An equilibrant is the single vector which would cancel out one or more other vectors

 Vectors may be added by placing them head to tail.  Resultant vector is from the tail of the 1 st vector to the tip of the last vector.  Resultant vector shows displacement

 Vectors are unchanged if moved to another location as long as their length and direction remain unchanged.  All vectors in the figure below are identical even though they are in different locations.

 moving the arrows has no effect on the vector sum. = =

Example : A wide receiver is running a pass pattern. He runs directly west for 15.0 feet, then 20° west of north for 15.0 feet, then 30° south of west for 10.0 feet. A)Determine the distance that he traveled. B) Determine his displacement graphically.

 Vectors can be added graphically but limited by how accurately vectors are drawn and measured.

 Vectors can be subtracted graphically.  The negative of a vector is represented by a vector of the same length as the original, but pointing in the opposite direction.  To subtract one vector from another, reverse the direction of one vector and add it to the other vector.

C To add or subtract vectors given in component form, just add or subtract the components separately.

C Example 1 A boat has a velocity B = (3i + 4j) m/s in still water. The river’s current has a velocity C = (2i – 1j) m/s. Calculate the resulting velocity V of the boat (V = B + C)

Example 2 An aircraft carrier identifies the positions of two jets, relative to the ship, in unit vectors. Calculate the displacement vector D from jet B to jet A (D = A – B) B A x y z Jet A = ( - 250i + 620j + 740k) m Jet B = ( + 125i - 240j + 319k) m

 the motion of one object relative to another object. Velocity of train worker relative to the ground: 1.2 m/s m/s = 16.2 m/s

Velocity of train worker relative to the ground: –1.2 m/s m/s = 13.8 m/s

 Vector representation of the scenario:  Using subscripts, as was done with the case of the worker on the train, helps clarify relative motion.

If two motions do not occur along a straight line, then the approach applied to the worker on the train still applies: The velocity of one object relative to another is found by adding the velocity vectors.

 In the figure below, the worker climbs up a ladder on a moving train. His velocity with respect to the ground is found by adding vectors head to tail.

 Ex: Train worker climbs the ladder with a speed of 0.20 m/s as the train coasts forward at 0.70 m/s. Find the speed and direction of the worker relative to the ground:

The relative velocity equation is:V OA = V OB + V BA A boat’s engine propels it horizontally at 1.50 m/s, while the river current flows at 3.00 m/s. What is the velocity of the boat relative to the shore? Give the direction as the angle from the river current. TRIANGLE