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Kinematics in Two Dimensions. Section 1: Adding Vectors Graphically. Adding Vectors Graphically Remember vectors have magnitude (length) and direction. When you add vectors you must maintain both magnitude and direction This information is represented by an arrow (vector).
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Adding Vectors Graphically • Remember vectors have magnitude (length) and direction. • When you add vectors you must maintain both magnitude and direction • This information is represented by an arrow (vector)
A vector has a magnitude and a direction • The length of a drawn vector represents magnitude. • The arrow represents the direction Larger Vector Smaller Vector
Graphical Representation of Vectors • Given Vector a: Draw 2a Draw -a
Problem set 1: • Which vector has the largest magnitude? • What would -b look like? • What would 2 c look like? a c b
Vectors • Three vectors a c b
a c b • When adding vectors graphically, align the vectors head-to-tail. • This means draw the vectors in order, matching up the point of one arrow with the end of the next, indicating the overall direction heading. • Ex. a + c • The starting point is called the origin c a origin
a c b • When all of the vectors have been connected, draw one straight arrow from origin to finish. This arrow is called the resultant vector. c a origin
a c b • Ex.1 Draw a + b
a c b • Ex.1 Draw a + b Resultant origin
a c b • Ex. 2 Draw a + b + c
a c b • Ex. 2 Draw a + b + c Resultant origin
a c b • Ex. 3 Draw 2a – b – 2c
a c b • Ex. 3 Draw 2a – b – 2c origin Resultant
Vector Direction Naming • How many degrees is this? N W E S
Vector Direction Naming • How many degrees is this? N 90º W E S
Vector Direction Naming • What is the difference between 15º North of East and 15 º East of North? N W E S
Vector Direction Naming • What is the difference between 15º North of East and 15º East of North? (can you tell now?) N N W E W E S S 15º North of East 15º East of North
Vector Direction Naming N 15º W S 15º North of what?
Vector Direction Naming N 15º W E S 15º North of East
15º W E S 15º East of What?
N 15º W E S 15º East of North
___ of ___ N E This is the baseline. It is the direction you look at first This is the direction you go from the baseline to draw your angle
Describing directions • 30º North of East • East first then 30º North • 40º South of East • East first then 30º South • 25º North of West • West first then 30º North • 30º South of West • West first then 30º South
Problem Set #2 (Name the angles) 30º 45º 20º 30º 20º
Intro: Get out your notes b • Draw the resultant of a – b + c 2. What would you label following angles a. b. 3. Draw the direction 15º S of W a c 28º 18º
Section 3: How do you add vectors mathematically (not projectile motion)
The Useful Right Triangle • Sketch a right triangle and label its sides c: hypotenuse a: opposite Ө b: adjacent The angle
The opposite (a) and adjacent (b) change based on the location of the angle in question • The hypotenuse is always the longest side Ө c: hypotenuse b: adjacent a: opposite
The opposite (a) and adjacent (b) change based on the location of the angle in question • The hypotenuse is always the longest side Ө c: hypotenuse b: adjacent a: opposite
To figure out any side when given two other sides • Use Pythagorean Theorem a2 + b2 =c2 c: hypotenuse a: opposite Ө b: adjacent The angle
Sometimes you need to use trig functions c: hypotenuse a: opposite Ө a: adjacent Opp Hyp Opp Adj Sin Ө = _____ Tan Ө = _____ Adj Hyp Cos Ө = _____
Sometimes you need to use trig functions c: hypotenuse a: opposite Ө a: adjacent Opp Hyp Opp Adj Sin Ө = _____ Tan Ө = _____ SOH CAH TOA Adj Hyp Cos Ө = _____
More used versions Opp Hyp Sin Ө = _____ Opp = (Sin Ө)(Hyp) Adj Hyp Cos Ө = _____ Adj = (Cos Ө)(Hyp) Opp Adj Opp Adj Ө = Tan-1 _____ Tan Ө = _____
To resolve a vector means to break it down into its X and Y components. Example: 85 m 25º N of W • Start by drawing the angle 25º
To resolve a vector means to break it down into its X and Y components. Example: 85 m 25º N of W • Start by drawing the angle • The magnitude given is always the hypotenuse 85 m 25º
To resolve a vector means to break it down into its X and Y components. Example: 85 m 25º N of W • this hypotenuse is made up of a X component (West) • and a Y component (North) 85 m • North 25º West
In other words: I can go so far west along the X axis and so far north along the Y axis and end up in the same place finish finish 85 m • North origin origin 25º West
If the question asks for the West component: Solve for that side • Here the west is the adjacent side Adj = (Cos Θ)(Hyp) 85 m 25º West or Adj.
If the question asks for the West component: Solve for that side • Here the west is the adjacent side Adj = (Cos Θ)(Hyp) Adj = (Cos 25º)(85) = 77 m W 85 m 25º West or Adj.
If the question asks for the North component: Solve for that side • Here the north is the opposite side Opp = (Sin Θ)(Hyp) 85 m • North • or • Opp. 25º
If the question asks for the North component: Solve for that side • Here the west is the opposite side Opp = (Sin Θ)(Hyp) Opp = (Sin 25º)(85) = 36 m N 85 m • North • or • Opp 25º
Resolving Vectors Into Components • Ex 4a. Find the west component of 45 m 19º S of W
Resolving Vectors Into Components • Ex 4a. Find the west component of 45 m 19º S of W
Remember the wording. These vectors are at right angles to each other. 5 m/s forward 5 m/s Redraw and it becomes 30 m/s Hypotenuse = Resultant speed velocity = 30 m/s down Right angle