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## Kinematics in Two Dimensions

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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