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Kinematics in Two Dimensions Slide 2 Section 1: Adding Vectors Graphically Slide 3 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) Slide 4 A vector has a magnitude and a direction The length of a drawn vector represents magnitude. The arrow represents the direction Larger VectorSmaller Vector Slide 5 Graphical Representation of Vectors Given Vector a: Draw 2aDraw -a Slide 6 Problem set 1: 1.Which vector has the largest magnitude? 2.What would -b look like? 3.What would 2 c look like? a b c Slide 7 Vectors Three vectors a b c Slide 8 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 a b c a c origin Slide 9 When all of the vectors have been connected, draw one straight arrow from origin to finish. This arrow is called the resultant vector. a c origin a b c Slide 10 Ex.1 Draw a + b a b c Slide 11 origin Resultant a b c Slide 12 Ex. 2 Draw a + b + c a b c Slide 13 origin Resultant a b c Slide 14 Ex. 3Draw 2a b 2c a b c Slide 15 origin Resultant a b c Slide 16 Section 2: How do you name vector directions? Slide 17 Vector Direction Naming How many degrees is this? W S E N Slide 18 Vector Direction Naming How many degrees is this? W S E N 90 Slide 19 Vector Direction Naming What is the difference between 15 North of East and 15 East of North? W S N E Slide 20 Vector Direction Naming What is the difference between 15 North of East and 15 East of North? (can you tell now?) W S E N 15 North of East W S E N 15 East of North Slide 21 Vector Direction Naming W S N 15 North of what? 15 Slide 22 Vector Direction Naming W S N E 15 15 North of East Slide 23 W S E 15 East of What? 15 Slide 24 W S E N 15 East of North 15 Slide 25 ___ of ___ 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 N E Slide 26 Describing directions 30 North of East East first then 30 North 40 South of East East first then 30 South 25 North of West West first then 30 North 30 South of West West first then 30 South Slide 27 Problem Set #2 (Name the angles) 20 30 45 Slide 28 Intro: Get out your notes 1.Draw the resultant of a b + c 2. What would you label following angles a.b. 3. Draw the direction 15 S of W a b c 28 18 Slide 29 Slide 30 Section 3: How do you add vectors mathematically (not projectile motion) Slide 31 The Useful Right Triangle Sketch a right triangle and label its sides a: opposite b: adjacent c: hypotenuse The angle Slide 32 The opposite (a) and adjacent (b) change based on the location of the angle in question The hypotenuse is always the longest side a: opposite b: adjacent c: hypotenuse Slide 33 The opposite (a) and adjacent (b) change based on the location of the angle in question The hypotenuse is always the longest side a: opposite b: adjacent c: hypotenuse Slide 34 To figure out any side when given two other sides Use Pythagorean Theorem a 2 + b 2 = c 2 a: opposite b: adjacent c: hypotenuse The angle Slide 35 Sometimes you need to use trig functions a: opposite a: adjacent c: hypotenuse Sin = _____ Cos = _____ Tan = _____ Opp Hyp Adj Hyp Opp Adj Slide 36 Sometimes you need to use trig functions a: opposite a: adjacent c: hypotenuse Sin = _____ Cos = _____ Tan = _____ Opp Hyp Adj Hyp Opp Adj SOH CAH TOA Slide 37 More used versions Sin = _____ Cos = _____ Tan = _____ Opp Hyp Adj Hyp Opp Adj Opp = (Sin )(Hyp) Adj = (Cos )(Hyp) = Tan -1 _____ Opp Adj Slide 38 To resolve a vector means to break it down into its X and Y components. Example: 85 m 25 N of W Start by drawing the angle 25 Slide 39 To resolve a vector means to break it down into its X and Y components. Example: 85 m 25 N of W Start by drawing the angle The magnitude given is always the hypotenuse 25 85 m Slide 40 To resolve a vector means to break it down into its X and Y components. Example: 85 m 25 N of W this hypotenuse is made up of a X component (West) and a Y component (North) 25 85 m West North Slide 41 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 85 m 25 West North origin finish Slide 42 If the question asks for the West component: Solve for that side Here the west is the adjacent side Adj = (Cos )(Hyp) 25 85 m West or Adj. Slide 43 If the question asks for the West component: Solve for that side Here the west is the adjacent side Adj = (Cos )(Hyp) Adj = (Cos 25)(85) = 77 m W 25 85 m West or Adj. Slide 44 If the question asks for the North component: Solve for that side Here the north is the opposite side Opp = (Sin )(Hyp) 25 85 m North or Opp. Slide 45 If the question asks for the North component: Solve for that side Here the west is the opposite side Opp = (Sin )(Hyp) Opp = (Sin 25)(85) = 36 m N 25 85 m North or Opp Slide 46 Resolving Vectors Into Components Ex 4a. Find the west component of 45 m 19 S of W Slide 47 Resolving Vectors Into Components Ex 4a. Find the west component of 45 m 19 S of W Slide 48 Ex 4a. Find the south component of 45 m 19 S of W Slide 49 Slide 50 5 m/s forward velocity = 30 m/s down Hypotenuse = Resultant speed 5 m/s 30 m/s Remember the wording. These vectors are at right angles to each other. Redraw and it becomes Right angle Slide 51 Section 4 (Solving for a resultant) Ex. 6Find the resultant of 35.0 m, N and 10.6 m, E. Start by drawing a vector diagram Then draw the resultant arrow Slide 52 Ex. 6Find the resultant of 35.0 m, N and 10.6 m, E. Then draw the resultant vector and angle The angle you find is in the triangle closest to the origin Slide 53 Now we use Pythagorean theorem to figure out the resultant (hypotenuse) Slide 54 Then inverse tangent to figure out the angle The answer needs a magnitude, angle, and direction Slide 55 Slide 56 Problem Set 3: Resolve the following vectors 1)48m, S and 25m, W 2)12.5m, S and 78m, N Slide 57 Problem Set #3 1)48m, S and 25m, W Slide 58 Slide 59 Section 4: How does projectile motion differ from 2D motion (without gravity)? Slide 60 Projectile Motion Slide 61 Projectile- Object that is launched by a force and continues to move by its own inertia Trajectory- parabolic path of a projectile Slide 62 Projectile motion involves an object moving in 2D (horizontally and vertically) but only vertically is influenced by gravity. The X and Y components act independently from each other and will be separated in our calculations. Slide 63 X and Y are independent X axis has uniform motion since gravity does not act upon it. Slide 64 X and Y are Independent Y axis will be accelerated by gravity -9.8 m/s 2 Slide 65 The equations for uniform acceleration, from unit one, can be written for either x or y variables: Slide 66 If we push the ball harder, giving it a greater horizontal velocity as it rolls off the table, the ball would take _________ time to fall to the floor. Slide 67 Horizontal and vertical movement is independent If we push the ball harder, giving it a greater horizontal velocity as it rolls off the table, the ball would: Y axis: take the same time to fall to the floor. X axis: It would just go further. Slide 68 Solving Simple Projectile Motion Problems You will have only enough information to deal with the y or x axis first You cannot use the Pythagorean theorem since X and Y-axes are independent Time will be the key: The time it took to fall is the same time the object traveled vertically. d x = (v x )(t) is the equation for the horizontal uniform motion. If you dont have 2 of three x variable you will have to solve for t using gravity and the y axis Slide 69 Equations Solving Simple Projectile Motion Problems Do not mix up y and x variables d y height (this is negative if falling down) d x range (displacement x) Slide 70 For all projectile motion problems Draw a diagram Separate the X and Y givens Something is falling in these problems X GivensY Givens d X =a = -9.8 m/s v X = t = Slide 71 Example Problem 8 A stone is thrown horizontally at 7.50 m/s from a cliff that is 68.4 m high. How far from the base of the cliff does the stone land? Slide 72 Write out your x and y givens separately A stone is thrown horizontally at 7.50 /s from a cliff that is 68.4 m high. How far from the base of the cliff does the stone land? X givensY givens Slide 73 A stone is thrown horizontally at 7.50 m/s from a cliff that is 68.4 m high. How far from the base of the cliff does the stone land? X givensY givens Slide 74 Ex. 9 A baseball is thrown horizontally with a velocity of 44 m/s. It travels a horizontal distance of 18, to the plate before it is caught. a)How long does the ball stay in the air? b)How far does it drop during its flight? Slide 75 A baseball is thrown horizontally with a velocity of 44 m/s. It travels a horizontal distance of 18, to the plate before it is caught. How long does the ball stay in the air? How far does it drop during its flight? X givensY givens Slide 76 A baseball is thrown horizontally with a velocity of 44 m/s. It travels a horizontal distance of 18, to the plate before it is caught. How long does the ball stay in the air? How far does it drop during its flight? X givensY givens Slide 77 A baseball is thrown horizontally with a velocity of 44 m/s. It travels a horizontal distance of 18, to the plate before it is caught. How long does the ball stay in the air? How far does it drop during its flight? X givensY givens Slide 78 Example 10 1. What is the initial vertical velocity of the ball? v oY = 0 m/s Same as if it was dropped from rest Slide 79 2. How much time is required to get to the ground? Since v oY = 0 m/s use 2(-10) -10 t = 1.4 s Slide 80 3. What is the vertical acceleration of the ball at point A? a oY = -10 m/s 2 always Slide 81 4. What is the vertical acceleration at point B? a oY = -10 m/s 2 always Slide 82 5. What is the horizontal velocity of the ball at point C? v X = 5 m/s (does not change) Slide 83 6. How far from the edge of the cliff does the ball land in the x plane? X givens v X = 5 m/s t = 1.4 d x = ? d x = (v X )(t) d x = (5)(1.4) = 7m Slide 84 What will happen if drops a package when the plane is directly over the target? Slide 85 The package has the same horizontal velocity as the plane and would land far away from the target. Slide 86 Section 5: What do you do different if you have projectile motion and V 0Y is not equal to 0 Slide 87 Projectile Motion Concepts Arrows represent x and y velocities (g always = 10 m/s 2 down) Slide 88 Key points in a projectiles path When a projectile is at its highest point its v fy = 0. This means it stopped moving up. Use v fy = 0 in a question that asks you to predict the vertical distance (how high) V oY = 0 m/s Slide 89 Key points in a projectiles path If an object lands at the same height its vertical velocities final magnitude equals its initial but is in the opposite direction (down) V oY = +30 m/s V fY = -30 m/s Slide 90 V oY = +30 m/s V fY = -30 m/s The time it takes to rise to the top equals the time it takes to fall. Givens to use to find time to the top: V oY = +30 m/s V fY = 0 m/s Givens to use to find time of entire flight: V oY = +30 m/s V fY = -30 m/s Slide 91 Key points in a projectiles path If a projectile lands below where it is launched the v fy magnitude will be greater than v oy and in the reverse direction Slide 92 It stays constant during the entire flight (no forces acting in the x direction) It accelerates (the force of gravity is pulling it to Earth) Ex. 11 A ball of m = 2kg is thrown from the ground with a horizontal velocity of 5 m/s and rises to a height of 45 m. 1.What happens to velocity in the x direction? Why? 2.What happens to velocity in the y direction? Why? Slide 93 3. Where is the projectile traveling the fastest? Why? 4. Where is the projectile traveling the slowest? What is its speed at this point? 5. Where is the acceleration of the projectile the greatest? Why? A and E (has the largest V Y component) C (has only V X component V Y =0) All (g stays -10m/s 2 ) Slide 94 6. What is the acceleration due to gravity at point B? 7. What is the initial vertical velocity the ball is thrown with? All (g stays -10m/s 2 ) Must solve a Y = -10m/s 2 d = 45m v o = ? V f = 0 v f 2 = v o 2 + 2ad v o = (v f 2 2ad) v o = (0 2 2(-10)(45) v o = 30 m/s up Slide 95 8. What is the time required to reach point C if thrown from the ground? Must solve Y givens a Y = -10m/s 2 v o = +30 m/s V f = 0 m/s t = ? V fY = -30 m/s V oY = +30 m/s v f = v o + at t = (v f v o ) a t = (0 30) -10 t = 3 s Slide 96 9. From point C, what is the time needed to reach the ground? Same as time it took to get to the top t = 3 s Slide 97 10. What is the horizontal velocity at point A? 11. What is the horizontal acceleration of the ball at point E? 5 m/s (never changes horizontally while in the air) a x = 0 m/s 2 (they asked for acceleration no horizontal acceleration) v x stays 5 m/s Slide 98 12. What is the vertical acceleration due to gravity at point E? a Y = -10 m/s 2 Slide 99 13. How far in the x plane (what is the range) does the ball travel? Must solve X givens t= 6 seconds total in air v X = 5 m/s d X = ? d X = (v X )(t) d X = (5)(6) = 30 m Slide 100 14. What would happen to the problem if the objects mass was 16 kg Nothing would change. The acceleration due to gravity is the same for any mass Slide 101 More complex projectile motion problems require you separate a resultant velocity vector into its components using soh-cah- toa A stone is thrown at 25 m/s at a 40 angle with the horizon. Start with the finding the v x and v oy Then solve the problem like we have v oy Slide 102 Example The punter on a football team tries to kick a football with an initial velocity of 25.0 m/s at an angle of 60.0 above the ground, what range (d x ) does it travel? Slide 103 Example The punter on a football team tries to kick a football with an initial velocity of 25.0 m/s at an angle of 60.0 above the ground, what range (d x ) does it travel? Slide 104 Example The punter on a football team tries to kick a football with an initial velocity of 25.0 m/s at an angle of 60.0 above the ground, what range (d x ) does it travel? Slide 105 Slide 106 Slide 107 45 will get you the greatest range Range is d x Horizontal displacement Slide 108 Besides 45, two sister angles will give you the same range 45 is would give you the greatest d x Any similar degree variation on either side of 45 would give you the same d x Ex these would give you the same d x. 40 and 50 30 and 60 15 would give you the same range as what? ___________ Slide 109 Classwork/Homework 2D motion Packet Pg 2 Exercise 10-16 Honors Addition: Book Pg 79 #16,17,18,20,22,27,31 Try 35