Vectors Graphical Method General discussion. Vector - A quantity which has magnitude and direction. Velocity, acceleration, Force, E Field, Mag Field, calar - A quantity which has magnitude only. (temp, pressure, time) This chapter: We mainly deal with Displacement D and Velocity = v Discussion is valid for any vector! 2 vectors NOT along the same line: D 1 = 10 km E, D 2 = 5 km N. Resultant =? If D 1 is perpendicular to D 2 D R = 11.2 km Adding vectors in same direction: Graphical Method Vector V is 2.50 cm long, scale is 1 cm = 12.2 m/s, find magnitude. V EEM IMPLE? = 30.5 m/s Graphical Method For 2 vectors NOT along same line, adding is more complicated: Example: D 1 = 10 km East, D 2 = 5 km North. What is the resultant (final) displacement? 2 methods of vector addition: Graphical (2 methods of this also!) Analytical (TRIGONOMETRY) Graphical Method (TIP TO TAIL) V = V 1 + V 2 1. Draw V 1 & V 2 to scale. 2. Place tail of V 2 at tip of V 1 3. Draw arrow from tail of V 1 to tip of V 2 This arrow is the resultant V (measure length and the angle it makes with the x-axis) 1
Graphical Method (TIP TO TAIL) Graphical Method (Parallelogram) Tip Tail Graphical Method (TIP TO TAIL) Consider the vectors A and B. Find A + B. A B A B A B C = A + B We can arrange the vectors as we want, as long as we maintain their length and direction!! Graphical Method econd graphical method of adding vectors (parallelogram). V = V 1 + V 2 1. Draw V 1 & V 2 to scale from common origin. 2. Construct parallelogram using V 1 & V 2 as 2 of the 4 sides. Resultant V = diagonal of parallelogram from common origin (measure length and the angle it makes with the x-axis) Graphical Method Adding (3 or more) vectors V = V 1 + V 2 + V 3 Correct Graphical Method Tip To Tail Parallelogram 2
Graphically determine the resultant of the following three vector displacements: (1) 34 m, 25º north of east; (2) 48 m, 33º east of north; and (3) 22 m, 56º west of south. The vectors for the problem are drawn approximately to scale. The resultant has a length of 58 m and a direction 48 o north of east. If you actually measured, the actual resultant should be 57.4 m at 47.5 o north of east. 22 ubtraction of Vectors 48 34 ubtraction of Vectors Graphical Method (TIP TO TAIL) B Deck of Card Activity 0 degrees from x axis A R = A + B R B A R = A - B A 90 degrees from x axis 180 degrees from x axis R -B 270 degrees from x axis ubtraction of Vectors First, define the negative of a vector: - V vector with the same magnitude (size) as V but with opposite direction. Math: V + (- V) 0 For 2 vectors, A - B A + (-B) 3
To do the graphical method we have to measure with ruler and protractor. Takes a lot of time The magnitude (length) of r is found using the Pythagorean theorem: We don t always have an graphic designer on staff to draw our vectors for us. r y o instead we resolve our vectors into its components (x,y) x Decomposition Example V is resolved into components: V x & V y V V x + V y (V x x axis, V y y axis) A coast guard cutter has taken a heading of 30 north of east to find a missing diver. Break up into it s X and Y d = displacement 500 m, 30º N of E Trig Functions to Decompose Vector determine the magnitude and direction of the resultant vector The direction is given by an angle of IN PROJECTE ALONG ONLY Y CO PROJECTE ALONG ONLY X Vy = sin V Vx = cos * V 47 o below the positive x-axis. 4
An airplane is travelling 735 km/hr in a direction 41.5º of north of west (a) Find the components of the velocity vector in the northerly and westerly directions. (b) How far north and how far west has the plane travelled after 3.00 h? x and y components of motion are independent. A man on a train tosses a ball straight up View this from two reference frames: Cart Reference frame on the moving train. Reference frame on the ground. Components are Independent A ball is thrown 23.0 with a velocity of 22.5 m/s. What is its vertical and horizontal velocity components? Conceptual Example 3-6 v 0x = 22.5 m/s cos 23.0 = 20.1 m/s = 22.5 m/s sin 23.0 = 8.79 m/s if v y is 12.4 m/s and v x is 15.4 m/s, find the angle. = 38.8 Demonstration!! Projectile Motion no air resistance a x = 0, v x = constant projectile s weight is only force acting a y = g = 9.8 m/s 2 v y = 0 at max height x, y motions are independent of each other resolve initial velocity into v x, v y and use our basic kinematic equations to solve problem 5
Projectile Motion PHYIC: y part of motion: v y = gt, y = (½)g t 2 AME as free fall motion!! An object projected horizontally will reach the ground at the same time as an object dropped vertically from the same point! (x & y motions are independent) DEMO INCE G I ALWAY DOWN, MAKE IT NEGATIVE FOR ALL PROJ MOTION AND POITIVE FOR FREE FALL Vertical: A small cart is rolling at constant velocity on a flat track. It fires a ball straight up into the air as it moves. After it is fired, what happens to the ball? A) it depends on how fast the cart is moving B) it falls behind the cart C) it falls in front of the cart D) it falls right back into the cart Horizontal: Your book says negative, just remember in most cases g will be opposite the initial velocity (so -9.8 m/s) (Except free fall) (no acceleration) E) it remains at rest Pre-Lecture Quiz 03 The same small cart is now rolling down an inclined track and accelerating. It fires a ball straight out of the cannon as it moves. After it is fired, what happens to the ball? A) it depends upon how much the track is tilted B) it falls behind the cart C) it falls in front of the cart D) it falls right back into the cart E) it remains at rest Pre-Lecture Quiz 03 6
You drop a package from a plane flying at constant speed in a straight line. Without air resistance, the package will: A) quickly lag behind the plane while falling B) remain vertically under the plane while falling C) move ahead of the plane while falling D) not fall at all How far below an initial straight-line path will a projectile fall in one second? 5 meters. Does your answer depend on the angle of launch or launch speed? No, vertical displacement below any line is ½ gt 2. Pre-Lecture Quiz 03 A rookie soldier in basic training drops his bullet that this same time that another soldiers shoots from his rifle (exactly horizontal) The bullet is dropped at the moment of the shot &no air res.) Which bullet touches the ground first? Galileo's space and time z z' v (x,y,z) (x',y',z') y y' The bullet touch down at the same time. x x' How many dimensions are there? Ball Rolls Across Table & Falls Off In this case I take down as positive since v yo=0 v yo =0 Time is the 4 th dimension, and it is shared by all the others. v y =gt 7
Driving off a cliff!! negative y (no acceleration) y is positive upward, y 0 = 0 at top. Also v y0 = 0 How fast must the motorcycle leave the cliff to land at x = 90 m, y = -50 m? v x0 =? A stone is thrown horizontally from the top of a cliff that is 40.0 m high. It has a horizontal velocity of 15.0 m/s. We want to find how long it takes the stone to fall to the deck and how far it will travel from the base of the cliff. Plug in the given values: Find x : x = v x t t =2.86 s x = 42.9 m A ball thrown horizontally at 22.2 m/s from the roof of a building lands 36.0 m from the base of the building. How tall is the building? ummary: Ball rolling across the table & falling. Vector velocity v has 2 components: Vector displacement D has 2 components:, The vertical displacement, which is the height of the building, is found by applying. For problems like this if we set y at the top = 0 then gravity is 9/8 m/s2 8
How fast does the stone hit in the previous problem? LAUNCHER Which does gravity have a stronger influence on? Does one ball go further than the other? Do both have the same Vo? Do both shots have the same time in air? Are they both free fall motion? Diagram in terms of vectors what is happening? You have a summer job with an insurance company and are helping to investigate a tragic "accident." At the scene, you see a road running straight down a hill. At the bottom of the hill, the road widens into a small, level parking lot overlooking a cliff. The cliff has a vertical drop of 400 m to the horizontal ground below where a car is wrecked 45 m from the base of the cliff. A witness claims that the car was parked on the hill and began coasting down the road, taking about 3 seconds to get down the hill (you calculate the average acceleration of 1.7m/s 2. ) The client insists that the emergency brake was on, as would be required on this hill. As you walk down inspect the wreckage you see the parking brake has been pushed in. Your boss drops a stone from the edge of the cliff and, from the sound of it hitting the ground below, determines that it takes 5.0 seconds to fall to the bottom. Obviously, your boss suspects foul play. Is he right? Does James Want V yo, How much? PLAY JB 5 CLIP HERE Initial velocity is 5.1m/s 9.03s to fall 45.9m from the cliff Projectile Motion 9
General Case: Projectile Motion a = g (down always) V yo =V o sin o V xo =V o cos If Bubba is traveling at 33m/s and the ramp is 30 deg will he make a ramp distance of. 2 possible Formulas http://www.youtube.com/watch?v=_gagjkkfbv4&feature=related&safety_mode=true&pe rsist_safety_mode=1&safe=active 10
A ball is given an initial velocity of 22. 7 m/s at an angle of 66.0 to the horizontal. Find how high the. ball will go? v y0 = v 0 sin 0 Find y: v y0 = 20.7 m/s and v y = 0. v y0 = 22.7 m/s (sin 66.0 ) v y0 = 20.7 m/s 2 possible Formulas A confused soldier fires his artillery piece at angle that maximizes Vy. His enemy utilizes an angle of 40 o. If both are in the air for 3 seconds (same height) what was the velocity of the enemy's ball. At Vy Max the Vxo =0, t to the top = 1.5s ince both are in the air for the same amount of time, both have same Vyo y y = 21.9 m 40 You throw a potato at an angle of 22.2. If it is in the air for 1.55 s, how far did it go? Find the vertical velocity: A naval gun fires a 16 inch projectile. The muzzle velocity (speed of the bullet) is 345 m/s with an angle of 32.0. What is the range of the shot? Hint since both are at y=o then you don t need the quadratic to solve for time Find the vertical velocity: Find the horizontal velocity: Find the time: Find range: x = 10904 m Find the horizontal displacement: t = 37.27 s The biker has a hang time of 4.5 s. He lands 48 m from the ramp, (assume no ramp height) what was the ramp angle? NEED TO FIND V y0 and V x0 vx 0 = 10.67 m/s A ball is thrown at some angle. The ball is in the air for 4.50 seconds before it hits. If it travels 45.0 meters before it hits the ground, what was the initial velocity of the ball (magnitude and direction please)? NEED TO FIND t at max height NEED TO FIND Vy t = 2.25 s NEED TO FIND t at max height Vy 0 = 22.05 m/s = 64 11
But what if the height is not the same after landing? A stone is thrown off the top of a building from a height of 45.0 m. The stone has a launch angle of 62.5 and a speed of 31.5 m/s. How long is the stone in flight What is its speed just before it hits the ground? What angle does it hit? Quadratic Equation Vf at bottom from 45 meters A husky kicker wants to provide as much hang time as possible For his coverage team to get down the field. If he kicks the ball with a velocity 40m/s at an angle of 35 degrees above horizontal, how much time does his team have to get down field and pound a cougar? At what distance from kick will we find a pounded cougar returner? Hint since both are at y=o then you don t need the quadratic to solve for time v y0 = v 0 sin v x0 = v 0 cos 35 v y0 = 40 sin 35 v x0 = 40 cos35 v y0 = 22.9 m/s v x0 = 32.7 m/s What is its speed just before it hits the ground? What angle does it hit? Remember that this 2.8 is negative (so will be below horizon) A football leaves a punters foot 1.0 m above the ground at 20.0 m/s. θ 0 = 37º. Find t then x. Vy 0 = 12.0 m/s, θ 0 = 37º FORMULA GIVEN ON AP Quadratic Equation 12