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Chapter 2 Vectors in One dimension Language of motion Origin: reference point Position, d : The straight-line distance between the origin and the object’s location including magnitude and direction. Ex 1: Position: 4.5 m [east of the origin] or 4.5 m [E]. Ex 2: Position: 2.0 m [south of the origin] or 2.0 m [S] Scalar vs. Vector quantities A scalar quantity has only magnitude. Ex: length, area, volume, time, speed, mass, temperature, etc. A vector quantity has both magnitude and direction. Ex: displacement, direction, velocity, acceleration, force, etc. 1. Time, t = 30 s 2. Displacement of 25 m North 3. Velocity, 4. Force, v = 20 m/s, Right F =6.50 x 107 N, 60o west of south Distance vs. Displacement Distance: the length of the path taken to move from one position to another-scalar Ex: distance of 70 m. Displacement: a straight line between initial and finial position; includes magnitude and direction-vector Ex: displacement of 30 m east Ex: Find distance and displacement Vectors vectors are drawn using arrows which show magnitude in the length as well as direction the resultant is the sum of two or more vectors to add vectors, arrange them head to tail then draw the resultant from the tail of the first vector to the head of the last vector + + = or Note that the resultant is the same regardless of the order of the vectors Sign conventions Ex 1: Contestants in a snowshoe race must move forward 10.0 m, untie a series of knots, move forward 5.0 m, solve a puzzle, and finally move forward 25.0 m to the finish line. Determine the resultant vector by adding the vectors graphically. 1. Choose an appropriate scale and reference direction. 2. Draw the first vector and label its magnitude and direction 3. Place the tail of the second vector at the tip of the first vector. The same for the rest of vectors. 4. Connect the tail of the first vector to the tip of the last vector. The new vector pointing toward the tip of the last vector is the resultant vector. 5. Find the magnitude of the resultant vector by measuring with a ruler, then convert the measured value using the scale. Ex 2: Find the distance and displacement travelled by the person. Ex 3: Find the distance and displacement of a dog that moved:[Use vector diagram and show on it the arrow that presents the displacement] - 10 m, west - 6.0 m south - 2.0 m east - 3.0 m, north - 8.0 m, east Ex 4: A car travels 300 km east and then 200 km west. Find the distance travelled by the car and its displacement.[show your sign convention, draw vector diagram, draw displacement arrow] Average speed vs. Average velocity Average speed = total distance/ time Average velocity = displacement/time Ex: A person travels 25 m east then 10 m west in 30 s. Find the average speed and average velocity. [Provide vector diagrams] Solve practice page 9#1-3 Solve Check and Reflect pg 10# 4-7 Vector Resultant (one dimension or linear) 1. Vectors in the Same Direction If two vectors have the same direction, their resultant has a magnitude equal to the sum of their magnitudes and will also have the same direction RESULTANT = SUM Ex 1: A person runs 25 m south and then another 15 m south. a) What is the distance travelled? b) What is the displacement? Ex 2: A plane has a velocity of 150 m/s [E]. The wind blows east at 75 m/s. Find the resultant velocity. 2. Vectors in Opposite Directions To add vector quantities that are in opposite directions. RESULTANT =DIFFERENCE Ex 1: A plane flies 200 km north and then turns around and comes back 150 km. a) What is the distance travelled? b) What is the displacement? Ex 2: The skier moves from A to B to C to D. Use the diagram to determine the distance traveled by the skier and the resulting displacement during these three minutes. Solve practice problems pg 73#1 & pg 74#1-3 & pg 75#6-9 Solve Page 1&2 workbook 3. Motion in two dimensions When you walk east, west, north, south, right, left, up, or down, you are walking in ONE DIMENSION. When walking in an angled movement, you are walking in TWO DIMENSIONS and you are making 2D vector. Components of a vector- is the path you take from the start (tail) to the end (head) using only +x, +y, -x, or –y. The shortest path from the start to the end is the resultant (sum). End ‘ 150 m Start 120 m ‘ +x direction y direction 90 m Ex: Write the x and y components of the following: y x Vector direction in 2-dimentions How to indicate the direction of a vector in 2-D. Cartesian Method-angles are measured from +x axis by moving counterclockwise. +x +x Ex: Draw vectors in two dimensions a) 25 km [120˚] b) 10 m/s [180˚] c) 225 m [75˚] Navigation Method- uses E, W, N, S to indicate the direction of the vector and an angle θ. Example 12.5 m [30˚ W of N] 30˚ To draw the vector arrow start from the origin then move towards the second direction, N, then move towards the first, W, to get to the head of the vector. Ex: Draw vectors in two dimensions: a) A boat sailing at 60 km [30˚ N of E] b) A force applied 450 N [S] c) The dog moved 10.2 m [25˚ W of S] Vectors that lie along the same straight line in one dimension are called collinear. Vectors not along a straight line, 2-D, are noncollinear. 35˚ Collinear vectors Non-Collinear vectors Resultant of 2-D vectors at 90˚ (finding resultant): 1. Graphically (diagramming) Draw vector 1 vR v2 Draw vector 2 so that its tail θ at the tip of vector 1 v1 Draw the resultant vector by v2 connecting the tail of vector 1 vR to the tip of vector 2 (head-to- v1 θ tail) State the resultant vector (both magnitude and direction) NOTE: The resultant angle,θ, is at the tail of the first vector and the tail of the resultant. Ex: A camper walks 2.0 km [S], then 4.0 km [E], and finally 1.0 km [N]. Find graphically her resultant displacement. 2. Mathematically Adding Vectors at 90˚ General equation for vector addition: vR Dv = v 1 +v 2 v2 v1 to add vectors that are at right angles to each other, you must calculate: a) the magnitude – use Pythagorean Theorem c 2 = a2 + b2 b) the direction – compass direction for example [35˚ N of E]… Ex 1: A plane flies 200 km north and then heads 150 km east. Calculate the displacement. (Draw a diagram!!!) Ex 2: A plane flies 350 km west, 200 km south and 100 km east. Calculate the displacement. Ex 3: A disoriented dog runs 30 m north then 20 m east then 10 m west. Find his displacement. 3. Determining Components first you need to be able to take any vector and break it down into its x and y components start by forming a right angle triangle from the vector…this shows each component eg) then use trigonometry to calculate the magnitude of the components ie) length of each side of the right angle triangle, given the vector Ex 1: A plane flies at an angle of 40.0 N of E at a velocity of 600 km/h. Find the east component and the north component of the velocity. Ex 2: A cyclist’s velocity is 10 m/s [245o]. Determine the x and y components of her velocity. now that you can break a vector into its components, you can add two or more vectors using the individual x and y components 1. Calculate the x and y components for each vector 2. Add all components in the x direction. Add all components in the y direction 3. Find the magnitude of the resultant vector (Pythagorean theorem) 4. Find the angle of the resultant vector (trigonometric ratios) Ex 1: Add the following vectors: 12.0 m [30o] and 9.0 m [155o]. Ex 2: Use components to determine the displacement of a cross-country skier who travelled 15.0 m [220o] and then 25.0 m [335o]. Ex 3: Give the magnitude and direction for the addition of the following vectors : 200 N, 20 N of E and 300 N, 60 S of W Ex 4: Add the following vectors: 200 N, 45 N of W and 100 N, 30 N of E. Ex 5: Add the following vectors: 20 N, 40 N of E; 30 N, due north; 50 N, 60 W of N Relative motion Objects sometimes move within a medium that is moving: Airplanes, kites, sailboats moving within moving air. http://www.physicsclassroom.com/mmedia/vectors/plane.html Boats, swimmers moving under the effect of water current. http://www.physicsclassroom.com/mmedia/vectors/rb.html The velocity of a moving object depends on the location of the observer: The observer is on the moving object (passenger riding on the bus) The observer is watching from a stationary position. Ex: The bus http://sasklearning.gov.sk.ca/branches/elearning/tsl/resources/subject_area/science/physics_30_resources/lesson_one/relative_motion.shtml Case 1: Collinear Relative motion (one dimension) Ex: A west jet airplane travels with an air velocity (velocity in still air) of 650 km/h [S] from Fort McMurray to Edmonton. It encounters a wind velocity of 43.2 km/h. What is the velocity of the airplane relative to an observer on the ground if, 1) the wind is south wind blowing to the north. 2) the wind is blowing towards the south. Case 2: Non-collinear relative motion (two dimensions) Relative motion in air Ex1: A plane travels north at an air speed of 300 km/h. The wind blows west at 50.0 km/h. a) At what angle should the plane be piloted such that it will go straight north? b) What is the velocity of the plane with respect to the ground? Relative motion in water Ex2: A boat is moving at 20 m/s in a river that has a velocity of 5.0 m/s south. Calculate the resultant velocity when: a) the boat is moving downstream (same direction) b) the boat is moving upstream (opposite direction) c) the boat is moving east Ex3: A boat moving 4 m/s east across a 80 m wide river encounters a current travelling 3.0 m/s north. Calculate: a) the resultant velocity of the boat b) how long it takes to reach the other side c) how far downstream the boat lands when it reaches the other side Projectile Motion a projectile is an object that travels in air. Force of gravity is the only force acting on it. it follows a curved path, called a trajectory, which is due to its horizontal and vertical velocity the horizontal distance the object travels is called the range characteristics of projectile motion: 1. the horizontal velocity is constant 2. the vertical velocity changes with the distance (height) the object falls 3. the horizontal velocity is independent of the vertical velocity it doesn’t matter what the horizontal speed is, gravity will affect it in the same way…that is, it will take the same time to fall at the point where the object starts to fall (y direction) the vertical velocity is 0 m/s to solve the problems, look at the x and y directions separately Ex 1: A stone is thrown horizontally at 15 m/s from the top of a cliff 44 m high. a) How long does it take to reach the bottom of the cliff? b) How far from the base of the cliff does the stone strike the ground? c) What is the vertical speed as the stone hits the ground? Ex 2: An object is thrown horizontally at a velocity of 10.0 m/s from the top of a 90.0 m high building. a) How long does it take to hit the ground? b) How far from the base of the building does it strike the ground? Ex 3: An object is thrown horizontally at 20.0 m/s from the top of a cliff. The object hits the ground 48.0 m from the base of the cliff: a) How high is the cliff? b) What is the vertical speed as the object hits the ground?