Vectors. Wind is blowing 15 m/s East. What is the magnitude of the wind s velocity? What is the direction?

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1 Physics R Scalar: Vector: Vectors Date: Examples of scalars and vectors: Scalars Vectors Wind is blowing 15 m/s East. What is the magnitude of the wind s velocity? What is the direction? Magnitude: Direction: Vectors are represented by an arrow. The length of the arrow represents while the way the arrow points represents direction. Drawing Vectors - Scaled Vector Diagrams Use a ruler and the scale given to draw the following vectors 1. 1 cm = 10 m 2. 1 cm = 5 m/s 3. 1 mm = 1 m Draw 30 m East Draw 20 m/s West Draw 25 m East Determine the scale for the following vectors cm = 1 cm = 1 cm = 1

2 Vector Direction A cow runs 10 South from West with a velocity of 6 m/s Start from West You are now 0 degrees from West What are the possible directions you could go from West? Which way should you go from West? Make a sketch of the vector 10 South from West, label it #1 Now make a sketch of a vector that is 10 West from South, label it #2. Is this the same as 10 South from West? 7. Measure (1) North from East Or East from North (2) (3) 8. Sketch a vector that is a. 20 North from East, label #1 b. 82 East from South, label #2 2

3 Vector Addition when (sometimes) Equilibrant Example: 9. Mr. Vigneaux pushes on a desk to the right with a force of 4 Newtons (the unit for force). A disgruntled student works against him by pushing it to the left with a force of 3 Newtons. What is the resultant force? The student and Mr. Vigneaux decide to work together to push the desk to the right, with the same magnitudes as before. What is the resultant force? Under what conditions can two vectors create the maximum resultant? When would they create the minimum resultant? Example: 10. Two forces, a 25 N force and a 15 N force act concurrently on an object. Which of the following could be the magnitude of the resultant force? (There can be more than one correct answer). 8 N 10 N 18 N 40 N 45 N 52 N Adding Vectors Mathematically When two vectors act in the same direction, When two vectors act in opposite directions, When two vectors act perpendicular to each other, 11. A plane is flying at 150 m/s due East while a wind blows it at 30 m/s. Find the magnitude of the resultant velocity when the wind is blowing a. North b. South c. East d. West 3

4 Adding Vectors Graphically Tip to Tail We can move vectors around so long as we keep the correct magnitude and direction. To determine the resultant from two vectors, we move the beginning (tail) of one vector to the end (tip) of the other vector. The resultant then points from start to finish. Move B to the tip of A, Move A to the tip of B, then sketch the resultant then sketch the resultant How do the results compare? Does it matter which vector we move? Tip-to-tail method Use the tip-to-tail method to sketch the resultant vector for each. Label the resultant Sketch a vector that could be added to vector A to create resultant R 14. Scaled Vector Diagrams 15. Example: Use a ruler and a scale of 1cm = 2m A man walks 10m East and then walks 4m West. Draw a scaled vector diagram of his walk. Then draw the resultant. *Remember that vectors add tip-to-tail and the resultant points start-to-finish.* 4

5 A. 2-Dimensional Vectors 16. Starting from point A, use a scale of 1cm = 5 N A force of 25 N East and a force of 10 N South both push on an object. Draw a scaled vector diagram Draw the resultant Measure the resultant in cm. What is this in N? Now calculate the resultant using the Pythagorean theorem. How does your result compare? 17. Starting from point B, use a scale of 1cm = 10 m A man walks 30 m East, 15 m North, and 10 m West Draw a scaled vector diagram Draw the resultant Measure the resultant and convert it to meters. Calculate his resultant displacement. B. How does the calculated displacement compare to the measured displacement? 18. A dog walks 8 meters due north and then 6 meters due east. 5

6 The Riverboat Problem 19. A motorboat traveling 7 m/s, East encounters a current traveling 3.0 m/s, North. a) What is the resultant velocity of the motorboat? b) If the width of the river is 80 meters wide, then how much time does it take the boat to travel shore to shore? c) What distance downstream does the boat reach the opposite shore? Speed/Velocity Distance Critical variable in multi-dimensional problems is TIME. We must consider each dimension SEPARATELY, using TIME as the only crossover VARIABLE. For problems with constant velocity, use the equation v = d / t or use d = v i t + ½ a t 2 and set the acceleration to zero, making the equation 6

7 20. A model airplane heads due east at 1.50 meters per second, while the wind blows due north at 0.70 meter per second. a) What is the resultant velocity of the airplane? (make a sketch) Distance Velocity b) If it s flying East across a highway which runs North-South which is 10 m wide, how long does it take to reach the other side? (make a sketch) c) How far down the highway does it drift in the wind? 21. An ant is crossing a treadmill at 0.05 m/s. The treadmill moves forward at 0.10 m/s a) What is the magnitude of the resultant velocity of the ant? b) If the treadmill is 0.50 m wide, how much time does it take the ant to cross? c) How far forward does the treadmill move as the ant crosses? d) What is the resultant displacement of the ant? 22. A motorboat, which has a speed of 5 meters per second in still water, is headed east as it crosses a river flowing south at 3.3 meters per second. What is the magnitude of the boat s resultant velocity with respect to the starting point? It takes 22 seconds to cross the river. How wide is the river? 23. A cruise ship is headed North at 10 m/s while a jogger runs across it at 3 m/s. What is the magnitude of his resultant velocity? If the ship is 40 m across, how much time will it take him to cross it? 7

8 Review Checklist Use the tip to tail method to add vectors graphically. Determine the resultant from adding two or more vectors graphically or algebraically. 1. Sketch the resultant for the following 2. Sketch the missing vector that when added to vector A would produce the resultant 3. The vector diagram below represents two forces, F 1 and F 2, simultaneously acting on an object. Which vector best represents the resultant of the two forces? 4. A model airplane heads due east at 1.50 meters per second, while the wind blows due north at 0.70 meters per second. The scaled diagram below represents these vector quantities. a. Using a ruler, determine the scale of the diagram. b. On the diagram, using a ruler and protractor, construct the resultant velocity of the airplane. c. Determine the magnitude of the resultant mathematically AND graphically. d. Determine the angle between North and the resultant velocity. 8

9 Determine the resultant speed/velocity in a two dimensional system. Determine an unknown variable in a two dimensional system with constant velocity in both directions. 5. A girl rows a boat with a velocity of 2.5 meters per second due east across a river. As she rows across the river a current of 1.5 meters per second pushes her boat due north. Sketch a set of vectors to represent these two velocities and calculate the resultant speed of the boat. 6. A plane flies north with a speed of 100 meters per second. At the same time, a crosswind pushes the plane east with a speed of 15 meters per second. In the time that the plane flies 300 meters north, how far east will it drift? 7. A swimmer moves across a 200 meter wide river at a velocity of 0.5 meter per second east. How long will it take the swimmer to get across? Now assume that as the swimmer moves across the river, a current pushes him downstream (south) at 0.1 meter per second. How far downstream will this push him? 8. A hockey player who is 5.0 meters in front of a 1.2 meter wide goal slides puck directly at its center. The player releases the puck at 2.5 meters per second. As the puck slides toward the goal it drifts to the right at 0.4 meter per second. How long does it take the puck to reach the goal? How far does it drift in this time? Is it a goal? Answers to Review Checklist (2) 4a. 1cm = m/s 4c m/s 4d. 65 degrees m/s m s, 40 m 8. 2 s, 0.8 m, nope 9