1. At what speed must you throw a ball vertically in order to reach the top of a building, 12m tall? Vectors

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1 Physics R Date: 1. At what speed must you throw a ball vertically in order to reach the top of a building, 12m tall? Scalar: Vectors Vectors Scalars Vector: Wind is blowing 15 m/s East. What is the magnitude of the wind s velocity? What is the direction? Drawing Vectors 2. A horse runs 10 North from West with a velocity of 60 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 North from West, label it #1 3. A plane flies with a velocity of 120 m/s 30 South from East Now make a sketch of a vector that is 10 West from North, label it #2. Is this the same as 10 North from West? Make a sketch of the vector. How else could you describe 30 South from East? 1

2 Vector Addition Example: 4. 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: 5. 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 graphically Tip-to-tail method Use the tip-to-tail method to sketch the resultant vector for each Sketch a vector that could be added to vector A to create resultant R 2

3 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, Scaled Vector Diagram Vectors are represented by an arrow. The length of the arrow represents while the way the arrow points represents direction. 6. 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.* 7. Use a scale of 1cm = 0.5 m/s A train is traveling at 3 m/s West while you walk to the front at 2 m/s West. Draw a scaled vector diagram of your velocity, then draw the resultant. A. 2-Dimensional Vectors 8. 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? 9. 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. B. Calculate his resultant displacement. How does the calculated displacement compare to the measured displacement? Calculate percent error. 3

4 The Riverboat Problem 10. 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. 11. 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) 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? 4

5 12. Tip-to-tail sketching practice Remember, we can move vectors around so long as we keep the correct magnitude and direction. 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? Use the tip-to-tail method to sketch the resultant Scaled Vector Diagram practice 13. A dog walks 8 meters due north and then 6 meters due east. Resultant minimum/maximum 14. What is the maximum resultant from combining a 10 N force and a 15 N force? What angle would give you that resultant? What is the minimum resultant from combining those two forces? What angle would give you that resultant? Using angles between degrees, the resultant can be anywhere between the minimum and maximum values 5

6 15. a) An ant is crossing a treadmill which is turned off. The treadmill is 0.5 m wide. How much time does it take the ant to cross the treadmill if the ant is traveling at 0.05 m/s? b) The treadmill is turned on and begins moving forward at 0.10 m/s. If the ant is crossing the treadmill again, what is the magnitude of his resultant velocity? c) In the time it takes him to cross the treadmill, the treadmill moves forward 1 meter. What is the magnitude of his resultant displacement? d) Use your answers from b) and c) to calculate how much time it takes the ant to cross the treadmill. How does this compare to your answer from part a? 16. 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? Draw a picture How much time will it take the boat if the river is 40m across? 17. 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? Draw a picture If the ship is 40 m across, how much time will it take him to cross it? 6

7 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. 7

8 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 8