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Vectors

Vectors. August 2011. Objectives. To define the difference between scalar and vectorial quantities. To understand and use vectorial addition. To state and understand the concept of relative velocities. Vector and Scalar quantities.

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Vectors

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  1. Vectors August 2011

  2. Objectives. • To define the difference between scalar and vectorial quantities. • To understand and use vectorial addition. • To state and understand the concept of relative velocities.

  3. Vector and Scalar quantities A quantity such as a velocity, which has a direction as well as magnitude is a vector quantity. Some example of vector quantities are force, displacementand momentum. However, many quantities have no direction like mass, time and temperature and are called scalar quantities. The number of grapes is a scalar quantity, the number of blocks and direction to a place is a vector quantity.

  4. Examples

  5. Adding Vectors (Graphical Method) When a vector quantity is handwritten its represented with an arrow over the letter representing it. Simple arithmetic cannot be used to add vectors that are not along the same line. They also must have the same units. Vectors can be added geometrically or algebraically. To add a vector to vector geometrically you must draw them to some scale. The resultant vector is the vector drawn from the tail of to the tip of as shown in the figure.

  6. Vector Components Adding vectors graphically is not sufficiently accurate and not useful for vectors in three dimensions. A vector can be expressed as the sum of two other vectors called the components. They are usually chosen to be along two perpendicular directions.

  7. Adding Vectors (Algebraically) Using trigonometric functions we can find the components are: We can now add vectors using components. First solve each vector into its components. Then the sum of the x components equals the x component of the resultant and similarly for y. So for =+

  8. The components of a vector form the sides of a right triangle having an hypotenuse with magnitude A that correspond to the magnitude of the resultant vector. Using the Pythagorean Theorem and the definition of a Tangent we find that: And to solve the angle of the resultant vector we can then use:

  9. Examples • A rural mail carrier leaves the post office and drives 22.0 km in a northerly direction. She then drives in a direction 60.0 degrees south of east for 47.0 km. What is her displacement from the post office? • Solution: 30.0 km. 38.5 degrees southeast. • An airplane trip involves three short trips with two stopovers. The first is due east for 620 km; the second is southeast (45 degrees) for 440 km; and the third is at 53 degrees south of west for 550 km. What is the plane’s total displacement? • Solution: 960 km. 51 degrees south of east.

  10. While exploring a cave, a spelunker starts at the entrance and moves the following distances: 75m north, 250m east, 125m at an angle 30 degrees north of east, and 150m south. Find the resultant displacement from the cave entrance. • Solution: 358.47m 1.998 degrees south of east • The eye of a hurricane passes over Grand Bahama Island in a direction of 60 degrees north of west with a speed of 41.0 km/h. Three hours later it shifts due north and its speed slows to 25.0 km/h. How far from grand Bahama is the hurricane after 4.5 hours? • Solution: 157 km.

  11. Activity: Treasure Hunting • Statistics from treasure hunting. • 12 Teams. • 2 treasure marks found. • 6 treasure marks not found but end point being within an average of 6.4 m. • 4 treasure marks not found due to wrong set of instructions. Why wasn’t the mark found? What were the most common errors on the activity? How can these mistakes be minimized?

  12. Activity: Vector Addition. Watch the following video about vector addition.After watching it. A set of questions will be given to you and you must answer them individually. http://www.youtube.com/watch?v=UeQNnfY0BQA http://www.youtube.com/watch?v=6zfMENV_tak&feature=relmfu

  13. What was the first problem the Myth Busters encountered when using a conveyor belt? • What is the grid pattern on the back for? • Why Grant decided to place a marker to know when to shoot ? • What was the reason for choosing the air pressure cannon over the other devices? • Briefly explain how Kari overcame the measuring problem of the truck speed. • Write a brief description of another way of proving vector addition you can think about.

  14. Relative Velocity The measured velocity of an object depends on the velocity of the observer with respect to the object. For example: On highways cars are moving in the same direction at high speed relative to earth, but with respect to each other they hardly move at all. But if velocities are not along the same line we must use vector addition to find the relative velocity. Velocity of the river with respect to earth. Velocity of the boat with respect to the river. Velocity of the boat with respect to earth.

  15. Examples • A train is traveling with a speed of 15m/s with respect to earth. A passenger standing at the end of the train throws a baseball with a speed of 15m/s relative to the train, in direction opposite the motion of the train. What is the velocity of the baseball relative to the earth? • Solution: 0 m/s • A rowboat crosses a river with a velocity of 3.3 mi/h at an angle of 62.5 degrees north of west relative to the water. The river is 0.505 mi wide and carries and eastward current of 1.25 mi/h. How far upstream is the boat when it reaches the opposite shore? • Solution: 0.26 miles upstream

  16. Two canoeists in identical canoes exert the same effort paddling and hence maintain the same speed relative to the water. One paddles directly upstream (and moves upstream), whereas the other paddles directly downstream. With downstream being the positive direction, an observer on the shore determines the velocities of the two canoes to be -1.2m/s and 2.9m/s, respectively. (a) What is the speed of the water relative to the shore? (b) What is the speed of each canoe relative to the water. • Solution: 0.85m/s , 2.05m/s • A jet airliner moving initially at 300mi/h due east enters a region where the wind is blowing at 100mi/h in a direction 30 degrees north of east. What is the new velocity of the aircraft relative to the ground? • Solution: 389.82 mi/h at 7.37 degrees north of east

  17. References • Giancoli , Douglas C. Physics Sixth Edition. USA Pearson 2005 • Serway, Raymond A. Essentials of College Physics. USA Thomson 2007

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