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Grade(s): 7-8
Introduction: Vectors are quantities which include a direction. As such, the addition of two or more vectors must take into account that the quantities being added have a directional characteristic. There are a number of methods for carrying out the addition of two or more vectors. The most common method is the “head-to-tail” method of vector addition. Using such a method, the first vector is drawn to scale in the appropriate direction. The second vector is then drawn such that its “tail” is positioned at the “head” (vector arrow) of the first vector. The sum of two such vectors is then represented by a third vector which stretches from the tail of the first vector to the head of the second vector. This third vector is known as the “resultant” - it is the result of adding the two vectors. Of course, the actual magnitude and direction of the resultant is dependent upon the direction which the two individual vectors have. Any two vectors can be added as long as they are the same vector quantity. Add two or more velocity vectors and the result is the resultant velocity; add two or more force vectors and the result is a resultant force; add two or more momentum vectors and the result is the resultant momentum.
Vectors are mathematical objects which have not only a magnitude, a size, the way ordinary numbers have, but also a direction in which they point. They represent arrows in any direction-- in the plane, in the boat, even in three dimensions. It is a new level of “numbers”, and that is the only way of looking at them. Vectors also allow us to represent velocities. We control a boat, for example, and meanwhile the wind pushes it sideways — how are progressing relative to the water surface? Vectors help answer that.
Learner Objective(s):
- The student will be able to understand the concept that both magnitude and direction are necessary when giving instructions for locating a place.
- The student will be able to resolve vectors into components along the directions of given axes, in two or more dimensions.
- The student will be able to add two or more vectors using components applicable to sea navigation.
Florida Sunshine State Standards(s): Science: SC.C.1.3.1 SC.C.2.3.2 SC.C.2.3.3; Math: MA.E.1.3.1 MA.D.1.3.2
Competency Based Curriculum: Science: M/J-I-8-A Math: M/J-1-V-2-A; Math: M/J-3-IV-6-A
Materials:
Classroom board
Miami Bayside transparency
Colored markers
Vector exercise sheet
Activity Procedure(s):
- Define and explain the following terms: vector, vector addition, vector components, magnitude of a vector, vector components parallel and perpendicular to a given direction.
- Sketch a map of the US on the board and demonstrate the principle of displacement by taking a marker and displacing it from New York to Chicago, then from Chicago to Seattle. The final effect is the same as if we displaced the marker from New York to Seattle. Explain vector sum during this time.
- Demonstrate the graphical method of adding two vectors on the board. Place the tail of the second at the head of the first--the sum is from the tail of the first to the head of the second. It does not make a difference which of the two is added first and which second. The two additions always forms a parallelogram, because each arrow appears twice, and the directions must be parallel. The sum is the diagonal of the parallelogram.
- Show students that vectors add like numbers when they are all along the same line. Vectors, however, along a line can have two directions. Draw examples that vectors in one direction are counted + and vectors in the opposite direction are counted -.
- Using the Miami Bayside transparency, the class is invited to take an architectural tour of the site. We meet at the main entrance and inquire as to the route that must be taken to reach the Miami Dade County Courthouse. The student must give both the number of blocks and the direction. The student will then draw on the board, arrows whose lengths represents the number of blocks and whose tips point in the correct direction, the route to be taken. This process can be repeated for several different locations departing from Bayside.
Student Assessment:
Allow student to answer critical thinking skills questions assigned by the teacher.
- Suppose you are given a vector on a flat plane. Would you be able to resolve it into its components?
- Apply the use of vectors to the navigation of a sea ship.
Have students complete the vector exercise sheet and let them discuss their results.
Activity Extension(s):
Allow students to calculate the speed and velocity of a 2-hour trip between two points on a map (Math).
Demonstrate how an object continually changes direction by using an arrow taped to a bicycle wheel (Physics).
Allow students to write about a scenario involving a ship lost at sea and the importance of having proper navigational tools for direction (Language Arts).
Vector Exercise Sheet
- Your airplane flies north at 120 m.p.h., while a wind blows from the west at 50 m.p.h. What is your “ground speed” V, relative to the land below?
Answer: V= 130 m.p.h.
- Your ship can make 10 miles per hour but the river flows at 5 m.p.h. What is your speed relative to the shore going (a) upstream and (b) downstream?
Answer: (a) 5 m.p.h. (b) 15 m.p.h.
- You run at 5mph on a treadmill but get nowhere. Explain why.
Answer: The treadmill is moving in the opposite direction at 5 m.p.h., therefore the total velocity is zero.
Home Learning Activity:
The student will find his/her vector displacement from the home to the school.
Vocabulary: vectors, resultant, resultant velocity, resultant force, resultant momentum
References/Related Links:
http://ww2010.atmos.uiuc.edu
www.sspboatsite.com
http://celestaire.com
http://www.glenbrook.k12.il.us
http://www-istp.gsfc.nasa.gov/stargaze/Lvector.htm
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