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VECTORS

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GPS SP1. Students will analyze the relationships between force, mass, gravity, and the motion of objects. b. Compare and contrast scalar and vector quantities.

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**SCALARS AND VECTORS Scalars only have magnitude (ex. 50 m)**

Vectors have magnitude and direction (ex. 50 m, North) When you combine two or more vectors the sum is called the resultant. For example in 1-D: 50 m North and 30 m South; the resultant is 20 m North (+50 m + (-30 m))

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VECTOR BASICS Images:

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**THE RESULTANT IN ONE DIMENSION**

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DIRECTION Images:

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**THE RESULTANT IN TWO DIMENSIONS (X AND Y)**

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PROPERTIES OF VECTORS Vectors can be moved parallel to themselves in a diagram. Vectors can be added in any order. For example, A + B is the same as B + A To subtract a vector, add its opposite. SIGNS (DIRECTION) ARE VERY IMPORTANT!!! Multiplying or dividing vectors by scalars results in vectors. For example: When you divide displacement (x or y) by time (s) the result is velocity (v).

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**RESULTANTS CAN BE DETERMINE GRAPHICALLY OR ALGEBRACIALLY**

When determining the resultant graphically you must be careful of several factors. Your scale must be determined and measured accurately with a ruler. Your angles (directions) must be done with a protractor. ALWAYS DRAW YOUR VECTORS FROM HEAD TO TAIL!!!!! The resultant is always from the head of your last vector to the tail of your first vector. TAIL HEAD

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DETERMINING SCALE

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**GRAPHICALLY DETERMINING A RESULTANT**

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**DETERMINING RESULTANTS BY ALGEBRA AND TRIGONOMETRY**

You must use the Pythagorean theorem and trigonometry to determine a resultant. WE ONLY USE DEGREES IN THIS CLASS!! NO RADIANS!!!! You must know SOHCAHTOA!! You must be able to use your calculator correctly! The resultant is always from the head of your last vector to the tail of your first vector. Direction is always from the tail of the first vector.

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SOHCAHTOA

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REAL LIFE VECTORS

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ANSWERS TO PRACTICE PRACTICE A: km at º W of N OR km at 63.44º N of W PRACTICE B: km at 53.13º S of W OR 50 km at 36.87º W of S

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PROBLEMS 1 and 2 Which of the following quantities are scalars, and which are vectors? (A) the acceleration of a plane as it takes off (B) the number of passengers on the plane (C) the duration of the flight (D) the displacement of the flight (E) the amount of fuel required for the flight? A roller coaster moves 85 m horizontally, then travels 45 m at an angle of 30° above the horizontal. What is its displacement from its starting point?(graphical techniques)

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ANSWERS (A) vector (B) scalar (C) scalar (D) vector (E) scalar 126 m at 10° above the horizontal 126 m at 10° above the horizontal RESULTANT 30°

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PROBLEMS 3 and 4 A novice pilot sets a plane’s controls, thinking the plane will fly at 250 km/hr to the north. If the wind blows at 75 km/hr toward the southeast, what is the plane’s resultant velocity? Use graphical techniques. While flying over the Grand Canyon, the pilot slows the plane’s engines down to one-half the velocity of the last problem. If the wind’s velocity is still 75 km/h toward the southeast, what will the plane’s new resultant velocity be?

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ANSWERS 204 km/h at 75° north of east 89 km/h at 54° north of east

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PROBLEM The water used in many fountains is recycled. For instance, a single water particle in a fountain travels through an 85 m system and then returns to the same point. What is the displacement of a water particle during one cycle?

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ANSWER ZERO

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