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VECTORS Level 1 Physics Objectives and Essential Questions Objectives Distinguish between basic trigonometric functions (SOH CAH TOA) Distinguish between vector and scalar quantities Add vectors using graphical and analytical methods Essential Questions What is a vector quantity? What is a scalar quantity? Give examples of each. SCALAR A SCALAR quantity is any quantity in physics that has MAGNITUDE ONLY Number value with units Scalar Example Magnitude Speed 35 m/s Distance 25 meters Age 16 years VECTOR A VECTOR quantity is any quantity in physics that has BOTH MAGNITUDE and DIRECTION r r r r x, v , a, F Vector Example Magnitude and Direction Velocity 35 m/s, North Acceleration 10 m/s2, South Force 20 N, East An arrow above the symbol illustrates a vector quantity. It indicates MAGNITUDE and DIRECTION VECTOR APPLICATION ADDITION: When two (2) vectors point in the SAME direction, simply add them together. EXAMPLE: A man walks 46.5 m east, then another 20 m east. Calculate his displacement relative to where he started. 46.5 m, E + 66.5 m, E 20 m, E MAGNITUDE relates to the size of the arrow and DIRECTION relates to the way the arrow is drawn VECTOR APPLICATION SUBTRACTION: When two (2) vectors point in the OPPOSITE direction, simply subtract them. EXAMPLE: A man walks 46.5 m east, then another 20 m west. Calculate his displacement relative to where he started. 46.5 m, E 20 m, W 26.5 m, E NON-COLLINEAR VECTORS When two (2) vectors are PERPENDICULAR to each other, you must use the PYTHAGOREAN THEOREM FINISH Example: A man travels 120 km east then 160 km north. Calculate his resultant displacement. the hypotenuse is called the RESULTANT 160 km, N c 2 a2 b2 c a2 b2 c resultant c 200km VERTICAL COMPONENT 120 160 2 2 S R T T A 120 km, E HORIZONTAL COMPONENT WHAT ABOUT DIRECTION? In the example, DISPLACEMENT asked for and since it is a VECTOR quantity, we need to report its direction. N W of N E of N N of E N of E N of W E W S of W NOTE: When drawing a right triangle that conveys some type of motion, you MUST draw your components HEAD TO TOE. S of E W of S E of S S NEED A VALUE – ANGLE! Just putting N of E is not good enough (how far north of east ?). We need to find a numeric value for the direction. To find the value of the angle we use a Trig function called TANGENT. 200 km 160 km, N oppositeside 160 Tan 1.333 adjacentside 120 N of E Tan1 (1.333) 53.1o 120 km, E So the COMPLETE final answer is : 200 km, 53.1 degrees North of East What are your missing components? Suppose a person walked 65 m, 25 degrees East of North. What were his horizontal and vertical components? H.C. = ? V.C = ? 25 65 m The goal: ALWAYS MAKE A RIGHT TRIANGLE! To solve for components, we often use the trig functions since and cosine. adjacent side opposite side sine hypotenuse hypotenuse adj hyp cos opp hyp sin cosine adj V .C. 65 cos 25 58.91m, N opp H .C. 65 sin 25 27.47m, E Example A bear, searching for food wanders 35 meters east then 20 meters north. Frustrated, he wanders another 12 meters west then 6 meters south. Calculate the bear's displacement. - 12 m, W - = 6 m, S 20 m, N 35 m, E 14 m, N R 23 m, E = 14 m, N R 14 2 232 26.93m 14 Tan .6087 23 Tan 1 (0.6087) 31.3 23 m, E The Final Answer: 26.93 m, 31.3 degrees NORTH or EAST Example A boat moves with a velocity of 15 m/s, N in a river which flows with a velocity of 8.0 m/s, west. Calculate the boat's resultant velocity with respect to due north. Rv 82 152 17 m / s 8.0 m/s, W 15 m/s, N Rv 8 Tan 0.5333 15 Tan 1 (0.5333) 28.1 The Final Answer : 17 m/s, @ 28.1 degrees West of North Example A plane moves with a velocity of 63.5 m/s at 32 degrees South of East. Calculate the plane's horizontal and vertical velocity components. adjacent side opposite side cosine sine hypotenuse hypotenuse adj hyp cos opp hyp sin H.C. =? 32 63.5 m/s V.C. = ? adj H .C. 63.5 cos 32 53.85 m / s, E opp V .C. 63.5 sin 32 33.64 m / s, S Example A storm system moves 5000 km due east, then shifts course at 40 degrees North of East for 1500 km. Calculate the storm's resultant displacement. 1500 km adjacent side opposite side sine hypotenuse hypotenuse V.C. adj hyp cos opp hyp sin cosine 40 5000 km, E H.C. adj H .C. 1500 cos 40 1149.1 km, E opp V .C. 1500 sin 40 964.2 km, N 5000 km + 1149.1 km = 6149.1 km R 6149.12 964.2 2 6224.2 km 964.2 0.157 6149.1 Tan1 (0.157) 8.92o Tan R 964.2 km The Final Answer: 6224.2 km @ 8.92 degrees, North of East 6149.1 km