advertisement

Vectors and Scalars Honors Physics Scalar A SCALAR is ANY quantity in physics that has MAGNITUDE, but NOT a direction associated with it. Magnitude – A numerical value with units. Scalar Example Magnitude Speed 20 m/s Distance 10 m Age 15 years Heat 1000 calories Vector A VECTOR is ANY quantity in physics that has BOTH MAGNITUDE and DIRECTION. v , x, a, F Vector Velocity Magnitude & Direction 20 m/s, N Acceleration 10 m/s/s, E Force 5 N, West Vectors are typically illustrated by drawing an ARROW above the symbol. The arrow is used to convey direction and magnitude. Applications of Vectors VECTOR ADDITION – If 2 similar vectors point in the SAME direction, add them. Example: A man walks 54.5 meters east, then another 30 meters east. Calculate his displacement relative to where he started? 54.5 m, E + 84.5 m, E 30 m, E Notice that the SIZE of the arrow conveys MAGNITUDE and the way it was drawn conveys DIRECTION. Applications of Vectors VECTOR SUBTRACTION - If 2 vectors are going in opposite directions, you SUBTRACT. Example: A man walks 54.5 meters east, then 30 meters west. Calculate his displacement relative to where he started? 54.5 m, E 30 m, W 24.5 m, E - Non-Collinear Vectors When 2 vectors are perpendicular, you must use the Pythagorean theorem. The hypotenuse in Physics is called the RESULTANT. A man walks 95 km, East then 55 km, north. Calculate his RESULTANT DISPLACEMENT. Finish c2 a2 b2 c a2 b2 55 km, N Horizontal Component Vertical Component c Resultant 952 552 c 12050 109.8 km 95 km,E Start The LEGS of the triangle are called the COMPONENTS BUT……what about the direction? In the previous example, DISPLACEMENT was asked for and since it is a VECTOR we should include a DIRECTION on our final answer. N W of N E of N N of E N of W N of E E W NOTE: When drawing a right triangle that conveys some type of motion, you MUST draw your components HEAD TO TOE. S of W S of E W of S E of S S BUT…..what about the VALUE of the angle??? Just putting North of East on the answer is NOT specific enough for the direction. We MUST find the VALUE of the angle. To find the value of the angle we use a Trig function called TANGENT. 109.8 km 55 km, N N of E 95 km,E opposite side 55 Tan 0.5789 adjacent side 95 Tan 1 (0.5789) 30 So the COMPLETE final answer is : 109.8 km, 30 degrees North of East What is you are missing a component? 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 hypotenuse adj hyp cos cosine opposite side hypotenuse opp hyp sin sine 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 14 2 232 26.93m 14 m, N R 23 m, E 14 .6087 23 Tan 1 (0.6087) 31.3 Tan 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 cosine hypotenuse adj hyp cos H.C. =? 32 63.5 m/s opposite side sine hypotenuse opp hyp sin 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 hypotenuse V.C. adj hyp cos cosine opposite side hypotenuse opp hyp sin sine 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 1500 km + 1149.1 km = 2649.1 km R 2649.12 964.2 2 2819.1 km 964.2 0.364 2649.1 Tan 1 (0.364) 20.0 Tan R 964.2 km 2649.1 km The Final Answer: 2819.1 km @ 20 degrees, East of North