Chapter 3 – Two Dimensional Motion and Vectors - MrTirjan.ppt

3-2 Vector Operations Objectives: Identify appropriate coordinate systems for solving problems with vectors. Apply the Pythagorean Theorem and tangent function to calculate the magnitude and direction of a resultant vector. Resolve vectors into components using the sine and cosine functions. Add vectors that are not perpendicular Geometry / Trigonometry Review Pythagorean Theorem – The square of the hypotenuse of a right triangle is equal to the sum of the squares of its legs c2 = a2 + b2 a b c Trigonometric Ratios: Hypotenuse (c) Leg (b) Leg (a) C A B Opposite side hypotenuse Adjacent side hypotenuse Opposite side Adjacent side Sin θ = Cos θ = Tan θ = Using Trig. Ratios Given an acute angle of a right triangle, to find the ratio of 2 specific sides of the triangle, enter the appropriate function (sine, cosine, tangent) of the angle in your calculator. a b c 20o Sin(20o)=b/c Cos(20o) = a/c Tan(20o) = b/a To find an acute angle of a right triangle, enter the inverse of the appropriate function of the ratio of the 2 corresponding sides. a b c θ θ = sin-1(a/c) θ = cos-1(b/c) θ = tan-1(a/b) Trigonometry Review Practice Worksheet Part III Vector Addition – Analytical Method Case #1 (easiest method): Adding 2 Vectors that are perpendicular R = magnitude of the resultant vector R = A2 + B2 Resultant, R A B θ The angle, θ, of the triangle can be found using the tan-1 function and THEN CONVERT it to the direction measured CCW from the +x - axis Example for Case #1 Add the following 2 velocity vectors. 5 m/s west (180o) and 8 m/s north (90o) 5 m/s 8 m/s R θ R2 = 52 + 82 R = 9.4 m/s θ = tan-1 (8/5) θ = 58o The direction (measured CCW from the +x – axis) is found by subtracting 180 – 58 = 122o R = 9.4 m/s 122o Case #2: Adding more than 2 perpendicular vectors First, find the vector sum of all of the horizontal vectors, call this Rx. Second, find the vector sum of all of the vertical vectors, call this Ry. Find the vector sum of Rx and Ry By following the method from