Week of Physics

Physics Monday, On a sheet of graph paper, in deference to those who prefer it, do the following problems. 1.A hiker walked 3 km north after which he turn west and walked another 4 km. a. What is the magnitude and direction of his displacement? b. What is the angle of the resultant in relation to the x-axis? 2. A ship cruising at 4 knots per hour traveled for 8 hours on a heading of southwest. The captain then received orders to turn east and sail for 6 hours. After these maneuvers, what was the ship's distance and approximate direction from its origin? What is the angle of the resultant in relation to the x-axis?

problem 1

32 kt 24 kt θ=45 o R Plot Note that “two” right triangles are formed. problem 2

Let: 1. y = perpendicular from the origin of the 1 st vector down to the 2 nd vector. 2. x = distance from the end of the 1 st vector to y. 3. x’ = 24 – x Cos θ = x/32 θ = 45 0 x = 32 cos 45 0 = 22.6 kt x’ = 24 – 22.6 = 1.4 kt Sin 45 0 = y/32; y = 22.6 kt (Note: internal angles are 45 0, so sides are equal.) R = sqrt( ) = 22.6 kt (rounded to nearest tenth) θ1 = Tan -1 x’/y = Tan -1 (1.4 / 22.6) = Angle of R from x-axis = = θ1θ1 x x’ y θ=45 o R

Physics Wednesday, More fun with vectors

23.5 cm 27.0 cm 49.5 cm θ 1 = 55 0 θ 2 = 33 0 θ R = 42 0 Results obtained by physical measurements. R = 49.5 cm θ R = 42 0

Results obtained by physical measurements of vectors and mathematical analysis of resultant. Determine the magnitude of R 1.Create two right triangles from the vectors’ tips down to the x-axis. 2.Determine the x- and y components of the triangles. Sum of x components = the x component of the resultant. Sum of the y components = the y component of the resultant. 3.For triangle 1, cos 55 0 = x 1 / 23.5; x 1 = 23.5 cos 55 0 = For triangle 2, cos 33 0 = x 2 / 27.0; x 1 = 27 cos 33 0 = x component of R = = y 1 = 23.5 sin 55 0 = y 2 = 27 sin 33 0 = y component of R = = 34 9.R = √ ( ) = 49.6

Results obtained by physical measurements of vectors and mathematical analysis of resultant. Determine the angular displacement of R 1.Dimensions of R: x R = 36.1 y R = 34 R = Use tangent function to determine θ R. 3.Tan θ R = (y R / x R ) = (34 / 36.1); θ R = Compare to measured values. Resultant R: Magnitude: Measured = 49.5; Calculated = 49.6 Angle: Measured = 42 deg.; Calculated = 43 deg.

Physics Thursday Evaluate the vectors in this example, and draw the resultant using the parallelogram method. Then mathematically determine the magnitude and direction (with respect to the negative x-axis ) of the resultant. Note that the y- components of the two right triangles are in opposite quadrants of the graph.