SECTION 9.4 Vectors 645 Finding the Actual Speed and Direction of an Aircraft A Boeing 767 aircraft maintains a constant airspeed of 500 miles per hour headed due south. The jet stream is 80 miles per hour in the northeasterly direction. (a) Express the velocity va of the 767 relative to the air and the velocity vw of the jet stream in terms of i and j. (b) Find the velocity of the 767 relative to the ground. (c) Find the actual speed and direction of the 767 relative to the ground. EXAMPLE 8 Figure 65 W N S vg 2500 E y x 500 vw va 5 2500j Orlando Naples Miami Wind N S W E Solution (a) Set up a coordinate system in which north (N) is along the positive y-axis. See Figure 65. The velocity of the 767 relative to the air is = − v j 500 . a The velocity of the jet stream vw has magnitude 80 and direction NE (northeast), so the angle between vw and i is ° 45 . Express vw in terms of i and j as ( ) ( ) = ° + ° = + ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ = + v i j i j i j 80 cos45 sin 45 80 2 2 2 2 40 2 w (b) The velocity of the 767 relative to the ground vg is ( ) ( ) = + = − + + = + − v v v j i j i j 500 40 2 40 2 40 2 500 w g a (c) The actual speed of the 767 is ( ) ( ) = + − ≈ v 40 2 40 2 500 447 miles per hour g 2 2 To find the actual direction of the 767 relative to the ground, determine the direction angle of v .g The direction angle is found by solving α = − tan 40 2 500 40 2 Then α ≈ − ° 82.7 . The 767 is traveling ° S7.3 E. Now Work PROBLEM 79 Finding the Weight of a Piano Two movers require a force of magnitude 300 pounds to push a piano up a ramp inclined at an angle of ° 20 from the horizontal. How much does the piano weigh? EXAMPLE 9 Solution Let F1 represent the force of gravity, F2 represent the force required to move the piano up the ramp, and F3 represent the force of the piano against the ramp. See Figure 66. The angle between the ground and the ramp is the same as the angle between F1 and F3 because triangles ABC and BDE are similar, so ∠ = ∠ = ° BAC DBE 20 . To find the magnitude of F1 (the weight of the piano), calculate ° = = F F F sin 20 300 2 1 1 = ° ≈ F 300 lb sin20 877 lb 1 The piano weighs approximately 877 pounds. In Figure 66, the triangle formed by the force vectors (in blue) is called a force diagram. An object is said to be in static equilibrium if the object is at rest and the sum of all forces acting on the object is zero—that is, if the resultant force is 0. Figure 66 208 208 A B D C E F3 F1 F2
RkJQdWJsaXNoZXIy NjM5ODQ=