810 CHAPTER 8 Applications of Trigonometry 51. Bearing and Ground Speed of a Plane An airline route from San Francisco to Honolulu is on a bearing of 233.0°. A jet flying at 450 mph on that bearing encounters a wind blowing at 39.0 mph from a direction of 114.0°. Find the resulting bearing and ground speed of the plane. 52. Path Traveled by a Plane The aircraft carrier Tallahassee is traveling at sea on a steady course with a bearing of 30° at 32 mph. Patrol planes on the carrier have enough fuel for 2.6 hr of flight when traveling at a speed of 520 mph. One of the pilots takes off on a bearing of 338° and then turns and heads in a straight line, so as to be able to catch the carrier and land on the deck at the exact instant that his fuel runs out. If the pilot left at 2 p.m., at what time did he turn to head for the carrier? 53. Airspeed and Ground Speed A pilot wants to fly on a bearing of 74.9°. By flying due east, he finds that a 42.0-mph wind, blowing from the south, puts him on course. Find the airspeed and the ground speed. 54. Bearing of a Plane A plane flies 650 mph on a bearing of 175.3°. A 25-mph wind, from a direction of 266.6°, blows against the plane. Find the resulting bearing of the plane. 55. Bearing and Ground Speed of a Plane A pilot is flying at 190.0 mph. He wants his flight path to be on a bearing of 64° 30′. A wind is blowing from the south at 35.0 mph. Find the bearing he should fly, and find the plane’s ground speed. 56. Bearing and Ground Speed of a Plane A pilot is flying at 168 mph. She wants her flight path to be on a bearing of 57° 40′. A wind is blowing from the south at 27.1 mph. Find the bearing she should fly, and find the plane’s ground speed. 57. Bearing and Airspeed of a Plane What bearing and airspeed are required for a plane to fly 400 mi due north in 2.5 hr if the wind is blowing from a direction of 328° at 11 mph? 58. Ground Speed and Bearing of a Plane A plane is headed due south with an airspeed of 192 mph. A wind from a direction of 78.0° is blowing at 23.0 mph. Find the ground speed and resulting bearing of the plane. 59. Ground Speed and Bearing of a Plane An airplane is headed on a bearing of 174° at an airspeed of 240 km per hr. A 30-km-per-hr wind is blowing from a direction of 245°. Find the ground speed and resulting bearing of the plane. 60. Velocity of a Star The space velocity v of a star relative to the sun can be expressed as the resultant vector of two perpendicular vectors—the radial velocity vr and the tangential velocity vt , where v = vr + vt. If a star is located near the sun and its space velocity is large, then its motion across the sky will also be large. Barnard’s Star is a relatively close star with a distance of 35 trillion mi from the sun. It moves across the sky through an angle of 10.34″ per year, which is the largest motion of any known star. Its radial velocity vr is 67 mi per sec toward the sun. (Data from Zeilik, M., S. Gregory, and E. Smith, Introductory Astronomy and Astrophysics, Second Edition, Saunders College Publishing; Acker, A., and C. Jaschek, Astronomical Methods and Calculations, John Wiley and Sons.) (a) Approximate the tangential velocity vt of Barnard’s Star. (Hint: Use the arc length formula s = ru.) (b) Compute the magnitude of v. 308 Plane N Carrier 3388 Barnard's Star Sun vr vt v NOT TO SCALE
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