25. Finding the Actual Speed and Direction of an Aircraft A Boeing 747 jumbo jet maintains an airspeed of 550 miles per hour in a southwesterly direction. The. velocity of the jet stream is a constant 80 miles per hour from the west. Find the actual speed and direction of the aircraft.

Step 1. Define a Coordinate System • Let east be the positive x–axis and north the positive y–axis. • Therefore, west is negative x and south is negative y. Step 2. Write the Airspeed Vector The Boeing 747’s airspeed is 550 mph on a southwesterly course. “Southwest” means that the plane’s velocity has equal components toward the south and the west. Thus, its components are:   West component = –550·cos(45°)   South component = –550·sin(45°) Since cos(45°) = sin(45°) = √2/2, we have:   West component = –550·(√2/2) ≈ –389.1 mph   South component = –550·(√2/2) ≈ –389.1 mph Step 3. Write the Wind (Jet Stream) Vector The jet stream is blowing from the west at 80 mph. “From the west” means the wind is moving toward the east. Its vector is:   East component = +80 mph   North component = 0 mph Step 4. Find the Ground Velocity (Resultant Vector) Add the corresponding components of the airspeed and wind vectors: • x–component (east–west):   = –389.1 + 80 = –309.1 mph   (This negative sign indicates a westward motion.) • y–component (north–south):   = –389.1 + 0 = –389.1 mph   (This negative sign indicates a southward motion.) Step 5. Compute the Magnitude (Actual Speed) The magnitude of the resultant vector (ground speed) is given by:   Speed = √[(–309.1)² + (–389.1)²]       = √(309.1² + 389.1²) Calculate:   309.1² ≈ 95486.8   389.1² ≈ 151351.0   Sum ≈ 95486.8 + 151351.0 = 246837.8   Speed ≈ √246837.8 ≈ 496.8 mph Step 6. Determine the Direction The plane’s ground velocity has components:   x = –309.1 (west)   y = –389.1 (south) To find the angle the resultant makes with due south, compute:   θ = arctan(|x| / |y|) = arctan(309.1 / 389.1)   θ ≈ arctan(0.794) ≈ 38.7° Since the motion is southward and westward, the direction is 38.7° west of due south. (Alternatively, one could say it is 51.3° south of due west, but the former is more natural since the vector is closer to south.) Step 7. State the Final Answer The actual (ground) speed of the aircraft is approximately 497 mph, and its direction is about 38.7° west of due south.