Final answer:
To find the airplane's ground speed and direction with an airspeed of 500 mph west and a wind blowing 150 mph south, vector addition is applied, resulting in a ground speed of approximately 522 mph in a direction of about 16.7 degrees south of west.
Step-by-step explanation:
Finding Airplane's Ground Speed and Direction
To determine the airplane's ground speed and the direction of travel, we need to apply vector addition. The plane has an airspeed of 500 miles per hour due west and there is also a 150 mph wind blowing due south. To find the resultant vector (which represents the plane's ground speed and direction), we calculate the vector sum of the plane's velocity and the wind's velocity.
Let the airplane's velocity be the vector A (500 mph, West) and the wind's velocity be the vector B (150 mph, South). The ground speed vector, C, is then A + B. We use the Pythagorean theorem to find the magnitude of C: |C| = sqrt(A^2 + B^2) = sqrt((500)^2 + (150)^2), which gives us |C| = sqrt(250000 + 22500) = sqrt(272500) = 522 mph.
To find the direction angle south of west, we use the arctan function of the y-component (wind speed) over the x-component (airplane speed). So, the angle θ = arctan(B/A) = arctan(150/500) which gives us an angle of approximately 16.7 degrees south of west. The angle indicates the direction relative to due west where the plane's actual travel direction is.
So, we have determined that the ground speed of the airplane is 522 mph in a direction approximately 16.7 degrees south of west.