Final answer:
The ground velocity of an airplane considering wind speed is calculated using vector addition by decomposing both the airplane's airspeed and the wind speed into Cartesian coordinates, summing them, and then finding the magnitude and direction of the resultant vector.
Step-by-step explanation:
The student asked about finding the ground velocity of an airplane and its direction of flight in Cartesian coordinates when the airplane is heading at a certain angle with a known airspeed and there is a wind blowing from a specific direction at a certain speed. To solve this problem, vector addition must be used, taking into account both the airspeed of the airplane and the wind speed. The airspeed vector of the airplane must be combined with the wind vector to obtain the ground velocity vector. This requires decomposing each vector into its Cartesian components (x for east/west and y for north/south), adding the corresponding components together, and then calculating the resultant vector's magnitude and direction.
For instance, if an airplane has an airspeed of 750 km/h heading at 225° (which is southwest), and the east wind is at 87 km/h, the wind's contribution to the plane's velocity would actually be in the negative x-direction (towards the west) when considering a Cartesian coordinate system. After calculating the x and y components of both vectors, the components are added to find the resultant x and y components of the ground velocity. The magnitude of the ground velocity vector is found using the Pythagorean theorem, and its direction with respect to the north can be calculated using the arctangent function.