Question

Wind velocity problem: A navigator on an airplane knows that the plane's velocity through the air is 250 km/h on a bearing of 237 measured clockwise from north. By observing the motion of the plane's shadow across the ground, she finds to her surprise that the plane's ground speed is only 52 km/h, and its direction is along a bearing of 15°. What wind velocity would explain this discrepancy?

Answer 1

To find the wind velocity causing the discrepancy in the airplane's ground speed versus airspeed, one must perform vector subtraction of the groundspeed vector from the plane's airspeed vector, accounting for direction expressed in bearings.

Step-by-step explanation:

To calculate the wind velocity that explains the discrepancy between the plane's velocity through the air and the ground speed, one must use vector subtraction. The plane's airspeed vector is given, as well as its groundspeed vector. By subtracting the groundspeed vector from the airspeed vector, the resulting vector represents the wind velocity (both speed and direction).

The plane's airspeed is 250 km/h on a bearing of 237°. We can envision this vector as pointing southwestward. The plane's ground speed is 52 km/h on a bearing of 15°, which points slightly northeastward. To find the wind velocity, we have to calculate the vector that, when added to the wind velocity vector, gives us the plane's airspeed vector.

Since these vectors are not aligned with the standard coordinate axes, the problem involves decomposing the vectors into their northerly and easterly components (north is positive y, east is positive x), calculating the difference, and then recomposing them to find the resultant vector representing the wind's velocity. To find the direction of the wind, we consider the arctangent of the ratio of northerly to easterly components.