You are flying in a light airplane spotting traffic for a radio station. Your flight carries you due east above a highway. Landmarks below tell you that your speed is 46.0 m/s relative to the ground and your air speed indicator also reads 46.0m/s. However, the nose of your airplane is pointed somewhat south of east and the station's weather person tells you that the wind is blowing with speed 25.0 m/s.In which direction is the wind blowing?
Let i be the unit vector easy and j be the unit vector north.
Velocity relative to ground = 46 i
wind velocity = 25 cosA i + 25 sinA j
(A is measured north from east)
velocity with respect to air =
46 cosB i + 46 sinB j
(B is direction the plane is pointed, north from east)
The ground velocity vector is the sum of the two that follow.
That would require that:
25 sinA + 46 sinB = 0
25 cosA + 46 cosB = 46
Solve for angles A and B. Looks messy
49.7
122
To determine the direction of the wind, we need to consider the relative motion between the airplane and the ground.
Given:
- Speed of the airplane relative to the ground (ground speed): 46.0 m/s
- Airspeed indicator reading (airspeed): 46.0 m/s
- Speed of the wind: 25.0 m/s
Since the ground speed and airspeed indicator are the same, it means there is no headwind or tailwind. The difference in velocity is solely due to the wind.
To find the wind direction, we can use vector addition. Let's assume the airplane's velocity relative to the air is A (east) and the wind's velocity is W (unknown direction). The resulting velocity (ground speed) is G (east).
G = A + W
In vector notation, it becomes:
G = 46.0 m/s (east)
A = 46.0 m/s (east)
W = ? m/s (unknown direction)
To find W, we need to find the difference between A and G:
W = G - A
Since the magnitudes of both A and G are equal, their difference is due to the wind:
W = 46.0 m/s (east) - 46.0 m/s (east)
W = 0 m/s (east)
Therefore, the wind is blowing in the eastward direction.