Determining the Resulting Velocity of an Airplane in a Windy Scenario

The purpose of solving problems involving airplane velocity and wind is to understand the concepts of vector addition and wind triangle in aviation, which are crucial for safe and efficient flight. The question posed here involves a scenario where an airplane traveling at 200 km/h is in the presence of a 50 km/h wind from the west, and its heading is set to [N40°E]. The goal is to determine the resulting velocity of the airplane and its ground track.

Understanding the Problem

Our starting point is the given data:

The airplane's airspeed is 200 km/h. The wind is blowing from the west at 50 km/h. The airplane is heading in the direction of [N40°E].

The key to solving this problem is to break down the velocities into their vector components, namely the north-south (NS) and east-west (EW) components. This involves using trigonometric functions to decompose the velocity vectors.

Breaking Down the Velocities

First, we break down the airplane's velocity into its components:

North-South (NS) Component: East-West (EW) Component:

For the airplane's velocity of 200 km/h in the direction of [N40°E], the north-south component can be calculated using the cosine function:

North-South Component 200 km/h * cos(40°)

The east-west component, on the other hand, involves the wind. Since the wind is blowing from the west, its effect on the east-west component is negative:

East-West Component -50 km/h

Calculating the Resultant Velocity

The next step is to combine these components to find the resultant velocity. We do this using the Pythagorean theorem and the arctangent function:

Resultant Velocity sqrt((North-South Component)^2 (East-West Component)^2)

And to find the direction relative to the east (theta) you can use:

theta arctan(East-West Component / North-South Component)

Example Calculation

Let's perform the necessary calculations step-by-step:

North-South Component: East-West Component: Resultant Velocity:

North-South Component:

North-South Component 200 km/h * cos(40°) ≈ 153 km/h

East-West Component:

East-West Component -50 km/h

Resultant Velocity:

Resultant Velocity sqrt((153 km/h)^2 (-50 km/h)^2) ≈ 165 km/h

Direction:

theta arctan(-50 / 153) ≈ -19.47° (which can be expressed as 19.47° south of east or 40.53° north of west)

Therefore, the airplanes resulting velocity is approximately 165 km/h in a direction of 40.53° north of west.

Key Considerations

It's important to note that the airplane's airspeed does not change regardless of the wind conditions. The wind only affects the direction of the plane's ground track. The pilot must correct for the wind's effect to maintain the desired ground track. This is why pilots use wind corrections in flight planning and navigation.

Conclusion

Determining the resulting velocity of an airplane in a windy scenario is a fundamental skill in aviation. By breaking down the velocities into vector components and using trigonometry, we can accurately predict the airplane's path over the ground, ensuring safe and efficient flight. Remember, the purpose of homework is to prepare you for real-world applications, and the process of solving these problems is what helps you learn.