Question

An airplane heads directly east, with an air speed of 140 mph while a wind of 50 mph is blowing from South 30° West. Find the direction and spped of the airplane over the ground

Answer 1

To find the direction and speed of the airplane over the ground, we use vector addition of the plane's airspeed vector and the wind's velocity vector.

Step-by-step explanation:

To determine the direction and speed of the airplane over the ground, we need to account for both the airplane's airspeed and the wind's velocity. This is a vector addition problem where the airspeed vector of the plane and the wind's velocity vector combine to give us the ground speed vector.

Step-by-step Solution

4. Convert the wind's direction from South 30° West to a standard bearing of 210° (since 180° + 30° = 210° from North).
8. Decompose the wind's velocity into x (east-west) and y (north-south) components:

• Wind's x-component (eastward) = 50 mph × cos(210°)
• Wind's y-component (southward) = 50 mph × sin(210°)

6. Add the wind's eastward component to the airplane's eastward velocity since the plane is heading directly east at 140 mph to find the ground speed's x-component.
10. Since the airplane's speed in the north-south direction (y-component) is 0 mph (it's heading east), the ground speed's y-component is just the wind's southward velocity.
14. Calculate the ground speed vector's magnitude using the Pythagorean theorem:
17. Ground speed = √(x-component² + y-component²).
22. Find the direction of the ground speed vector by calculating the arctangent of the ratio of the y-component to the x-component (atan(y/x)). Add 180° if the vector points westwards.

This process will yield the ground speed of the airplane in mph and the direction of travel over the ground as a standard bearing or as an angle from east.