Final answer:
The velocity of the airplane relative to the Earth is approximately 272 m/s at an angle of 8.0° south of west. The answers are consistent with expectations because the wind adds to the airplane's velocity in the x-direction, but it also adds some velocity in the y-direction due to its angle. This results in a slightly slower velocity and a slightly more southern direction.
Step-by-step explanation:
To find the velocity of the airplane relative to the Earth, we need to find the resultant velocity. We can use vector addition to solve this problem. First, we need to convert the velocities to their x and y components. The air speed of the airplane is 260 m/s at an angle of 5.0° south of west.
This can be broken down into x = 260 cos(5.0°) and y = -260 sin(5.0°).
The velocity of the wind is 35.0 m/s at an angle of 15° south of east.
This can be broken down into x = 35.0 cos(15°) and y = -35.0 sin(15°).
Now we can add the x and y components of the airplane's velocity and the wind's velocity to find the resultant velocity. Finally, we can find the magnitude and direction of the resultant velocity using the Pythagorean theorem and trigonometry.
The resultant velocity of the airplane relative to the Earth is approximately 272 m/s at an angle of 8.0° south of west. This is the velocity of the airplane relative to the Earth.
The answers are consistent with expectations because the wind adds to the airplane's velocity in the x-direction, but it also adds some velocity in the y-direction due to its angle. This results in a slightly slower velocity and a slightly more southern direction.