Question

(A) An airplane is flying in a jet stream that is blowing at 45.0 m/s in a direction 26° south of east. Its direction of motion relative to the earth is 45.0° south of west, while its direction of travel relative to the air is 9° south of west. What is the airplane's speed relative to the air mass in meters per second?

(B) What is the airplane's speed relative to the Earth in meters per second?

Answer 1

In my experience it is unusual for the question to be posed like this, and this version is tougher than most. I think unit vectors are required.

25º south of east = -26º → or 334º if you prefer, same same
45º south of west = 225º
9º south of west = 189º

V plane w/r/t ground = V plane w/r/t air + V air w/r/t ground
Vcos225 i + Vsin225 j = vcos189 i + vsin189 j + 45.0cos-26 i + 45.0sin-26 j
where V is the speed w/r/t earth and v is w/r/t the air mass

-0.7071V i - 0.7071V j = -0.9877v i - 0.1564v j + 40.7839 i - 19.0178 j

Now let's just look at the x-components ("i"):
-0.7071V = -0.9877v + 40.7839

And now the y:
-0.7071V = -0.1564v - 19.0178

These simultaneous equations solve to
v = 71.9 m/s ≈ 72 m/s ◄ plane w/r/t air mass
V = 42.8 m/s ≈ 43 m/s ◄ plane w/r/t ground