Is there anyone out there who can check my workings on this practice question.
A plane is flying due East at 400km/hr into a Northwest headwind at 50km/hr.
Find the direction the plane ends up flying and its groundspeed?
The plane vector heading East is
$\vec{v}=400\vec{i}$
The wind vector is
$\vec{\omega}=50\cos{45}\vec{i}+50\sin{45}\vec{j}$
$\vec{\omega}=35.4\vec{i}+35.4\vec{j}$
Now since the wind is going against the flight direction the velocity relative to the ground is
$\vec{v}-\vec{\omega}=400\vec{i}-(35.4\vec{i}+35.4\vec{j})=364.64\vec{i}-35.4\vec{j}$
So the direction relative to the ground is
$\tan\theta={\frac{35.4}{364.64}}$
$\theta= \tan^{-1}{\frac{35.4}{364.64}}=5.55^{\circ}$
Therefore direction is E $5.55^{\circ}N$
So will the ground speed be
$\sqrt{364.64^2+35.4^2}=366.35 km/hr$
or is it
$\sqrt{364.64^2-35.4^2}=362.91 km/hr$
Best Answer
A diagram often helps with problems like these.
When the question says the aircraft is flying due east at $400$ km/hr, I suppose it means the aircraft itself is pointing due east and its speed relative to the air mass around it (what aviators call "true airspeed") is $400$ km/hr. This gives us a vector represented by the blue arrow in the figure below.
Often when we see "northwest wind" it means the air mass is coming from the northwest and moving toward the southeast. But since the problem statement says "headwind" it seems reasonable to suppose it means the air mass is moving toward the northwest, since the other direction would be a tailwind for this aircraft. The vector for this is represented by the purple arrow in the figure below.
The resulting velocity of the aircraft relative to the ground is the vector shown as a red arrow. As you already determined (by interpreting "headwind" the same way I did), the direction of the ground velocity is slightly north of east.
The groundspeed is the magnitude of the ground velocity, that is, the length of the red arrow. That should make it clear which formula to use. (Apply the Pythagorean Theorem if there is any remaining doubt.)
By the way, using $\vec\imath$ as the unit east vector, I would have set $\vec\jmath$ to the unit north vector, so I would have written the wind vector as $-35.4\vec\imath + 35.4\vec\jmath$ rather than $35.4\vec\imath + 35.4\vec\jmath$ or even $-(35.4\vec\imath + 35.4\vec\jmath)$. But in this particular question, as it happens, the choice of $35.4\vec\jmath$ or $-35.4\vec\jmath$ doesn't actually change the answer. In another case it might, which is why I mention it.