How Should a Pilot Adjust Course in Wind to Maintain Direction?

[mattst88]

Homework Statement

An airplane is traveling at 30 m/s and wishes to travel to a point 8000 m NE (45 degrees). If there is a constant 10m/s wind blowing west:
A) In what direction must the pilot aim the plane in degrees?
B) How long will the trip take?

Homework Equations

Basic kinematic equations and trigonometry.

The Attempt at a Solution

Since I know only the magnitude of the velocity vector, and have to find the direction, I'm having trouble.

I've tried taking the arcsin of 10/30 (Opposite over Hypotenuse) and got 19.47 degrees. Using the Law of Sines, I can calculate the other angles and the other side length.

Side Length (m/s) Angle (Degrees)
10 19.47
30 58.4
29.33 102

Obviously, the 102 degrees doesn't make sense, since it is not opposite the largest side.

Am I making this much more difficult than it really is?

Please advise.

 

[LowlyPion]

Homework Statement

An airplane is traveling at 30 m/s and wishes to travel to a point 8000 m NE (45 degrees). If there is a constant 10m/s wind blowing west:
A) In what direction must the pilot aim the plane in degrees?
B) How long will the trip take?

Since I know only the magnitude of the velocity vector, and have to find the direction, I'm having trouble.

I've tried taking the arcsin of 10/30 (Opposite over Hypotenuse) and got 19.47 degrees. Using the Law of Sines, I can calculate the other angles and the other side length.

Side Length (m/s) Angle (Degrees)
10 19.47
30 58.4
29.33 102

Obviously, the 102 degrees doesn't make sense, since it is not opposite the largest side.

Am I making this much more difficult than it really is?

Please advise.

Likely you aren't making it difficult enough.

What you do have is a vector addition. Except this one involves certain variables. I would recommend that you construct the vectors and their components, and then add them as you know they must be added to end at your destination.

For instance let A be your wind speed blowing West. Withe East being positive X and H being the time to get there:

$$\vec{A} = -10*H*\hat{x}$$

Likewise for the Plane:

$$\vec{B} = 30*H*Cos \theta * \hat{x} + 30*H*Sin \theta *\hat{y}$$

And then you have your Destination vector:

$$\vec{D} = 8000*Cos45*\hat{x} + 8000*Sin45 * \hat{y}$$

Since you know

$$\vec{D} = \vec{A} + \vec{B}$$

Then solve for the angle.