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:

[tex] \vec{A} = -10*H*\hat{x} [/tex]

Likewise for the Plane:

[tex] \vec{B} = 30*H*Cos \theta * \hat{x} + 30*H*Sin \theta *\hat{y} [/tex]

And then you have your Destination vector:

[tex] \vec{D} = 8000*Cos45*\hat{x} + 8000*Sin45 * \hat{y} [/tex]

Since you know

[tex] \vec{D} = \vec{A} + \vec{B} [/tex]

Then solve for the angle.

 

[baba89]

I would like to clarify a few things before providing a response. First, we need to assume that the airplane is traveling in a straight line and the wind is also blowing in a straight line, as these are simplifications that allow us to use basic kinematic equations and trigonometry to solve the problem. Second, we need to clarify the direction in which the wind is blowing. In the problem statement, it is mentioned that the wind is blowing west, but it is not specified whether it is blowing to the west or from the west. For the sake of simplicity, let's assume that the wind is blowing to the west, which means it is blowing in the opposite direction of the airplane's desired direction of travel (NE).

Now, let's address the first part of the problem, which asks us to find the direction in which the pilot must aim the plane. To do this, we can use the concept of vector addition. The velocity of the airplane can be represented by a vector of magnitude 30 m/s in the NE direction. The wind velocity can be represented by a vector of magnitude 10 m/s in the west direction. To find the resultant velocity (the velocity of the airplane relative to the ground), we can use the Pythagorean theorem and the trigonometric function tangent to find the angle.

Using the Pythagorean theorem, we can find the magnitude of the resultant velocity:

Resultant velocity = √(30^2 + 10^2) = √(900 + 100) = √1000 = 31.62 m/s

Next, we can use the trigonometric function tangent to find the angle:

tan θ = opposite/adjacent = 10/30 = 1/3

θ = arctan(1/3) = 18.43 degrees

Therefore, the direction in which the pilot must aim the plane is 18.43 degrees east of north (or 71.57 degrees north of east).

Moving on to the second part of the problem, which asks us to find the time it will take for the airplane to travel 8000 m, we can use the equation d = vt, where d is the distance, v is the velocity, and t is the time. Since we have already calculated the resultant velocity to be 31.62 m/s, we can plug it into the equation along with the distance of 8000